20.2: I.2 Spherical coordinates
- Page ID
- 18119
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let \(\psi\) be a scalar and \(\vec{u}\) be a vector expressed as a linear combination of the cylindrical basis vectors: \(\vec{u} = u_r\hat{e}^{(r)} +u_\theta \hat{e}^{(\theta)} +u_\phi \hat{e}^{(\phi)}\).
Gradient of a scalar
\[\vec{\nabla} \psi=\hat{e}^{(r)} \frac{\partial \psi}{\partial r}+\hat{e}^{(\theta)} \frac{1}{r} \frac{\partial \psi}{\partial \theta}+\hat{e}^{(\phi)} \frac{1}{r \sin \theta} \frac{\partial \psi}{\partial \phi}\label{eqn:1} \]
Divergence of a vector
\[\vec{\nabla} \cdot \vec{u}=\frac{1}{r^{2}} \frac{\partial\left(r^{2} u_{r}\right)}{\partial r}+\frac{1}{r \sin \theta} \frac{\partial\left(u_{\theta} \sin \theta\right)}{\partial \theta}+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial u_{\phi}}{\partial \phi}\label{eqn:2} \]
Curl of a vector
\[\vec{\nabla} \times \vec{u}=\hat{e}^{(r)} \frac{1}{r \sin \theta}\left[\frac{\partial\left(u_{\phi} \sin \theta\right)}{\partial \theta}-\frac{\partial u_{\theta}}{\partial \phi}\right]+\hat{e}^{(\theta)} \frac{1}{r}\left[\frac{1}{\sin \theta} \frac{\partial u_{r}}{\partial \phi}-\frac{\partial\left(r u_{\phi}\right)}{\partial r}\right]+\hat{e}^{(\phi)} \frac{1}{r}\left[\frac{1}{r} \frac{\partial\left(r u_{\theta}\right)}{\partial r}-\frac{\partial u_{r}}{\partial \theta}\right]\label{eqn:3} \]
Laplacian of a scalar
\[\nabla^{2} \psi=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \psi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} \psi}{\partial \phi^{2}}\label{eqn:4} \]
Laplacian of a vector
\[\begin{equation}\begin{aligned}
\nabla^{2} \vec{u} &=\hat{e}^{(r)}\left[\nabla^{2} u_{r}-\frac{2 u_{r}}{r^{2}}-\frac{2}{r^{2} \sin \theta} \frac{\partial\left(u_{\theta} \sin \theta\right)}{\partial \theta}-\frac{2}{r^{2} \sin \theta} \frac{\partial u_{\phi}}{\partial \phi}\right] \\
&+\hat{e}^{(\theta)}\left[\nabla^{2} u_{\theta}+\frac{2}{r^{2}} \frac{\partial u_{r}}{\partial \theta}-\frac{u_{\theta}}{r^{2} \sin ^{2} \theta}-\frac{2}{r^{2}} \frac{\cos \theta}{\sin ^{2} \theta} \frac{\partial u_{\phi}}{\partial \phi}\right] \\
&+\hat{e}^{(\phi)}\left[\nabla^{2} u_{\phi}+\frac{2}{r^{2} \sin ^{2} \theta} \frac{\partial u_{r}}{\partial \phi}+\frac{2}{r^{2}} \frac{\cos \theta}{\sin ^{2} \theta} \frac{\partial u_{\theta}}{\partial \phi}-\frac{u_{\theta}}{r^{2} \sin ^{2} \theta}\right]
\end{aligned}\label{eqn:5}\end{equation} \]
Material derivative
\[\frac{D}{D t}=\frac{\partial}{\partial t}+u_{r} \frac{\partial}{\partial r}+\frac{u_{\theta}}{r} \frac{\partial}{\partial \theta}+\frac{u_{\phi}}{r \sin \theta} \frac{\partial}{\partial \phi}\label{eqn:6} \]
Centrifugal acceleration
\[\vec{a}_{C}=\hat{e}^{(r)} \frac{u_{\theta}^{2}+u_{\phi}^{2}}{r}+\hat{e}^{(\theta)}\left(-\frac{u_{r} u_{\theta}}{r}+\frac{u_{\phi}^{2}}{r \tan \theta}\right)+\hat{e}^{(\phi)}\left(-\frac{u_{r} u_{\phi}}{r}-\frac{u_{\theta} u_{\phi}}{r \tan \theta}\right). \nonumber \]