Vektorová analýza

Dyády

V dvourozměrném Euklidově prostoru a kartézském souřadnicovém systému se vektory \(\boldsymbol{a}\) a \(\boldsymbol{b}\) mohou zapsat následovně, [1], [2]

(1)\[\begin{split}\begin{equation} \begin{split} \boldsymbol{a} &= a_1\boldsymbol{i}+a_2\boldsymbol{j}, \\ \boldsymbol{b} &= b_1\boldsymbol{i}+b_2\boldsymbol{j}, \end{split} \end{equation}\end{split}\]

kde \(\boldsymbol{i}\) a \(\boldsymbol{j}\) jsou bázové vektory. S použitím stejné symboliky se dyadický součin může zapsat jako součet čtyř dyád

(2)\[\boldsymbol{ab}=a_1b_1\boldsymbol{i}\boldsymbol{i}+a_1b_2\boldsymbol{i}\boldsymbol{j} +a_2b_1\boldsymbol{j}\boldsymbol{i}+a_2b_2\boldsymbol{j}\boldsymbol{j}.\]

Používá se také maticového zápisu

(3)\[\begin{split}\boldsymbol{ab}\equiv\boldsymbol{a}\otimes\boldsymbol{b} \equiv\boldsymbol{ab}^{T} =\left[ \begin{array}{c} a_1\\ a_2 \end{array} \right] \left[ \begin{array}{cc} b_1 & b_2 \end{array} \right] =\left[ \begin{array}{cc} a_1b_1 & a_1b_2\\ a_2b_1 & a_2b_2 \end{array} \right],\end{split}\]

takže dyády \(\boldsymbol{ii}\), \(\boldsymbol{ij}\), \(\boldsymbol{ji}\) a \(\boldsymbol{jj}\) tvoří bázi

(4)\[\begin{split}\boldsymbol{ii}=\left[ \begin{array}{cc} 1 & 0\\ 0 & 0 \end{array} \right],\quad \boldsymbol{ij}=\left[ \begin{array}{cc} 0 & 1\\ 0 & 0 \end{array} \right],\quad \boldsymbol{ji}=\left[ \begin{array}{cc} 0 & 0\\ 1 & 0 \end{array} \right],\quad \boldsymbol{jj}=\left[ \begin{array}{cc} 0 & 0\\ 0 & 1 \end{array}\right].\end{split}\]

Základní vlastnosti dyád

Jestliže \(\alpha\) je skalár a \(\boldsymbol{a}\), \(\boldsymbol{b}\) a \(\boldsymbol{c}\) jsou vektory, pak pro dyády \(\boldsymbol{ab}\), \(\boldsymbol{bc}\) a \(\boldsymbol{ac}\) platí

(5)\[(\alpha\boldsymbol{a})\boldsymbol{b}=\boldsymbol{a}(\alpha\boldsymbol{b}) =\alpha(\boldsymbol{ab})\]

a

(6)\[\begin{split}\begin{equation} \begin{split} \boldsymbol{a}(\boldsymbol{b}+\boldsymbol{c}) &= \boldsymbol{ab}+\boldsymbol{ac}, \\ (\boldsymbol{a}+\boldsymbol{b})\boldsymbol{c} &= \boldsymbol{ac}+\boldsymbol{bc}. \end{split} \end{equation}\end{split}\]

Dyadická algebra

Pro skalární součin vektoru \(\boldsymbol{c}\) a dyády \(\boldsymbol{ab}\) platí

(7)\[\begin{split}\begin{equation} \begin{split} \boldsymbol{c}\cdot\boldsymbol{ab} &= (\boldsymbol{c}\cdot\boldsymbol{a})\boldsymbol{b}, \\ (\boldsymbol{ab})\cdot\boldsymbol{c} &= \boldsymbol{a}(\boldsymbol{b}\cdot\boldsymbol{c}). \end{split} \end{equation}\end{split}\]

Pro vektorový součin vektoru \(\boldsymbol{c}\) a dyády \(\boldsymbol{ab}\) platí

(8)\[\begin{equation} \begin{split} \boldsymbol{c}\times\boldsymbol{ab} &= (\boldsymbol{c}\times\boldsymbol{a})\boldsymbol{b}, (\boldsymbol{ab})\times\boldsymbol{c} &= \boldsymbol{a}(\boldsymbol{b}\times\boldsymbol{c}). \end{split} \end{equation}\]

Tečkový součin dvou dyád \(\boldsymbol{ab}\) a \(\boldsymbol{cd}\) je

(9)\[\boldsymbol{ab}\cdot\boldsymbol{cd}=\boldsymbol{a}(\boldsymbol{b}\cdot\boldsymbol{c})\boldsymbol{d} =(\boldsymbol{b}\cdot\boldsymbol{c})\boldsymbol{ad}.\]

A nakonec dvoutečkový součin dvou dyád \(\boldsymbol{ab}\) a \(\boldsymbol{cd}\) je

(10)\[\boldsymbol{ab}::\boldsymbol{cd}=(\boldsymbol{a}\cdot\boldsymbol{d})(\boldsymbol{b}\cdot\boldsymbol{c})\]

nebo

(11)\[\boldsymbol{ab}::\boldsymbol{cd}=\boldsymbol{c}\cdot\boldsymbol{ab}\cdot\boldsymbol{d} =(\boldsymbol{a}\cdot\boldsymbol{c})(\boldsymbol{b}\cdot\boldsymbol{d}).\]

Jsou i další součiny, ale zatím jich není třeba.

Speciální dyády

Jednotková dyáda \(\boldsymbol{I}\) je v kartézském souřadnicovém systému definována vztahem

(12)\[\begin{split}\boldsymbol{I}=\boldsymbol{ii}+\boldsymbol{jj} =\left[ \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \right]\end{split}\]

s těmito vlastnostmi

(13)\[\boldsymbol{I}\cdot\boldsymbol{a} = \boldsymbol{a}\cdot\boldsymbol{I}=\boldsymbol{a},\]

(14)\[(\boldsymbol{a}\times\boldsymbol{I})\cdot(\boldsymbol{b}\times\boldsymbol{I}) =\boldsymbol{ab}-(\boldsymbol{a}\cdot\boldsymbol{b})\boldsymbol{I},\]

(15)\[\boldsymbol{I}::\boldsymbol{ab}=(\boldsymbol{I}\cdot\boldsymbol{a})\cdot\boldsymbol{b} =\boldsymbol{a}\cdot\boldsymbol{b}=\mathrm{tr}(\boldsymbol{ab}).\]

Polární a kartézské souřadnicové systémy

Pěkný zdroj jsou webové stránky [3], je však nutné v odkazu dodržet velká a malá písmena. Souřadnice bodu v polárních a kartézských jsouřadnicích jsou svázány vztahem

(16)\[\begin{split}\begin{equation} \begin{split} x &= r\cos\varphi, \\ y &= r\sin\varphi. \end{split} \end{equation}\end{split}\]

Inverzní transformace má tvar

(17)\[\begin{split}\begin{equation} \begin{split} r &= \sqrt{x^2+y^2}, \\ \varphi &= \arctan\frac{y}{x}. \end{split} \end{equation}\end{split}\]

Základní transformační vztahy mezi vektory a tenzory v polárním a kartézském souřadnicovém systému

Vektor

(18)\[\boldsymbol{u}=u_r\boldsymbol{e}_r+u_\varphi\boldsymbol{e}_\varphi\]

v polárním souřadnicovém systému je reprezentován bází tvořenou vektory

(19)\[\begin{split}\begin{equation} \begin{split} \boldsymbol{e}_r &= (\cos\varphi,\sin\varphi), \\ \boldsymbol{e}_\varphi &= \boldsymbol{k}\times\boldsymbol{e}_r=(-\sin\varphi,\cos\varphi), \end{split} \end{equation}\end{split}\]

kde \(\boldsymbol{k}\) je kolmý na rovinu vektoru \(\boldsymbol{e}_r\). Podobně jako vektor, může být polárními souřadnicemi reprezentován i tenzor

(20)\[\begin{split}\begin{equation} \begin{split} \boldsymbol{S} &= S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r + S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_{\varphi} \\ &+ S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_r + S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \\ &= \left[ \begin{array}{cc} S_{rr} & S_{r\varphi}\\ S_{\varphi r} & S_{\varphi\varphi} \end{array} \right], \end{split} \end{equation}\end{split}\]

kde význam vnějšího nebo též tenzorového součinu je vysvětlen v odstavci Diadická algebra. Jestliže v kartézských souřadnicích má vektor \(\boldsymbol{u}\) souřadnice \(u_x\) a \(u_y\), v polárních má souřadnice

(21)\[\begin{split}\begin{equation} \begin{split} u_r &= u_x\cos\varphi+u_y\sin\varphi, \\ u_\varphi &= -u_x\sin\varphi+u_y\cos\varphi, \end{split} \end{equation}\end{split}\]

nebo-li

(22)\[\begin{split}\left[ \begin{array}{c} u_r\\ u_\varphi \end{array} \right]=\left[ \begin{array}{cc} \cos\varphi & \sin\varphi\\ -\sin\varphi & \cos\varphi \end{array} \right]\left[ \begin{array}{c} u_x\\ u_y \end{array} \right]\end{split}\]

a naopak

(23)\[\begin{split}\begin{equation} \begin{split} u_x &= u_r\cos\varphi-u_\varphi\sin\varphi, \\ u_y &= u_r\sin\varphi+u_\varphi\cos\varphi, \end{split} \end{equation}\end{split}\]

nebo-li

(24)\[\begin{split}\left[ \begin{array}{c} u_x\\ u_y \end{array} \right]=\left[ \begin{array}{cc} \cos\varphi & -\sin\varphi\\ \sin\varphi & \cos\varphi \end{array} \right]\left[ \begin{array}{c} u_r\\ u_\varphi \end{array} \right]\end{split}\]

Pro tenzor třeba třetího řádu, v polárních souřadnicích zapsaný např. následovně

(25)\[\begin{split}\begin{equation} \begin{split} \boldsymbol{S} &= S_{rrr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{rr\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{r\varphi r}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ S_{r\varphi\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +S_{\varphi rr}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{\varphi r\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &+ S_{\varphi\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi, \end{split} \end{equation}\end{split}\]

platí transformace

(26)\[S_{ijk}(r,\varphi)=c_{il}(\varphi)c_{jm}(\varphi)c_{kn}(\varphi)S_{lmn}(x,y),\]

kde \(c_{ij}(\varphi)\) jsou prvky transformačních matice ze vztahu (22). Inverzní transformace

(27)\[S_{lmn}(x,y)=c_{li}(\varphi)c_{mj}(\varphi)c_{nk}(\varphi)S_{ijk}(r,\varphi),\]

má \(c_{ji}(\varphi)\) prvky matice ze vztahu (24). U tenzoru druhého řádu se transformační vztahy mohou zapsat i maticově

(28)\[\begin{split}\left[ \begin{array}{cc} S_{xx} & S_{xy}\\ S_{yx} & S_{yy} \end{array} \right] = \left[ \begin{array}{cc} \cos\varphi & -\sin\varphi\\ \sin\varphi & \cos\varphi \end{array} \right]\left[ \begin{array}{cc} S_{rr} & S_{r\varphi}\\ S_{\varphi r} & S_{\varphi\varphi} \end{array}\right]\left[ \begin{array}{cc} \cos\varphi & \sin\varphi\\ -\sin\varphi & \cos\varphi \end{array} \right],\end{split}\]

(29)\[\begin{split}\begin{equation} \begin{split} S_{xx} &= \cos^2\varphi S_{rr}-\cos\varphi\sin\varphi S_{\varphi r}-\cos\varphi\sin\varphi S_{r\varphi} +\sin^2\varphi S_{\varphi\varphi}, \\ S_{xy} &= \cos\varphi\sin\varphi S_{rr}-\sin^2\varphi S_{\varphi r}+\cos^2\varphi S_{r\varphi} -\cos\varphi\sin\varphi S_{\varphi\varphi}, \\ S_{yx} &= \cos\varphi\sin\varphi S_{rr}+\cos^2\varphi S_{\varphi r}-\sin^2\varphi S_{r\varphi} -\cos\varphi\sin\varphi S_{\varphi\varphi}, \\ S_{yy} &= \sin^2\varphi S_{rr}+\cos\varphi\sin\varphi S_{\varphi r}+\cos\varphi\sin\varphi S_{r\varphi} +\cos^2\varphi S_{\varphi\varphi} \end{split} \end{equation}\end{split}\]

a inverzně

(30)\[\begin{split}\left[ \begin{array}{cc} S_{rr} & S_{r\varphi}\\ S_{\varphi r} & S_{\varphi\varphi} \end{array} \right] &= \left[ \begin{array}{cc} \cos\varphi & \sin\varphi\\ -\sin\varphi & \cos\varphi \end{array} \right]\left[ \begin{array}{cc} S_{xx} & S_{xy}\\ S_{yx} & S_{yy} \end{array} \right]\left[ \begin{array}{cc} \cos\varphi & -\sin\varphi\\ \sin\varphi & \cos\varphi \end{array} \right],\end{split}\]

(31)\[\begin{split}\begin{equation} \begin{split} S_{rr} &= \cos^2\varphi S_{xx}+\cos\varphi\sin\varphi S_{yx} +\cos\varphi\sin\varphi S_{xy}+\sin^2\varphi S_{yy}, \\ S_{r\varphi} &= -\cos\varphi\sin\varphi S_{xx}-\sin^2\varphi S_{yx} +\cos^2\varphi S_{xy}+\cos\varphi\sin\varphi S_{yy}, \\ S_{\varphi r} &= -\cos\varphi\sin\varphi S_{xx}+\cos^{2}\varphi S_{yx} -\sin^2\varphi S_{xy}+\cos\varphi\sin\varphi S_{yy}, \\ S_{\varphi\varphi} &= \sin^2\varphi S_{xx}-\cos\varphi\sin\varphi S_{yx} -\cos\varphi\sin\varphi S_{xy}+\cos^2\varphi S_{yy}. \end{split} \end{equation}\end{split}\]

Derivace v polárním souřadnicovém systému

Problémem derivování v polárním souřadnicovém systému je závislost vektorů \(\boldsymbol{e}_r\) a \(\boldsymbol{e}_\varphi\) na poloze, tj.

(32)\[\begin{split}\begin{equation} \begin{split} \partial_{\varphi}\boldsymbol{e}_r &= \boldsymbol{e}_\varphi, \\ \partial_{\varphi}\boldsymbol{e}_\varphi &= -\boldsymbol{e}_r. \end{split} \end{equation}\end{split}\]

Ostatní derivace jsou nulové. Nicméně lze derivace skalární, vektorové a tenzorové funkce vyjádřit elegantně pomocí operátoru nabla, který má v polárních souřadnicích tvar

(33)\[\nabla\equiv\boldsymbol{e}_r\partial_r+\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi =\left[\partial_{r},\frac{1}{r}\partial_\varphi\right]^T.\]

Gradient skalární funkce

Jestliže je \(f(r,\varphi)\) skalární funkce, její gradient lze zapsat jako

(34)\[\nabla f=\left(\boldsymbol{e}_r\partial_r +\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi\right)f =\boldsymbol{e}_r\partial_rf+\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi f\]

nebo maticově

(35)\[\nabla f=\left[\partial_{r}f,\frac{1}{r}\partial_\varphi f\right]^T.\]

Gradient vektorové funkce

Jestliže \(\boldsymbol{v}(r,\varphi)\equiv v_r\boldsymbol{e}_r+v_\varphi\boldsymbol{e}_\varphi\) je vektorová funkce, potom její gradient je tenzor, který může být vyjádřen pomocí dyadického (tenzorového, vnějšího) součinu, viz odstavec Dyadická algebra,

(36)\[\boldsymbol{v}\otimes\nabla=(v_r\boldsymbol{e}_r+v_\varphi\boldsymbol{e}_\varphi) \otimes\left(\boldsymbol{e}_r\partial_r +\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi\right).\]

Pozor, je nutné derivovat i bázové vektory podle (32), takže předchozí vztah má tvar

(37)\[\begin{split}\begin{equation} \begin{split} \boldsymbol{v}\otimes\nabla &= \partial_rv_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rv_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}\partial_\varphi(v_r\boldsymbol{e}_r)\otimes\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi(v_\varphi\boldsymbol{e}_\varphi)\otimes\boldsymbol{e}_\varphi \\ &= \partial_rv_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rv_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}\partial_\varphi v_r\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}v_r\partial_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi v_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\frac{1}{r}v_\varphi\partial_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \\ &= \partial_rv_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rv_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}\partial_\varphi v_r\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}v_r\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi v_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi -\frac{1}{r}v_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &= \partial_{r}v_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rv_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \frac{1}{r}\left(\partial_\varphi v_\varphi+v_r\right)\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\frac{1}{r}\left(\partial_\varphi v_r-v_\varphi\right)\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi, \end{split} \end{equation}\end{split}\]

nebo-li

(38)\[\begin{split}\boldsymbol{v}\otimes\nabla\equiv\left[ \begin{array}{cc} \partial_rv_r & r^{-1}\left(\partial_\varphi v_r-v_\varphi\right)\\ \partial_rv_\varphi & r^{-1}\left(\partial_\varphi v_\varphi+v_r\right) \end{array} \right].\end{split}\]

Divergence vektorové funkce

Divergence vektorové funkce \(\boldsymbol{v}(r,\varphi)\equiv v_r\boldsymbol{e}_r+v_\varphi\boldsymbol{e}_\varphi\) se dostane jako její skalární součin s operátorem nabla,

(39)\[\begin{split}\begin{equation} \begin{split} \nabla\cdot\boldsymbol{v} &= \left(\boldsymbol{e}_r\partial_r +\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi\right) \cdot(v_r\boldsymbol{e}_r+v_\varphi\boldsymbol{e}_\varphi) \\ &= \partial_rv_r\boldsymbol{e}_r\cdot\boldsymbol{e}_r +\partial_rv_\varphi\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi(v_r\boldsymbol{e}_r) \cdot\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi(v_\varphi\boldsymbol{e}_\varphi)\cdot\boldsymbol{e}_\varphi \\ &= \partial_rv_r+\frac{1}{r}\partial_\varphi v_r\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi +\frac{1}{r}v_r\partial_\varphi\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi v_\varphi\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_\varphi +\frac{1}{r}v_\varphi\partial_\varphi\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_\varphi \\ &= \partial_{r}v_r+\frac{1}{r}v_r\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi v_\varphi-\frac{1}{r}v_\varphi\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi \\ &= \partial_{r}v_r+\frac{1}{r}v_r+\frac{1}{r}\partial_\varphi v_\varphi. \end{split} \end{equation}\end{split}\]

Takže

(40)\[\nabla\cdot\boldsymbol{v} = \partial_{r}v_r+\frac{1}{r}v_r+\frac{1}{r}\partial_\varphi v_\varphi.\]

Divergence tenzorové funkce

Divergence \(\nabla\cdot\boldsymbol{S}(r,\varphi)\) tenzorové funkce

(41)\[\boldsymbol{S}(r,\varphi) \equiv S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\]

je vektor

(42)\[\begin{split}\begin{equation} \begin{split} \nabla\cdot\boldsymbol{S} &= \left( \boldsymbol{e}_r\partial_r +\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi \right) \cdot\left( S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \right) \\ &= \partial_rS_{rr}\boldsymbol{e}_r\cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rS_{r\varphi}\boldsymbol{e}_r\cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\partial_rS_{\varphi r}\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \partial_rS_{\varphi\varphi}\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi \cdot\frac{1}{r}\partial_\varphi\left( S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r \right) +\boldsymbol{e}_\varphi \cdot\frac{1}{r}\partial_\varphi\left( S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \right) \\ &+ \boldsymbol{e}_\varphi \cdot\frac{1}{r}\partial_\varphi\left( S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \right) +\boldsymbol{e}_\varphi \cdot\frac{1}{r}\partial_\varphi\left( S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \right) \\ &= \partial_rS_{rr}\boldsymbol{e}_r+\partial_rS_{r\varphi}\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi S_{rr}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\frac{1}{r}S_{rr}\boldsymbol{e}_\varphi \cdot\partial_\varphi\left( \boldsymbol{e}_r\otimes\boldsymbol{e}_r \right) \\ &+ \frac{1}{r}\partial_\varphi S_{r\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{r\varphi}\boldsymbol{e}_\varphi \cdot\partial_\varphi\left( \boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \right) \\ &+ \frac{1}{r}\partial_\varphi S_{\varphi r}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}S_{\varphi r}\boldsymbol{e}_\varphi \cdot\partial_\varphi\left( \boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \right) \\ &+ \frac{1}{r}\partial_\varphi S_{\varphi\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_\varphi \cdot\partial_\varphi\left( \boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \right) \\ &= \partial_rS_{rr}\boldsymbol{e}_r+\partial_rS_{r\varphi}\boldsymbol{e}_\varphi +\frac{1}{r}S_{rr}\boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\frac{1}{r}S_{rr}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\partial_\varphi\boldsymbol{e}_r \\ &+ \frac{1}{r}\partial_\varphi S_{r\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{r\varphi}\boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{r\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\partial_\varphi\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi S_{\varphi r}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}S_{\varphi r}\boldsymbol{e}_\varphi \cdot\partial_{\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}S_{\varphi r}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\partial_\varphi\boldsymbol{e}_r \\ &+ \frac{1}{r}\partial_\varphi S_{\varphi\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\partial_\varphi\boldsymbol{e}_\varphi \\ &= \partial_rS_{rr}\boldsymbol{e}_r +\partial_rS_{r\varphi}\boldsymbol{e}_\varphi +\frac{1}{r}S_{rr}\boldsymbol{e}_r +\frac{1}{r}S_{r\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi S_{\varphi r}\boldsymbol{e}_r \\ &- \frac{1}{r}S_{\varphi r}\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_{r} +\frac{1}{r}S_{\varphi r}\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi S_{\varphi\varphi}\boldsymbol{e}_\varphi -\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi -\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_r \\ &= \partial_rS_{rr}\boldsymbol{e}_r +\partial_rS_{r\varphi}\boldsymbol{e}_\varphi +\frac{1}{r}S_{rr}\boldsymbol{e}_r+\frac{1}{r}S_{r\varphi}\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi S_{\varphi r}\boldsymbol{e}_r \\ &+ \frac{1}{r}S_{\varphi r}\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi S_{\varphi\varphi}\boldsymbol{e}_\varphi -\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_r \\ &= \left(\partial_rS_{rr}+\frac{1}{r}S_{rr}+\frac{1}{r}\partial_\varphi S_{\varphi r} -\frac{1}{r}S_{\varphi\varphi}\right)\boldsymbol{e}_r \\ &+ \left(\partial_rS_{r\varphi}+\frac{1}{r}S_{r\varphi}+\frac{1}{r}S_{\varphi r} +\frac{1}{r}\partial_\varphi S_{\varphi\varphi}\right)\boldsymbol{e}_\varphi. \end{split} \end{equation}\end{split}\]

Zapsáno vektorově

(43)\[\begin{split}\nabla\cdot\boldsymbol{S}\equiv\left[ \begin{array}{c} \partial_rS_{rr}+r^{-1}S_{rr} +r^{-1}\partial_\varphi S_{\varphi r}-r^{-1}S_{\varphi\varphi}\\ \partial_rS_{r\varphi}+r^{-1}S_{r\varphi}+r^{-1}S_{\varphi r} +r^{-1}\partial_{\varphi}S_{\varphi\varphi} \end{array} \right].\end{split}\]

Gradient tenzorové funkce

Gradient tenzorové funkce

(44)\[\boldsymbol{S}(r,\varphi) \equiv S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\]

je tenzor třetího řádu a dostane se jako tenzorový součin \(\boldsymbol{S}(r,\varphi)\) s operátorem nabla

(45)\[\begin{split}\boldsymbol{S}\otimes\nabla &= (S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi )\otimes\left( \boldsymbol{e}_r\partial_r +\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi \right) \\ &= \partial_r(S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r)\otimes\boldsymbol{e}_r +\partial_r(S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi)\otimes\boldsymbol{e}_r \\ &+ \partial_r(S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r)\otimes\boldsymbol{e}_r +\partial_r( S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi )\otimes\boldsymbol{e}_r \\ &+ \frac{1}{r}\partial_\varphi( S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r )\otimes\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi( S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi )\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi( S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r )\otimes\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi( S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi )\otimes\boldsymbol{e}_\varphi \\ &= \partial_rS_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{rr}\partial_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{rr}\boldsymbol{e}_r\otimes\partial_r\boldsymbol{e}_r\otimes\boldsymbol{e}_{r} \\ &+ \partial_rS_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{r\varphi}\partial_r\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{r\varphi}\boldsymbol{e}_r\otimes\partial_r\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \partial_rS_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{\varphi r}\partial_r\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{\varphi r}\boldsymbol{e}_\varphi\otimes\partial_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r \\ &+ \partial_rS_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi}\partial_r\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\partial_r\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \frac{1}{r}\partial_\varphi S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{rr}\partial_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{rr}\boldsymbol{e}_r\otimes\partial_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{r\varphi}\partial_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{r\varphi}\boldsymbol{e}_r\otimes\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{\varphi r}\partial_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{\varphi r}\boldsymbol{e}_\varphi\otimes\partial_\varphi\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{\varphi\varphi}\partial_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi \\ &= \partial_rS_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rS_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \partial_rS_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rS_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r \\ &+ \frac{1}{r}\partial_\varphi S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{rr}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{r\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_{\varphi} -\frac{1}{r}S_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_\varphi S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi -\frac{1}{r}S_{\varphi r}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_{\varphi}S_{\varphi\varphi}\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi -\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi -\frac{1}{r}S_{\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi.\end{split}\]

Takže

(46)\[\begin{split}\boldsymbol{S}\otimes\nabla &= \partial_rS_{rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rS_{r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \partial_{r}S_{\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_{r}S_{\varphi\varphi}\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \frac{1}{r}( \partial_\varphi S_{rr}-S_{r\varphi}-S_{\varphi r} )\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}( S_{rr}-S_{\varphi\varphi}+\partial_\varphi S_{\varphi r} )\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}( S_{rr}-S_{\varphi\varphi}+\partial_\varphi S_{r\varphi} )\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\frac{1}{r}( S_{r\varphi}+S_{\varphi r}+\partial_\varphi S_{\varphi\varphi} )\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi.\end{split}\]

Divergence triády

Divergence tenzorové funkce

(47)\[\begin{split}\boldsymbol{S}(r,\varphi) &= S_{rrr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{rr\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{r\varphi r}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ S_{r\varphi\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +S_{\varphi rr}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{\varphi r\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &+ S_{\varphi\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi\end{split}\]

je dyáda a dostane se jako vektorový součin \(\boldsymbol{S}(r,\varphi)\) s operátorem nabla

(48)\[\begin{split}\nabla\cdot\boldsymbol{S} &= \left( \boldsymbol{e}_r\partial_r+\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi \right)\cdot\left( S_{rrr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{rr\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{r\varphi r}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \right. \\ &+ S_{r\varphi\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +S_{\varphi rr}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{\varphi r\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &+ \left. S_{\varphi\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi \right) \\ &= \partial_rS_{rrr}\boldsymbol{e}_r \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rS_{rr\varphi}\boldsymbol{e}_r \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\partial_rS_{r\varphi r}\boldsymbol{e}_r \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \partial_rS_{r\varphi\varphi}\boldsymbol{e}_r \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\partial_rS_{\varphi rr}\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rS_{\varphi r\varphi}\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \\ &+ \partial_rS_{\varphi\varphi r}\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\partial_rS_{\varphi\varphi\varphi}\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\left[ \partial_{\varphi}S_{rrr}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r \right. \\ &+ \left. S_{rrr}\left( \boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\partial_\varphi\boldsymbol{e}_r \otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r \otimes\partial_\varphi\boldsymbol{e}_r \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{rr\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \right. \\ &+ \left. S_{rr\varphi}\left( \boldsymbol{e}_\varphi \cdot\partial_{\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\partial_\varphi\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r \otimes\partial_{\varphi}\boldsymbol{e}_\varphi \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{r\varphi r}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r \right. \\ &+ \left. S_{r\varphi r}\left( \boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r \otimes\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\partial_\varphi\boldsymbol{e}_r \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{r\varphi\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi \right. \\ &+ \left. S_{r\varphi\varphi}\left( \boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\partial_\varphi\boldsymbol{e}_\varphi \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi rr}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r \right. \\ &+ \left. S_{\varphi rr}\left( \boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\boldsymbol{e}_{\varphi} \cdot\boldsymbol{e}_\varphi \otimes\partial_\varphi\boldsymbol{e}_r \otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \otimes\partial_\varphi\boldsymbol{e}_r \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi r\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi \right. \\ &+ \left. S_{\varphi r\varphi}\left( \boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_{\varphi} \cdot\boldsymbol{e}_\varphi\otimes\partial_\varphi\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \otimes\partial_\varphi\boldsymbol{e}_\varphi \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi\varphi r}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r \right. \\ &+ \left. S_{\varphi\varphi r}\left( \boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi \otimes\partial_{\varphi}\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi \otimes\partial_\varphi\boldsymbol{e}_r \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi} \cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \otimes\boldsymbol{e}_{\varphi} \right. \\ &+ \left. S_{\varphi\varphi\varphi}\left( \boldsymbol{e}_\varphi \cdot\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi \otimes\partial_\varphi\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi \otimes\partial_\varphi\boldsymbol{e}_\varphi \right) \right] \\ &= \partial_rS_{rrr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rS_{rr\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\partial_rS_{r\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\partial_rS_{r\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}S_{rrr}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\frac{1}{r}S_{rr\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\otimes\boldsymbol{e}_{\varphi} \\ &+ \frac{1}{r}S_{r\varphi r}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}S_{r\varphi\varphi}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{\varphi rr}\left( -\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{\varphi r\varphi}\left( -\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_r \otimes\boldsymbol{e}_\varphi +\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi -\boldsymbol{e}_r\otimes\boldsymbol{e}_r \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi r}\left( -\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_r -\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi} \otimes\boldsymbol{e}_\varphi +S_{\varphi\varphi\varphi}\left( -\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_\varphi -\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi -\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \right) \right] \\ &= \partial_rS_{rrr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rS_{rr\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\partial_rS_{r\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\partial_rS_{r\varphi\varphi}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}S_{rrr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\frac{1}{r}S_{rr\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +\frac{1}{r}S_{r\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}S_{r\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi rr}\boldsymbol{e}_r\otimes\boldsymbol{e}_r +S_{\varphi rr}\left( \boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\boldsymbol{e}_r\otimes\boldsymbol{e}_{\varphi} \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi r\varphi}\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi +S_{\varphi r\varphi}\left( \boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi -\boldsymbol{e}_r\otimes\boldsymbol{e}_r \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi\varphi r}\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +S_{\varphi\varphi r}\left( -\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \right) \right] \\ &+ \frac{1}{r}\left[ \partial_\varphi S_{\varphi\varphi\varphi}\boldsymbol{e}_\varphi \otimes\boldsymbol{e}_{\varphi} +S_{\varphi\varphi\varphi} \left(-\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi -\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \right) \right]\end{split}\]

Takže

(49)\[\begin{split}\nabla\cdot\boldsymbol{S} &= \left( \partial_rS_{rrr}+\frac{1}{r}S_{rrr}+\frac{1}{r}\partial_\varphi S_{\varphi rr} -\frac{1}{r}S_{\varphi r\varphi}-\frac{1}{r}S_{\varphi\varphi r} \right)\boldsymbol{e}_r\otimes\boldsymbol{e}_r \\ &+ \left( \partial_rS_{rr\varphi}+\frac{1}{r}S_{rr\varphi}+\frac{1}{r}S_{\varphi rr} +\frac{1}{r}\partial_\varphi S_{\varphi r\varphi} -\frac{1}{r}S_{\varphi\varphi\varphi} \right)\boldsymbol{e}_{r}\otimes\boldsymbol{e}_\varphi \\ &+ \left( \partial_rS_{r\varphi r}+\frac{1}{r}S_{r\varphi r}+\frac{1}{r}S_{\varphi rr} +\frac{1}{r}\partial_\varphi S_{\varphi\varphi r} -\frac{1}{r}S_{\varphi\varphi\varphi} \right)\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \\ &+ \left( \partial_rS_{r\varphi\varphi}+\frac{1}{r}S_{r\varphi\varphi} +\frac{1}{r}S_{\varphi r\varphi}+\frac{1}{r}S_{\varphi\varphi r} +\frac{1}{r}\partial_\varphi S_{\varphi\varphi\varphi} \right)\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi.\end{split}\]

Druhé derivace v polárním souřadnicovém systému

Podobně jako v části Derivace v polárním souřadnicovém systému se snadno odvodí velmi důležité vztahy pro Laplacián ze sklární a vektorové funkce v polárním souřadnicovém systému.

Laplacián ze skalární funkce

Pro skalární funkci \(f(r,\varphi)\) platí

(50)\[\begin{split}\begin{equation} \begin{split} \Delta f \equiv \nabla^2f &= \nabla\cdot(\nabla f) =\left( \boldsymbol{e}_r\partial_r+\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi \right) \cdot\left( \boldsymbol{e}_r\partial_rf+\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi f \right) \\ &=\boldsymbol{e}_r\cdot\partial_r(\boldsymbol{e}_r\partial_rf) +\boldsymbol{e}_r\cdot\partial_r\left( \boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi f \right) +\frac{1}{r}\boldsymbol{e}_\varphi\cdot\partial_\varphi(\boldsymbol{e}_r\partial_rf) +\frac{1}{r}\boldsymbol{e}_\varphi \cdot\partial_\varphi\left( \boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi f \right) \\ &=\boldsymbol{e}_r\cdot\boldsymbol{e}_r\partial_{rr}f -\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi\frac{1}{r^2}\partial_\varphi f +\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi\frac{1}{r}\partial_{r\varphi}f +\frac{1}{r}\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_\varphi\partial_rf \\ &+\frac{1}{r}\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_r\partial_{r\varphi}f -\frac{1}{r}\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_r\frac{1}{r}\partial_\varphi f +\frac{1}{r}\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi\frac{1}{r}\partial_{\varphi\varphi}f \\ &=\partial_{rr}f+\frac{1}{r}\partial_rf+\frac{1}{r^2}\partial_{\varphi\varphi}f \end{split} \end{equation}\end{split}\]

Tedy přehledněji

(51)\[\Delta f=\partial_{rr}f+\frac{1}{r}\partial_rf+\frac{1}{r^2}\partial_{\varphi\varphi}f\]

Laplacián z vektorové funkce

Laplacián z vektorové funkce \(\boldsymbol{v}(r,\varphi)\) se dovodí pomocí (38) následovně

(52)\[\begin{split}\begin{equation} \begin{split} \Delta\boldsymbol{v} \equiv \nabla^2\boldsymbol{v} &= (\boldsymbol{v}\otimes\nabla)\cdot\nabla \\ &= \left[ \partial_rv_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r +\partial_rv_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r +\frac{1}{r}\left(\partial_\varphi v_\varphi+v_r\right)\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \right. \\ &+ \left. \frac{1}{r}\left(\partial_\varphi v_r-v_\varphi\right)\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \right]\cdot\left( \boldsymbol{e}_r\partial_r+\boldsymbol{e}_\varphi\frac{1}{r}\partial_\varphi \right) \\ &= \partial_r(\partial_rv_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r)\cdot\boldsymbol{e}_r +\partial_r(\partial_rv_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r) \cdot\boldsymbol{e}_r \\ &+ \partial_r\left[ \frac{1}{r}\left( \partial_\varphi v_\varphi+v_r \right)\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \right]\cdot\boldsymbol{e}_r +\partial_r\left[ \frac{1}{r}\left( \partial_\varphi v_r-v_\varphi \right)\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \right]\cdot\boldsymbol{e}_r \\ &+ \frac{1}{r}\partial_\varphi(\partial_rv_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r)\cdot\boldsymbol{e}_\varphi +\frac{1}{r}\partial_\varphi(\partial_rv_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r) \cdot\boldsymbol{e}_\varphi \\ &+ \partial_\varphi\left[ \frac{1}{r}\left( \partial_\varphi v_\varphi+v_r \right) \boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \right]\cdot\frac{1}{r}\boldsymbol{e}_\varphi +\partial_\varphi\left[ \frac{1}{r}\left( \partial_\varphi v_r-v_\varphi \right) \boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \right]\cdot\frac{1}{r}\boldsymbol{e}_\varphi \\ &= \partial_{rr}v_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r\cdot\boldsymbol{e}_r +\partial_{rr}v_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\cdot\boldsymbol{e}_r \\ &- \frac{1}{r^2}\left( \partial_\varphi v_\varphi+v_r \right)\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_r +\frac{1}{r}\left( \partial_{r\varphi}v_\varphi +\partial_rv_r \right)\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_r \\ &- \frac{1}{r^2}\left( \partial_\varphi v_r-v_\varphi \right)\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_r +\frac{1}{r}\left( \partial_{r\varphi}v_r -\partial_rv_\varphi \right)\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_r \\ &+ \frac{1}{r}\partial_{r\varphi}v_r\boldsymbol{e}_r\otimes\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi +\frac{1}{r}\partial_rv_r\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi +\frac{1}{r}\partial_rv_r\boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi \\ &+ \frac{1}{r}\partial_{r\varphi}v_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi -\frac{1}{r}\partial_rv_\varphi\boldsymbol{e}_r\otimes\boldsymbol{e}_r \cdot\boldsymbol{e}_\varphi +\frac{1}{r}\partial_rv_\varphi\boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi \cdot\boldsymbol{e}_\varphi \\ &+ \frac{1}{r^2}\left( \partial_{\varphi\varphi} v_\varphi+\partial_\varphi v_r \right) \boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_\varphi -\frac{1}{r^2}\left( \partial_\varphi v_\varphi+v_r \right) \boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_\varphi \\ &- \frac{1}{r^2}\left( \partial_\varphi v_\varphi+v_r \right) \boldsymbol{e}_\varphi\otimes\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi +\frac{1}{r^2}\left( \partial_{\varphi\varphi}v_r-\partial_\varphi v_\varphi \right) \boldsymbol{e}_r\otimes\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_\varphi \\ &+ \frac{1}{r^2}\left( \partial_\varphi v_r-v_\varphi \right) \boldsymbol{e}_\varphi\otimes\boldsymbol{e}_\varphi\cdot\boldsymbol{e}_\varphi -\frac{1}{r^2}\left( \partial_\varphi v_r-v_\varphi \right) \boldsymbol{e}_r\otimes\boldsymbol{e}_r\cdot\boldsymbol{e}_\varphi \\ &= \left(\partial_{rr}v_r+\frac{1}{r}\partial_rv_r -\frac{2}{r^2}\partial_\varphi v_\varphi-\frac{1}{r^2}v_r +\frac{1}{r^2}\partial_{\varphi\varphi}v_r \right)\boldsymbol{e}_r \\ &+ \left( \partial_{rr}v_\varphi\boldsymbol{e}_\varphi +\frac{1}{r}\partial_rv_\varphi +\frac{1}{r^2}\partial_{\varphi\varphi} v_\varphi+\frac{2}{r^2}\partial_\varphi v_r -\frac{1}{r^2}v_\varphi \right)\boldsymbol{e}_\varphi. \end{split} \end{equation}\end{split}\]

Přehlednější zápis ve vektorovém tvaru

(53)\[\begin{split}\Delta\boldsymbol{v} = \left[ \begin{array}{c} \partial_{rr}v_r+r^{-1}\partial_rv_r -2r^{-2}\partial_\varphi v_\varphi-r^{-2}v_r +r^{-2}\partial_{\varphi\varphi}v_r \\ \partial_{rr}v_\varphi\boldsymbol{e}_\varphi +r^{-1}\partial_rv_\varphi +r^{-2}\partial_{\varphi\varphi} v_\varphi+2r^{-2}\partial_\varphi v_r -r^{-2}v_\varphi \end{array} \right].\end{split}\]

Transformace tenzoru napětí \(\boldsymbol{\tau}\) do polárních souřadnic

Tenzor napětí \(\boldsymbol{\tau}\) se do polárních souřadnic přepočítává klasicky podle transformace tenzorů při natočení souřadnic, viz také Polární a kartézské souřadnicové systémy. Takže pro \(\tau_{rr}\) platí

(54)\[\tau_{rr}=\cos^2\varphi\tau_{xx}+2\cos\varphi\sin\varphi\tau_{xy}+\sin^2\varphi\tau_{yy}.\]

Dosadí se z Hookeova zákona

(55)\[\begin{split}\tau_{xx} &= (\lambda+2\mu)\varepsilon_{xx}+\lambda\varepsilon_{yy}, \\ \tau_{xy} &= 2\mu\varepsilon_{xy}, \\ \tau_{yy} &= (\lambda+2\mu)\varepsilon_{yy}+\lambda\varepsilon_{xx},\end{split}\]

za \(\tau_{xx}\), \(\tau_{xy}\) a \(\tau_{yy}\), tj.

(56)\[\tau_{rr} = \cos^2\varphi[(\lambda+2\mu)\varepsilon_{xx}+\lambda\varepsilon_{yy}] +4\cos\varphi\sin\varphi\mu\varepsilon_{xy}+\sin^2\varphi[(\lambda+2\mu)\varepsilon_{yy} +\lambda\varepsilon_{xx}]\]

a následně upraví

(57)\[\begin{split}\tau_{rr} &= (\lambda+2\mu)[\cos^2\varphi\varepsilon_{xx}+\sin^2\varphi\varepsilon_{yy}] +\lambda\cos^2\varphi\varepsilon_{yy}+4\cos\varphi\sin\varphi\mu\varepsilon_{xy} \\ &+ \lambda\sin^2\varphi\varepsilon_{xx} +(2\lambda-2\lambda)\cos\varphi\sin\varphi\varepsilon_{xy} \\ &= (\lambda+2\mu)[\cos^2\varphi\varepsilon_{xx}+2\cos\varphi\sin\varphi\varepsilon_{xy} +\sin^2\varphi\varepsilon_{yy}] \\ &+ \lambda[\cos^2\varphi\varepsilon_{yy}-2\cos\varphi\sin\varphi\varepsilon_{xy} +\sin^2\varphi\varepsilon_{xx}] \\ &= (\lambda+2\mu)[\cos^2\varphi\varepsilon_{xx}+2\cos\varphi\sin\varphi\varepsilon_{xy} +\sin^2\varphi\varepsilon_{yy}] \\ &+ \lambda[\cos^2\varphi\varepsilon_{yy}-2\cos\varphi\sin\varphi\varepsilon_{xy} +\sin^2\varphi\varepsilon_{xx}].\end{split}\]

Výrazy v hranatých závorkách jsou transformační vztahy pro tenzory \(\varepsilon_{rr}\) a \(\varepsilon_{\varphi\varphi}\). Takže výsledný vztah je

(58)\[\tau_{rr}=(\lambda+2\mu)\varepsilon_{rr}+\lambda\varepsilon_{\varphi\varphi}.\]

Stejným způsobem se pokračuje v případě dalších složek tenzoru \(\boldsymbol{\tau}\)

(59)\[\begin{split}\tau_{r\varphi} &= -\cos\varphi\sin\varphi\tau_{xx}+(\cos^2\varphi-\sin^2\varphi)\tau_{xy} +\cos\varphi\sin\varphi\tau_{yy} \\ &= -\cos\varphi\sin\varphi[(\lambda+2\mu)\varepsilon_{xx}+\lambda\varepsilon_{yy}] +2(\cos^2\varphi-\sin^2\varphi)\mu\varepsilon_{xy} \\ &+ \cos\varphi\sin\varphi[(\lambda+2\mu)\varepsilon_{yy} +\lambda\varepsilon_{xx}] \\ &+ (\lambda-\lambda)(\cos^2\varphi-\sin^2\varphi)\varepsilon_{xy} \\ &= (\lambda+2\mu)[-\cos\varphi\sin\varphi\varepsilon_{xx} +(\cos^2\varphi-\sin^2\varphi)\varepsilon_{xy}+\cos\varphi\sin\varphi\varepsilon_{yy}] \\ &+ \lambda[-\cos\varphi\sin\varphi\varepsilon_{yy} -(\cos^2\varphi-\sin^2\varphi)\varepsilon_{xy}+\cos\varphi\sin\varphi\varepsilon_{xx}] \\ &= (\lambda+2\mu)\varepsilon_{\varphi r}-\lambda\varepsilon_{\varphi r} \\ &= 2\mu\varepsilon_{\varphi r}, \\ \tau_{\varphi\varphi} &= \sin^2\varphi\tau_{xx}-2\cos\varphi\sin\varphi\tau_{yx}+\cos^2\varphi\tau_{yy} \\ &= \sin^2\varphi[(\lambda+2\mu)\varepsilon_{xx}+\lambda\varepsilon_{yy}] -4\cos\varphi\sin\varphi\mu\varepsilon_{xy} +\cos^2\varphi[(\lambda+2\mu)\varepsilon_{yy}+\lambda\varepsilon_{xx}] \\ &= (\lambda+2\mu)(\sin^2\varphi\varepsilon_{xx}+\cos^2\varphi\varepsilon_{yy}) +\lambda(\sin^2\varepsilon_{yy}+\cos^2\varphi\varepsilon_{xx}) -4\mu\cos\varphi\sin\varphi\varepsilon_{xy} \\ &+ (2\lambda-2\lambda)\cos\varphi\sin\varphi\varepsilon_{xy} \\ &= (\lambda+2\mu)(\sin^2\varphi\varepsilon_{xx} -2\cos\varphi\sin\varphi\varepsilon_{xy} +\cos^2\varphi\varepsilon_{yy}) \\ &+ \lambda(\sin^2\varepsilon_{yy}+2\cos\varphi\sin\varphi\varepsilon_{xy} +\cos^2\varphi\varepsilon_{xx}) \\ &= (\lambda+2\mu)\varepsilon_{\varphi\varphi}+\lambda\varepsilon_{rr}.\end{split}\]

Vektorová algebra pro válcové souřadnice

Vektor ve válcových souřadnicích se může zapsat ve tvaru

(60)\[\boldsymbol{u}=u_{r}\boldsymbol{e}_{r}+u_{\varphi}\boldsymbol{e}_{\varphi}+u_{z}\boldsymbol{k},\]

kde

(61)\[\begin{split}\boldsymbol{e}_{r} &= (\cos\varphi,\sin\varphi,0), \\ \boldsymbol{e}_{\varphi} &= \boldsymbol{k}\times\boldsymbol{e}_{r}=(-\sin\varphi,\cos\varphi,0), \\ \boldsymbol{k} &= (0,0,1).\end{split}\]

Derivace vektorů \(\boldsymbol{e}_{r}\), \(\boldsymbol{e}_{\varphi}\) a \(\boldsymbol{k}\) jsou

(62)\[\boldsymbol{e}_{r,r}=0,\quad\boldsymbol{e}_{r,\varphi}=\boldsymbol{e}_{\varphi}, \quad\boldsymbol{e}_{r,z}=0,\]

(63)\[\boldsymbol{e}_{\varphi,r}=0,\quad\boldsymbol{e}_{\varphi,\varphi}=-\boldsymbol{e}_{r}, \quad\boldsymbol{e}_{\varphi,z}=0,\]

(64)\[\boldsymbol{k}_{,r}=0,\quad\boldsymbol{k}_{,\varphi}=0,\quad\boldsymbol{k}_{,z}=0.\]

Operátor \(\nabla\) má ve válcových souřadnicích tvar

(65)\[\nabla\equiv\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi} +\boldsymbol{k}\partial_{z}=[\partial_{r},r^{-1}\partial_{\varphi},\partial_{z}]^{T}\]

Gradient skalární funkce

Jestliže je \(f(r,\varphi,z)\) skalární funkce, pak její gradient má tvar

(66)\[\nabla f=(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi} +\boldsymbol{k}\partial_{z})f =\boldsymbol{e}_{r}f_{,r}+\boldsymbol{e}_{\varphi}r^{-1}f_{,\varphi}+\boldsymbol{k}f_{,z}.\]

Zapsáno maticově

(67)\[\nabla f=[f_{,r},r^{-1}f_{,\varphi},f_{,z}].\]

Gradient vektorové funkce

Jestliže je \(\boldsymbol{v}(r,\varphi,z)\equiv v_{r}\boldsymbol{e}_{r}+v_{\varphi}\boldsymbol{e}_{\varphi}+v_{z}\boldsymbol{k}\) vektorová funkce, potom její gradient má tvar

(68)\[\begin{split}\boldsymbol{v}\otimes\nabla &= (v_{r}\boldsymbol{e}_{r}+v_{\varphi}\boldsymbol{e}_{\varphi}+v_{z}\boldsymbol{k}) \otimes(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi} +\boldsymbol{k}\partial_{z}) \\ &=(v_{r}\boldsymbol{e}_{r})_{,r}\otimes\boldsymbol{e}_{r}+(v_{\varphi}\boldsymbol{e}_{\varphi})_{,r}\otimes\boldsymbol{e}_{r} +(v_{z}\boldsymbol{k})_{,r}\otimes\boldsymbol{e}_{r} \\ &+r^{-1}(v_{r}\boldsymbol{e}_{r})_{,\varphi}\otimes\boldsymbol{e}_{r} +r^{-1}(v_{\varphi}\boldsymbol{e}_{\varphi})_{,\varphi}\otimes\boldsymbol{e}_{r} +r^{-1}(v_{z}\boldsymbol{k})_{,\varphi}\otimes\boldsymbol{e}_{r} \\ &+(v_{r}\boldsymbol{e}_{r})_{,z}\otimes\boldsymbol{k}+(v_{\varphi}\boldsymbol{e}_{\varphi})_{,z}\otimes\boldsymbol{k} +(v_{z}\boldsymbol{k})_{,z}\otimes\boldsymbol{k} \\ &=v_{r,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+v_{\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +v_{z,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r} \\ &+r^{-1}v_{r,\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+r^{-1}v_{r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}v_{\varphi,\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} -r^{-1}v_{\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+r^{-1}v_{z,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} \\ &+v_{r,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k}+v_{\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} +v_{z,z}\boldsymbol{k}\otimes\boldsymbol{k} \\ &=v_{r,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+r^{-1}(v_{r,\varphi}-v_{\varphi})\boldsymbol{e}_{r} \otimes\boldsymbol{e}_{\varphi}+v_{r,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+v_{\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+r^{-1}(v_{r}+v_{\varphi,\varphi})\boldsymbol{e}_{\varphi} \otimes\boldsymbol{e}_{\varphi}+v_{\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+v_{z,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r}+r^{-1}v_{z,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +v_{z,z}\boldsymbol{k}\otimes\boldsymbol{k},\end{split}\]

nebo-li

(69)\[\begin{split}\boldsymbol{v}\otimes\nabla= \left[\begin{array}{ccc} v_{r,r} & r^{-1}(v_{r,\varphi}-v_{\varphi}) & v_{r,z}\\ v_{\varphi,r} & r^{-1}(v_{r}+v_{\varphi,\varphi}) & v_{\varphi,z}\\ v_{z,r} & r^{-1}v_{z,\varphi} & v_{z,z} \end{array}\right].\end{split}\]

Gradient tenzorové funkce

Jestliže je \(\boldsymbol{S}(r,\varphi,z)\) tenzorová funkce, tj.

(70)\[\begin{split}\boldsymbol{S}(r,\varphi,z) &\equiv S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} +S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{rz}\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{zz}\boldsymbol{k}\otimes\boldsymbol{k},\end{split}\]

pak platí

(71)\[\begin{split}\boldsymbol{S}\otimes\nabla &=(S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +S_{rz}\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} +S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +S_{zz}\boldsymbol{k}\otimes\boldsymbol{k}) \\ &\otimes(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi}+\boldsymbol{k}\partial_{z}) \\ &=(S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,r}\otimes\boldsymbol{e}_{r} +(S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,r}\otimes\boldsymbol{e}_{r} +(S_{rz}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,r}\otimes\boldsymbol{e}_{r} \\ &+(S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,r}\otimes\boldsymbol{e}_{r} +(S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,r}\otimes\boldsymbol{e}_{r} +(S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,r}\otimes\boldsymbol{e}_{r} \\ &+(S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,r}\otimes\boldsymbol{e}_{r} +(S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,r}\otimes\boldsymbol{e}_{r} +(S_{zz}\boldsymbol{k}\otimes\boldsymbol{k})_{,r}\otimes\boldsymbol{e}_{r} \\ &+r^{-1}(S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,\varphi}\otimes\boldsymbol{e}_{\varphi} +r^{-1}(S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}(S_{rk}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,\varphi}\otimes\boldsymbol{e}_{\varphi} +r^{-1}(S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}(S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\otimes\boldsymbol{e}_{\varphi} +r^{-1}(S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}(S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,\varphi}\otimes\boldsymbol{e}_{\varphi} +r^{-1}(S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}(S_{zz}\boldsymbol{k}\otimes\boldsymbol{k})_{,\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+(S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,z}\otimes\boldsymbol{k} +(S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,z}\otimes\boldsymbol{k} +(S_{rz}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,z}\otimes\boldsymbol{k} \\ &+(S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,z}\otimes\boldsymbol{k} +(S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,z}\otimes\boldsymbol{k} +(S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,z}\otimes\boldsymbol{k} \\ &+(S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,z}\otimes\boldsymbol{k} +(S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,z}\otimes\boldsymbol{k} +(S_{zz}\boldsymbol{k}\otimes\boldsymbol{k})_{,z}\otimes\boldsymbol{k} \\ &=S_{rr,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} +S_{r\varphi,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +S_{rz,r}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r} \\ &+S_{\varphi r,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} +S_{\varphi\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +S_{\varphi z,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r} \\ &+S_{zr,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} +S_{z\varphi,r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +S_{zz,r}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r} \\ &+r^{-1}S_{rr,\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +r^{-1}S_{rr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +r^{-1}S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{r\varphi,\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} +r^{-1}S_{r\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} -r^{-1}S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{rz,\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +r^{-1}S_{rz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{\varphi r,\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} -r^{-1}S_{\varphi r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +r^{-1}S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{\varphi\varphi,\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} -r^{-1}S_{\varphi\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} -r^{-1}S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{\varphi z,\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} -r^{-1}S_{\varphi z}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{zr,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +r^{-1}S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{z\varphi,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} -r^{-1}S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{zz,\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} \\ &+S_{rr,z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} +S_{r\varphi,z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} +S_{rz,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{\varphi r,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} +S_{\varphi\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} +S_{\varphi z,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{zr,z}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} +S_{z\varphi,z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} +S_{zz,z}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &=S_{rr,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{rr,\varphi}-S_{r\varphi}-S_{\varphi r})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +S_{rr,z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{r\varphi,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{rr}+S_{r\varphi,\varphi}-S_{\varphi\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{r\varphi,z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{rz,r}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r}+r^{-1}(S_{rz,\varphi} -S_{\varphi z})\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +S_{rz,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{\varphi r,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{rr}+S_{\varphi r,\varphi}-S_{\varphi\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +S_{\varphi r,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{r\varphi}+S_{\varphi r} +S_{\varphi\varphi,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} +S_{\varphi\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{\varphi z,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{rz}+S_{\varphi z,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +S_{\varphi z,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{zr,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+r^{-1}(S_{zr,\varphi} -S_{z\varphi})\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +S_{zr,z}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{z\varphi,r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{zr}+S_{z\varphi,\varphi})\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} +S_{z\varphi,z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{zz,r}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r} +r^{-1}S_{zz,\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +S_{zz,z}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{k}.\end{split}\]

Takže

(72)\[\begin{split}\boldsymbol{S}\otimes\nabla &=S_{rr,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{rr,\varphi}-S_{r\varphi}-S_{\varphi r})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +S_{rr,z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{r\varphi,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+r^{-1}(S_{rr}+S_{r\varphi,\varphi} -S_{\varphi\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} +S_{r\varphi,z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{rz,r}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r}+r^{-1}(S_{rz,\varphi} -S_{\varphi z})\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +S_{rz,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{\varphi r,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+r^{-1}(S_{rr}+S_{\varphi r,\varphi} -S_{\varphi\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +S_{\varphi r,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{r\varphi}+S_{\varphi r} +S_{\varphi\varphi,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} +S_{\varphi\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{\varphi z,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r} +r^{-1}(S_{rz}+S_{\varphi z,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +S_{\varphi z,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{zr,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+r^{-1}(S_{zr,\varphi} -S_{z\varphi})\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} +S_{zr,z}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{z\varphi,r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+r^{-1}(S_{zr} +S_{z\varphi,\varphi})\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} +S_{z\varphi,z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{zz,r}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r} +r^{-1}S_{zz,\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} +S_{zz,z}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{k}.\end{split}\]

Divergence vektorové funkce

Jestliže je \(\boldsymbol{v}(r,\varphi,z) \equiv v_{r}\boldsymbol{e}_{r}+v_{\varphi}\boldsymbol{e}_{\varphi}+v_{z}\boldsymbol{k}\) vektorová funkce, potom její divergence je

(73)\[\begin{split}\nabla\cdot\boldsymbol{v} &=(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi}+\boldsymbol{k}\partial_{z})\cdot(v_{r}\boldsymbol{e}_{r}+v_{\varphi}\boldsymbol{e}_{\varphi}+v_{z}\boldsymbol{k}) \\ &=(v_{r}\boldsymbol{e}_{r})_{r}\cdot\boldsymbol{e}_{r}+(v_{\varphi}\boldsymbol{e}_{\varphi})_{r}\cdot\boldsymbol{e}_{r}+(v_{z}\boldsymbol{k})_{r}\cdot\boldsymbol{e}_{r} \\ &+r^{-1}(v_{r}\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-1}(v_{\varphi}\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-1}(v_{z}\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(v_{r}\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(v_{\varphi}\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(v_{z}\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &=v_{r,r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}+v_{\varphi,r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}+v_{z,z}\boldsymbol{k}\cdot\boldsymbol{e}_{r} \\ &+r^{-1}v_{r,\varphi}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}+r^{-1}v_{r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-1}v_{\varphi,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}-r^{-1}v_{\varphi}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}+r^{-1}v_{z,\varphi}\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi} \\ &+v_{r,z}\boldsymbol{e}_{r}\cdot\boldsymbol{k}+v_{\varphi,z}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}+v_{z,z}\boldsymbol{k}\cdot\boldsymbol{k} \\ &=v_{r,r}+r^{-1}v_{r}+r^{-1}v_{\varphi,\varphi}+v_{z,z}.\end{split}\]

Takže

(74)\[\nabla\cdot\boldsymbol{v}=v_{r,r}+r^{-1}v_{r}+r^{-1}v_{\varphi,\varphi}+v_{z,z}.\]

Divergence tenzorové funkce

Jestliže

(75)\[\begin{split}\boldsymbol{S}(r,\varphi,z) &\equiv S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{rk}\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi k}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{kr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{k\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{kk}\boldsymbol{k}\otimes\boldsymbol{k},\end{split}\]

je tenzor, pak jeho divergence je

(76)\[\begin{split}\nabla\cdot\boldsymbol{S} &=(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi}+\boldsymbol{k}\partial_{z})\cdot(S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{rz}\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}+S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{zz}\boldsymbol{k}\otimes\boldsymbol{k}) \\ &=\boldsymbol{e}_{r}\cdot(S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,r}+\boldsymbol{e}_{r}\cdot(S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,r}+\boldsymbol{e}_{r}\cdot(S_{rz}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,r} \\ &+\boldsymbol{e}_{r}\cdot(S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,r}+\boldsymbol{e}_{r}\cdot(S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,r}+\boldsymbol{e}_{r}\cdot(S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,r} \\ &+\boldsymbol{e}_{r}\cdot(S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,r}+\boldsymbol{e}_{r}\cdot(S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,r}+\boldsymbol{e}_{r}\cdot(S_{zz}\boldsymbol{k}\otimes\boldsymbol{k})_{r} \\ &+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{rz}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,\varphi} \\ &+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,\varphi} \\ &+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot(S_{zz}\boldsymbol{k}\otimes\boldsymbol{k})_{,\varphi} \\ &+\boldsymbol{k}\cdot(S_{rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,z}+\boldsymbol{k}\cdot(S_{r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,z}+\boldsymbol{k}\cdot(S_{rz}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,z} \\ &+\boldsymbol{k}\cdot(S_{\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,z}+\boldsymbol{k}\cdot(S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,z}+\boldsymbol{k}\cdot(S_{\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,z} \\ &+\boldsymbol{k}\cdot(S_{zr}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,z}+\boldsymbol{k}\cdot(S_{z\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,z}+\boldsymbol{k}\cdot(S_{zz}\boldsymbol{k}\otimes\boldsymbol{k})_{,z} \\ &=S_{rr,r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{r\varphi,r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{rz,r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi r,r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{\varphi\varphi,r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi z,r}\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{zr,r}\boldsymbol{e}_{r}\cdot\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{z\varphi,r}\boldsymbol{e}_{r}\cdot\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{zz,r}\boldsymbol{e}_{r}\cdot\boldsymbol{k}\otimes\boldsymbol{k} \\ &+r^{-1}S_{rr,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+r^{-1}S_{rr}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+r^{-1}S_{rr}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{r\varphi,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+r^{-1}S_{r\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}-r^{-1}S_{r\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} \\ &+r^{-1}S_{rz,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{k}+r^{-1}S_{rz}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+r^{-1}S_{\varphi r,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}-r^{-1}S_{\varphi r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+r^{-1}S_{\varphi r}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{\varphi\varphi,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}-r^{-1}S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}-r^{-1}S_{\varphi\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} \\ &+r^{-1}S_{\varphi z,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}-r^{-1}S_{\varphi z}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+r^{-1}S_{zr,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}\otimes\boldsymbol{e}_{r}+r^{-1}S_{zr}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} \\ &+r^{-1}S_{z\varphi,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}-r^{-1}S_{z\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}\otimes\boldsymbol{e}_{r} \\ &+r^{-1}S_{zz,\varphi}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{rr,z}\boldsymbol{k}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{r\varphi,z}\boldsymbol{k}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{rz,z}\boldsymbol{k}\cdot\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi r,z}\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{\varphi\varphi,z}\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi z,z}\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{zr,z}\boldsymbol{k}\cdot\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{z\varphi,z}\boldsymbol{k}\cdot\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{zz,z}\boldsymbol{k}\cdot\boldsymbol{k}\otimes\boldsymbol{k} \\ &=(S_{rr,r}+r^{-1}S_{rr}+r^{-1}S_{\varphi r,\varphi}-r^{-1}S_{\varphi\varphi}+S_{zr,z})\boldsymbol{e}_{r} \\ &+(S_{r\varphi,r}+r^{-1}S_{r\varphi}+r^{-1}S_{\varphi r}+r^{-1}S_{\varphi\varphi,\varphi}+S_{z\varphi,z})\boldsymbol{e}_{\varphi} \\ &+(S_{rz,r}+r^{-1}S_{rz}+r^{-1}S_{\varphi z,\varphi}+S_{zz,z})\boldsymbol{k}\end{split}\]

nebo-li

(77)\[\begin{split}\nabla\cdot\boldsymbol{S}= \left[\begin{array}{c} S_{rr,r}+r^{-1}S_{rr}+r^{-1}S_{\varphi r,\varphi}-r^{-1}S_{\varphi\varphi}+S_{zr,z}\\ S_{r\varphi,r}+r^{-1}S_{r\varphi}+r^{-1}S_{\varphi r}+r^{-1}S_{\varphi\varphi,\varphi}+S_{z\varphi,z}\\ S_{rz,r}+r^{-1}S_{rz}+r^{-1}S_{\varphi z,\varphi}+S_{zz,z} \end{array}\right]\end{split}\]

Jestliže

(78)\[\begin{split}\boldsymbol{S}(r,\varphi,z) &\equiv S_{rrr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{rr\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{rrz}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{r\varphi r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{r\varphi\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{r\varphi z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{rzr}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{rz\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{rzz}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{\varphi rr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi rz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{\varphi zr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{\varphi z\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi zz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{zrr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{zr\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{zrz}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{z\varphi r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{z\varphi\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{z\varphi z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{zzr}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{zz\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{zzz}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{k}\end{split}\]

je tenzor, pak jeho divergence

(79)\[\begin{split}\boldsymbol{S}\cdot\nabla je\boldsymbol{S}\cdot\nabla &=(S_{rrr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{rr\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{rrz}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{r\varphi r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{r\varphi\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{r\varphi z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{rzr}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{rz\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{rzz}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{\varphi rr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi rz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{\varphi zr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{\varphi z\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{\varphi zz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{k} \\ &+S_{zrr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+S_{zr\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+S_{zrz}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+S_{z\varphi r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+S_{z\varphi\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+S_{z\varphi z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+S_{zzr}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r}+S_{zz\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+S_{zzz}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{k}) \\ &\cdot(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi}+\boldsymbol{k}\partial_{z}) \\ &=(S_{rrr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{rr\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{rrz}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(S_{r\varphi r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{r\varphi\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{r\varphi z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(S_{rzr}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{rz\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{rzz}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(S_{\varphi rr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{\varphi rz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(S_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{\varphi\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(S_{\varphi zr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{\varphi z\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{\varphi zz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(S_{zrr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{zr\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{zrz}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(S_{z\varphi r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{z\varphi\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{z\varphi z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(S_{zzr}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(S_{zz\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(S_{zzz}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+r^{-1}[(S_{rrr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{rr\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{rrz}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(S_{r\varphi r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{r\varphi\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{r\varphi z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(S_{rzr}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{rz\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{rzz}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(S_{\varphi rr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{\varphi rz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(S_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{\varphi\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(S_{\varphi zr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{\varphi z\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{\varphi zz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(S_{zrr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{zr\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{zrz}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(S_{z\varphi r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{z\varphi\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{z\varphi z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(S_{zzr}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{zz\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+(S_{zzz}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}] \\ &+(S_{rrr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{rr\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{rrz}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(S_{r\varphi r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{r\varphi\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{r\varphi z}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(S_{rzr}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{rz\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{rzz}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(S_{\varphi rr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{\varphi r\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{\varphi rz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(S_{\varphi\varphi r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{\varphi\varphi\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{\varphi\varphi z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(S_{\varphi zr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{\varphi z\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{\varphi zz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(S_{zrr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{zr\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{zrz}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(S_{z\varphi r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{z\varphi\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{z\varphi z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(S_{zzr}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(S_{zz\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(S_{zzz}\boldsymbol{k}\otimes\boldsymbol{k}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &=(S_{rrr,r}+r^{-1}S_{rrr}+r^{-1}S_{rr\varphi,\varphi}-r^{-1}S_{r\varphi\varphi}-r^{-1}S_{\varphi r\varphi}+S_{rrz,z})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} \\ &+(S_{r\varphi r,r}+r^{-1}S_{rr\varphi}+r^{-1}S_{r\varphi r}+r^{-1}S_{r\varphi\varphi,\varphi}-r^{-1}S_{\varphi\varphi\varphi}+S_{r\varphi z,z})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} \\ &+(S_{rzr,r}+r^{-1}S_{rzr}+r^{-1}S_{rz\varphi,\varphi}-r^{-1}S_{\varphi z\varphi}+S_{rzz,z})\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+(S_{\varphi rr,r}+r^{-1}S_{rr\varphi}+r^{-1}S_{\varphi rr}+r^{-1}S_{\varphi r\varphi,\varphi}-r^{-1}S_{\varphi\varphi\varphi}+S_{\varphi rz,z})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} \\ &+(S_{\varphi\varphi r,r}+r^{-1}S_{r\varphi\varphi}+r^{-1}S_{\varphi r\varphi}+r^{-1}S_{\varphi\varphi r}+r^{-1}S_{\varphi\varphi\varphi,\varphi}+S_{\varphi\varphi z,z})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+(S_{\varphi zr,r}+r^{-1}S_{rz\varphi}+r^{-1}S_{\varphi zr}+r^{-1}S_{\varphi z\varphi,\varphi}+S_{\varphi zz,z})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+(S_{zrr,r}+r^{-1}S_{zrr}+r^{-1}S_{zr\varphi,\varphi}-r^{-1}S_{z\varphi\varphi}+S_{zrz,z})\boldsymbol{k}\otimes\boldsymbol{e}_{r} \\ &+(S_{z\varphi r,r}+r^{-1}S_{zr\varphi}+r^{-1}S_{z\varphi r}+r^{-1}S_{z\varphi\varphi,\varphi}+S_{z\varphi z,z})\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} \\ &+(S_{zzr,r}+r^{-1}S_{zzr}+r^{-1}S_{zz\varphi,\varphi}+S_{zzz,z})\boldsymbol{k}\otimes\boldsymbol{k}\end{split}\]

Takže

(80)\[\begin{split}\boldsymbol{S}\cdot\nabla &=(S_{rrr,r}+r^{-1}S_{rrr}+r^{-1}S_{rr\varphi,\varphi}-r^{-1}S_{r\varphi\varphi}-r^{-1}S_{\varphi r\varphi}+S_{rrz,z})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r} \\ &+(S_{r\varphi r,r}+r^{-1}S_{rr\varphi}+r^{-1}S_{r\varphi r}+r^{-1}S_{r\varphi\varphi,\varphi}-r^{-1}S_{\varphi\varphi\varphi}+S_{r\varphi z,z})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi} \\ &+(S_{rzr,r}+r^{-1}S_{rzr}+r^{-1}S_{rz\varphi,\varphi}-r^{-1}S_{\varphi z\varphi}+S_{rzz,z})\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+(S_{\varphi rr,r}+r^{-1}S_{rr\varphi}+r^{-1}S_{\varphi rr}+r^{-1}S_{\varphi r\varphi,\varphi}-r^{-1}S_{\varphi\varphi\varphi}+S_{\varphi rz,z})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r} \\ &+(S_{\varphi\varphi r,r}+r^{-1}S_{r\varphi\varphi}+r^{-1}S_{\varphi r\varphi}+r^{-1}S_{\varphi\varphi r}+r^{-1}S_{\varphi\varphi\varphi,\varphi}+S_{\varphi\varphi z,z})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi} \\ &+(S_{\varphi zr,r}+r^{-1}S_{rz\varphi}+r^{-1}S_{\varphi zr}+r^{-1}S_{\varphi z\varphi,\varphi}+S_{\varphi zz,z})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+(S_{zrr,r}+r^{-1}S_{zrr}+r^{-1}S_{zr\varphi,\varphi}-r^{-1}S_{z\varphi\varphi}+S_{zrz,z})\boldsymbol{k}\otimes\boldsymbol{e}_{r} \\ &+(S_{z\varphi r,r}+r^{-1}S_{zr\varphi}+r^{-1}S_{z\varphi r}+r^{-1}S_{z\varphi\varphi,\varphi}+S_{z\varphi z,z})\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi} \\ &+(S_{zzr,r}+r^{-1}S_{zzr}+r^{-1}S_{zz\varphi,\varphi}+S_{zzz,z})\boldsymbol{k}\otimes\boldsymbol{k}\end{split}\]

Laplacián ze skalární funkce

Pro skalární funkci \(f(r,\varphi,z)\equiv f\) platí

(81)\[\begin{split}\Delta f\equiv\nabla^{2}f=\nabla\cdot(\nabla f) &=(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi}+\boldsymbol{k}\partial_{z})\cdot(\boldsymbol{e}_{r}f_{,r}+\boldsymbol{e}_{\varphi}r^{-1}f_{,\varphi}+\boldsymbol{k}f_{,z}) \\ &=\boldsymbol{e}_{r}\cdot(\boldsymbol{e}_{r}f_{,r})_{,r}+\boldsymbol{e}_{r}\cdot(\boldsymbol{e}_{\varphi}r^{-1}f_{,\varphi})_{,r}+\boldsymbol{e}_{r}\cdot(\boldsymbol{k}f_{,z})_{,r} \\ &+r^{-1}\boldsymbol{e}_{\varphi}\cdot(\boldsymbol{e}_{r}f_{,r})_{,\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot(\boldsymbol{e}_{\varphi}r^{-1}f_{,\varphi})_{,\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot(\boldsymbol{k}f_{,z})_{,\varphi} \\ &+\boldsymbol{k}\cdot(\boldsymbol{e}_{r}f_{,r})_{,z}+\boldsymbol{k}\cdot(\boldsymbol{e}_{\varphi}r^{-1}f_{,\varphi})_{,z}+\boldsymbol{k}\cdot(\boldsymbol{k}f_{,z})_{,z} \\ &=\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}f_{,rr}-\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}r^{-2}f_{,\varphi}+\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}r^{-1}f_{,\varphi r}+\boldsymbol{e}_{r}\cdot\boldsymbol{k}f_{,zr} \\ &+r^{-1}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}f_{,r}+r^{-1}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}f_{,r\varphi}-r^{-1}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}r^{-1}f_{,\varphi} \\ &+r^{-1}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}r^{-1}f_{,\varphi\varphi}+r^{-1}\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}f_{,z\varphi} \\ &+\boldsymbol{k}\cdot\boldsymbol{e}_{r}f_{,rz}+\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi}r^{-1}f_{,\varphi z}+\boldsymbol{k}\cdot\boldsymbol{k}f_{,zz} \\ &=f_{,rr}+r^{-1}f_{,r}+r^{-2}f_{,\varphi\varphi}+f_{,zz}.\end{split}\]

Tedy

(82)\[\Delta f=f_{,rr}+r^{-1}f_{,r}+r^{-2}f_{,\varphi\varphi}+f_{,zz}.\]

Laplacián z vektorové funkce

Pro vektorovou funkci \(\boldsymbol{v}(r,\varphi,z)\equiv v_{r}\boldsymbol{e}_{r}+v_{\varphi}\boldsymbol{e}_{\varphi}+v_{z}\boldsymbol{k}\) platí

(83)\[\begin{split}\Delta\boldsymbol{v}\equiv\nabla^{2}\boldsymbol{v} &=(\boldsymbol{v}\otimes\nabla)\cdot\nabla=[v_{r,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}+r^{-1}(v_{r,\varphi}-v_{\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}+v_{r,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k} \\ &+v_{\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}+r^{-1}(v_{r}+v_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}+v_{\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k} \\ &+v_{z,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r}+r^{-1}v_{z,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}+v_{z,z}\boldsymbol{k}\otimes\boldsymbol{k}]\cdot(\boldsymbol{e}_{r}\partial_{r}+\boldsymbol{e}_{\varphi}r^{-1}\partial_{\varphi}+\boldsymbol{k}\partial_{z}) \\ &=(v_{r,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+[r^{-1}(v_{r,\varphi}-v_{\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}]_{,r}\cdot\boldsymbol{e}_{r}+(v_{r,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(v_{\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+[r^{-1}(v_{r}+v_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}]_{,r}\cdot\boldsymbol{e}_{r}+(v_{\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+(v_{z,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,r}\cdot\boldsymbol{e}_{r}+(r^{-1}v_{z,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,r}\cdot\boldsymbol{e}_{r}+(v_{z,z}\boldsymbol{k}\otimes\boldsymbol{k})_{,r}\cdot\boldsymbol{e}_{r} \\ &+r^{-1}(v_{r,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-1}[r^{-1}(v_{r,\varphi}-v_{\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}]_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-1}(v_{r,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-1}(v_{\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-1}[r^{-1}(v_{r}+v_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}]_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-1}(v_{\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-1}(v_{z,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-1}(r^{-1}v_{z,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-1}(v_{z,z}\boldsymbol{k}\otimes\boldsymbol{k})_{,\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+(v_{r,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+[r^{-1}(v_{r,\varphi}-v_{\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}]_{,z}\cdot\boldsymbol{k}+(v_{r,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(v_{\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+[r^{-1}(v_{r}+v_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}]_{,z}\cdot\boldsymbol{k}+(v_{\varphi,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &+(v_{z,r}\boldsymbol{k}\otimes\boldsymbol{e}_{r})_{,z}\cdot\boldsymbol{k}+(r^{-1}v_{z,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi})_{,z}\cdot\boldsymbol{k}+(v_{z,z}\boldsymbol{k}\otimes\boldsymbol{k})_{,z}\cdot\boldsymbol{k} \\ &=v_{r,rr}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}-r^{-2}(v_{r,\varphi}-v_{\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r} \\ &+r^{-1}(v_{r,\varphi r}-v_{\varphi,r})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}+v_{r,zr}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\cdot\boldsymbol{e}_{r} \\ &+v_{\varphi,rr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}-r^{-2}(v_{r}+v_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r} \\ &+r^{-1}(v_{r,r}+v_{\varphi,\varphi r})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}+v_{\varphi,zr}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\cdot\boldsymbol{e}_{r} \\ &+v_{z,rr}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{r}-r^{-2}v_{z,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}+r^{-1}v_{z,\varphi r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{r}+v_{z,zr}\boldsymbol{k}\otimes\boldsymbol{k}\cdot\boldsymbol{e}_{r} \\ &+r^{-1}v_{r,r\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}+r^{-1}v_{r,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}+r^{-1}v_{r,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-2}(v_{r,\varphi\varphi}-v_{\varphi,\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}+r^{-2}(v_{r,\varphi}-v_{\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}-r^{-2}(v_{r,\varphi}-v_{\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-1}v_{r,z\varphi}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi}+r^{-1}v_{r,z}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-1}v_{\varphi,r\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}-r^{-1}v_{\varphi,r}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}+r^{-1}v_{\varphi,r}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-2}(v_{r,\varphi}+v_{\varphi,\varphi\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}-r^{-2}(v_{r}+v_{\varphi,\varphi})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}-r^{-2}(v_{r}+v_{\varphi,\varphi})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-1}v_{\varphi,z\varphi}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi}-r^{-1}v_{\varphi,z}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-1}v_{z,r\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}+r^{-1}v_{z,r}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi} \\ &+r^{-2}v_{z,\varphi\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{e}_{\varphi}-r^{-2}v_{z,\varphi}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{e}_{\varphi}+r^{-1}v_{z,z\varphi}\boldsymbol{k}\otimes\boldsymbol{k}\cdot\boldsymbol{e}_{\varphi} \\ &+v_{r,rz}\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{k}+r^{-1}(v_{r,\varphi z}-v_{\varphi z})\boldsymbol{e}_{r}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}+v_{r,zz}\boldsymbol{e}_{r}\otimes\boldsymbol{k}\cdot\boldsymbol{k} \\ &+v_{\varphi,rz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{k}+r^{-1}(v_{r,z}+v_{\varphi,\varphi z})\boldsymbol{e}_{\varphi}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}+v_{\varphi,zz}\boldsymbol{e}_{\varphi}\otimes\boldsymbol{k}\cdot\boldsymbol{k} \\ &+v_{z,rz}\boldsymbol{k}\otimes\boldsymbol{e}_{r}\cdot\boldsymbol{k}+r^{-1}v_{z,\varphi z}\boldsymbol{k}\otimes\boldsymbol{e}_{\varphi}\cdot\boldsymbol{k}+v_{z,zz}\boldsymbol{k}\otimes\boldsymbol{k}\cdot\boldsymbol{k} \\ &=[v_{r,rr}+r^{-1}v_{r,r}+r^{-2}(v_{r,\varphi\varphi}-v_{\varphi,\varphi})-r^{-2}(v_{r}+v_{\varphi,\varphi})+v_{r,zz}]\boldsymbol{e}_{r} \\ &+[v_{\varphi,rr}+r^{-2}(v_{r,\varphi}-v_{\varphi})+r^{-1}v_{\varphi,r}+r^{-2}(v_{r,\varphi}+v_{\varphi,\varphi\varphi})+v_{\varphi,zz}]\boldsymbol{e}_{\varphi} \\ &+[v_{z,rr}+r^{-1}v_{z,r}+r^{-2}v_{z,\varphi\varphi}+v_{z,zz}]\boldsymbol{k}.\end{split}\]

Tedy

(84)\[\begin{split}\Delta\boldsymbol{v}= \left[\begin{array}{c} v_{r,rr}+r^{-1}v_{r,r}-2r^{-2}v_{\varphi,\varphi}-r^{-2}v_{r}+r^{-2}v_{r,\varphi\varphi}+v_{r,zz}\\ v_{\varphi,rr}+r^{-1}v_{\varphi,r}+r^{-2}v_{\varphi,\varphi\varphi}+2r^{-2}v_{r,\varphi}-r^{-2}v_{\varphi}+v_{\varphi,zz}\\ v_{z,rr}+r^{-1}v_{z,r}+r^{-2}v_{z,\varphi\varphi}+v_{z,zz} \end{array}\right].\end{split}\]

Literatura

[1]	https://en.wikipedia.org/wiki/Dyadics

[2]	Brand, L. Vector and tensor analysis. John Wiley & sons, 1947.

[3]	https://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Polar_Coords/Polar_Coords.htm