Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

1.17.4: The Del-operator

Last updated

Nov 20, 2023
Save as PDF
- 1.17.3: SI Units
- Index

Johan Wevers

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

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

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

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

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

$\newcommand{\id}{\mathrm{id}}$

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

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

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

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

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

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

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

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

The $\nabla$ -operator

In cartesian coordinates $(x,y,z)$ :

$\vec{\nabla}=\frac{\partial }{\partial x}\vec{e}_{x}+\frac{\partial }{\partial y}\vec{e}_{y}+\frac{\partial }{\partial z}\vec{e}_{z}~~,~~ {\rm grad}f=\vec{\nabla}f=\frac{\partial f}{\partial x}\vec{e}_{x}+\frac{\partial f}{\partial y}\vec{e}_{y}+\frac{\partial f}{\partial z}\vec{e}_{z}$

${\rm div}~\vec{a}=\vec{\nabla}\cdot\vec{a}=\frac{\partial a_x}{\partial x}+\frac{\partial a_y}{\partial y}+\frac{\partial a_z}{\partial z}~~,~~ \nabla^2 f=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2}$

${\rm rot}~\vec{a}=\vec{\nabla}\times\vec{a}= \left(\frac{\partial a_z}{\partial y}-\frac{\partial a_y}{\partial z}\right)\vec{e}_{x}+ \left(\frac{\partial a_x}{\partial z}-\frac{\partial a_z}{\partial x}\right)\vec{e}_{y}+ \left(\frac{\partial a_y}{\partial x}-\frac{\partial a_x}{\partial y}\right)\vec{e}_{z}$ In cylinder coordinates

$(r,\varphi,z)$ holds:

$\vec{\nabla}=\frac{\partial }{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial }{\partial \varphi}\vec{e}_{\varphi}+\frac{\partial }{\partial z}\vec{e}_{z}~~,~~ {\rm grad}f=\frac{\partial f}{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial f}{\partial \varphi}\vec{e}_{\varphi}+\frac{\partial f}{\partial z}\vec{e}_{z}$

${\rm div}~\vec{a}=\frac{\partial a_r}{\partial r}+\frac{a_r}{r}+\frac{1}{r}\frac{\partial a_\varphi}{\partial \varphi}+\frac{\partial a_z}{\partial z}~~,~~ \nabla^2 f=\frac{\partial^2 f}{\partial r^2}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{\partial^2 f}{\partial \varphi^2}+\frac{\partial^2 f}{\partial z^2}$

${\rm rot}~\vec{a}=\left(\frac{1}{r}\frac{\partial a_z}{\partial \varphi}-\frac{\partial a_\varphi}{\partial z}\right)\vec{e}_{r}+ \left(\frac{\partial a_r}{\partial z}-\frac{\partial a_z}{\partial r}\right)\vec{e}_{\varphi}+ \left(\frac{\partial a_\varphi}{\partial r}+\frac{a_\varphi}{r}-\frac{1}{r}\frac{\partial a_r}{\partial \varphi}\right)\vec{e}_{z}\\$

In spherical coordinates $(r,\theta,\varphi)$ :

$\begin{aligned} \vec{\nabla} &=&\frac{\partial }{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial }{\partial \theta}\vec{e}_{\theta}+\frac{1}{r\sin\theta}\frac{\partial }{\partial \varphi}\vec{e}_{\varphi}\\ {\rm grad}f &=&\frac{\partial f}{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial f}{\partial \theta}\vec{e}_{\theta}+\frac{1}{r\sin\theta}\frac{\partial f}{\partial \varphi}\vec{e}_{\varphi}\\ {\rm div}~\vec{a}&=&\frac{\partial a_r}{\partial r}+\frac{2a_r}{r}+\frac{1}{r}\frac{\partial a_\theta}{\partial \theta}+\frac{a_\theta}{r\tan\theta}+\frac{1}{r\sin\theta}\frac{\partial a_\varphi}{\partial \varphi}\\ {\rm rot}~\vec{a}&=&\left(\frac{1}{r}\frac{\partial a_\varphi}{\partial \theta}+\frac{a_\theta}{r\tan\theta}-\frac{1}{r\sin\theta}\frac{\partial a_\theta}{\partial \varphi}\right)\vec{e}_{r}+ \left(\frac{1}{r\sin\theta}\frac{\partial a_r}{\partial \varphi}-\frac{\partial a_\varphi}{\partial r}-\frac{a_\varphi}{r}\right)\vec{e}_{\theta}+\\ &&\left(\frac{\partial a_\theta}{\partial r}+\frac{a_\theta}{r}-\frac{1}{r}\frac{\partial a_r}{\partial \theta}\right)\vec{e}_{\varphi}\\ \nabla^2 f &=&\frac{\partial^2 f}{\partial r^2}+\frac{2}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}+\frac{1}{r^2\tan\theta}\frac{\partial f}{\partial \theta}+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \varphi^2}\end{aligned}$

General orthonormal curvelinear coordinates $(u,v,w)$ can be obtained from cartesian coordinates by the transformation $\vec{x}=\vec{x}(u,v,w)$ . The unit vectors are then given by:

$\vec{e}_{u}=\frac{1}{h_1}\frac{\partial \vec{x}}{\partial u}~,~~\vec{e}_{v}=\frac{1}{h_2}\frac{\partial \vec{x}}{\partial v}~,~~ \vec{e}_{w}=\frac{1}{h_3}\frac{\partial \vec{x}}{\partial w}$ where the factors

$h_i$ set the norm to 1. Then holds:

$\begin{aligned} {\rm grad}f &=&\frac{1}{h_1}\frac{\partial f}{\partial u}\vec{e}_{u}+\frac{1}{h_2}\frac{\partial f}{\partial v}\vec{e}_{v}+\frac{1}{h_3}\frac{\partial f}{\partial w}\vec{e}_{w}\\ {\rm div}~\vec{a}&=&\frac{1}{h_1h_2h_3}\left(\frac{\partial }{\partial u}(h_2h_3a_u)+\frac{\partial }{\partial v}(h_3h_1a_v)+\frac{\partial }{\partial w}(h_1h_2a_w)\right)\\ {\rm rot}~\vec{a}&=&\frac{1}{h_2h_3}\left(\frac{\partial (h_3a_w)}{\partial v}-\frac{\partial (h_2a_v)}{\partial w}\right)\vec{e}_{u}+ \frac{1}{h_3h_1}\left(\frac{\partial (h_1a_u)}{\partial w}-\frac{\partial (h_3a_w)}{\partial u}\right)\vec{e}_{v}+\\ &&\frac{1}{h_1h_2}\left(\frac{\partial (h_2a_v)}{\partial u}-\frac{\partial (h_1a_u)}{\partial v}\right)\vec{e}_{w}\\ \nabla^2 f &=&\frac{1}{h_1h_2h_3}\left[\frac{\partial }{\partial u}\left(\frac{h_2h_3}{h_1}\frac{\partial f}{\partial u}\right)+ \frac{\partial }{\partial v}\left(\frac{h_3h_1}{h_2}\frac{\partial f}{\partial v}\right)+ \frac{\partial }{\partial w}\left(\frac{h_1h_2}{h_3}\frac{\partial f}{\partial w}\right)\right]\end{aligned}$

The ∇\nabla-operator

Support Center

How can we help?

The $\nabla$ -operator