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