10.5: Mathematical Formulas - Vector Operators
- Page ID
- 24320
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This section contains a summary of vector operators expressed in each of the three major coordinate systems:
- Cartesian (\(x\),\(y\),\(z\))
- cylindrical (\(\rho\),\(\phi\),\(z\))
- spherical (\(r\),\(\theta\),\(\phi\))
Associated basis vectors are identified using a caret (\(\hat{~}\)) over the symbol. The vector operand \({\bf A}\) is expressed in terms of components in the basis directions as follows:
- Cartesian: \({\bf A} = \hat{\bf x}A_x + \hat{\bf y}A_y + \hat{\bf z}A_z\)
- cylindrical: \({\bf A} = \hat{\bf \rho}A_{\rho} + \hat{\bf \phi}A_{\phi} + \hat{\bf z}A_z\)
- spherical: \({\bf A} = \hat{\bf r}A_r + \hat{\bf \theta}A_{\theta} + \hat{\bf \phi}A_{\phi}\)
Gradient
Gradient in Cartesian coordinates:
\[\nabla f = \hat { \mathbf { x } } \frac { \partial f } { \partial x } + \hat { \mathbf { y } } \frac { \partial f } { \partial y } + \hat { \mathbf { z } } \frac { \partial f } { \partial z } \nonumber \]
Gradient in cylindrical coordinates:
\[\nabla f = \hat { \rho } \frac { \partial f } { \partial \rho } + \hat { \phi } \frac { 1 } { \rho } \frac { \partial f } { \partial \phi } + \hat { \mathbf { z } } \frac { \partial f } { \partial z } \nonumber \]
Gradient in spherical coordinates:
\[\nabla f = \hat { \mathbf { r } } \frac { \partial f } { \partial r } + \hat { \theta } \frac { 1 } { r } \frac { \partial f } { \partial \theta } + \hat { \phi } \frac { 1 } { r \sin \theta } \frac { \partial f } { \partial \phi } \nonumber \]
Divergence
Divergence in Cartesian coordinates:
\[\nabla \cdot \mathbf { A } = \frac { \partial A _ { x } } { \partial x } + \frac { \partial A _ { y } } { \partial y } + \frac { \partial A _ { z } } { \partial z } \nonumber \]
Divergence in cylindrical coordinates:
\[\begin{aligned} \nabla \cdot {\bf A} &= \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho A_{\rho}\right) +\frac{1}{\rho}\frac{\partial A_{\phi}}{\partial \phi} + \frac{\partial A_z}{\partial z} & \end{aligned} \nonumber \] Divergence in spherical coordinates: \[\begin{aligned} \nabla \cdot {\bf A} &= ~~\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2 A_r\right) & \nonumber \\ &~~ +\frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left(A_{\theta}\sin\theta\right)& \nonumber \\ &~~ +\frac{1}{r\sin\theta}\frac{\partial A_{\phi}}{\partial \phi} & \end{aligned} \nonumber \]
Curl
Curl in Cartesian coordinates:
\[\begin{aligned} \nabla \times \mathbf { A } = & \hat { \mathbf { x } } \left( \frac { \partial A _ { z } } { \partial y } - \frac { \partial A _ { y } } { \partial z } \right) \\ & + \hat { \mathbf { y } } \left( \frac { \partial A _ { x } } { \partial z } - \frac { \partial A _ { z } } { \partial x } \right) \\ & + \hat { \mathbf { z } } \left( \frac { \partial A _ { y } } { \partial x } - \frac { \partial A _ { x } } { \partial y } \right) \end{aligned} \nonumber \]
Curl in cylindrical coordinates:
\[\begin{aligned} \nabla \times \mathbf { A } = & \hat { \rho } \left( \frac { 1 } { \rho } \frac { \partial A _ { z } } { \partial \phi } - \frac { \partial A _ { \phi } } { \partial z } \right) \\ & + \hat { \phi } \left( \frac { \partial A _ { \rho } } { \partial z } - \frac { \partial A _ { z } } { \partial \rho } \right) \\ & + \hat { \mathbf { z } } \frac { 1 } { \rho } \left[ \frac { \partial } { \partial \rho } \left( \rho A _ { \phi } \right) - \frac { \partial A _ { \rho } } { \partial \phi } \right] \end{aligned} \nonumber \]
Curl in spherical coordinates:
\[\begin{aligned} \nabla \times \mathbf { A } & = \hat { \mathbf { r } } \frac { 1 } { r \sin \theta } \left[ \frac { \partial } { \partial \theta } \left( A _ { \phi } \sin \theta \right) - \frac { \partial A _ { \theta } } { \partial \phi } \right] \\ & + \hat { \theta } \frac { 1 } { r } \left[ \frac { 1 } { \sin \theta } \frac { \partial A _ { r } } { \partial \phi } - \frac { \partial } { \partial r } \left( r A _ { \phi } \right) \right] \\ & + \hat { \phi } \frac { 1 } { r } \left[ \frac { \partial } { \partial r } \left( r A _ { \theta } \right) - \frac { \partial A _ { r } } { \partial \theta } \right] \end{aligned} \nonumber \]
Laplacian
Laplacian in Cartesian coordinates:
\[\nabla ^ { 2 } f = \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } + \frac { \partial ^ { 2 } f } { \partial z ^ { 2 } } \nonumber \]
Laplacian in cylindrical coordinates:
\[\nabla ^ { 2 } f = \frac { 1 } { \rho } \frac { \partial } { \partial \rho } \left( \rho \frac { \partial f } { \partial \rho } \right) + \frac { 1 } { \rho ^ { 2 } } \frac { \partial ^ { 2 } f } { \partial \phi ^ { 2 } } + \frac { \partial ^ { 2 } f } { \partial z ^ { 2 } } \nonumber \]
Laplacian in spherical coordinates:
\[\begin{aligned} \nabla ^ { 2 } f = & \frac { 1 } { r ^ { 2 } } \frac { \partial } { \partial r } \left( r ^ { 2 } \frac { \partial f } { \partial r } \right) \\ & + \frac { 1 } { r ^ { 2 } \sin \theta } \frac { \partial } { \partial \theta } \left( \frac { \partial f } { \partial \theta } \sin \theta \right) \\ & + \frac { 1 } { r ^ { 2 } \sin ^ { 2 } \theta } \frac { \partial ^ { 2 } f } { \partial \phi ^ { 2 } } \end{aligned} \nonumber \]