# 10.5: Mathematical Formulas - Vector Operators

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \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: $${\bf A} = \hat{\bf r}A_r + \hat{\bf \theta}A_{\theta} + \hat{\bf \phi}A_{\phi}$$

$\nabla f = \hat { \mathbf { x } } \frac { \partial f } { \partial x } + \hat { \mathbf { y } } \frac { \partial f } { \partial y } + \hat { \mathbf { z } } \frac { \partial f } { \partial z } \nonumber$

$\nabla f = \hat { \rho } \frac { \partial f } { \partial \rho } + \hat { \phi } \frac { 1 } { \rho } \frac { \partial f } { \partial \phi } + \hat { \mathbf { z } } \frac { \partial f } { \partial z } \nonumber$

$\nabla f = \hat { \mathbf { r } } \frac { \partial f } { \partial r } + \hat { \theta } \frac { 1 } { r } \frac { \partial f } { \partial \theta } + \hat { \phi } \frac { 1 } { r \sin \theta } \frac { \partial f } { \partial \phi } \nonumber$

## 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

This page titled 10.5: Mathematical Formulas - Vector Operators is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Steven W. Ellingson (Virginia Tech Libraries' Open Education Initiative) .