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Coordinate Systems and Formulae

Cartesian Coordinates

\begin{displaymath}
f = f(x,y,z), \vec{F} = (F_1, F_2, F_3)\end{displaymath}

\scalebox {0.5}{\includegraphics{cart.ps}}

\begin{displaymath}
\nabla \cdot \vec{F} = \nabla \cdot (F_1(x,y,z)\hat{i} + F_2(x,y,z)\hat{j}
+ F_3(x,y,z)\hat{k}) = (F_1)_x + (F_2)_y + (F_3)_z\end{displaymath}

\begin{displaymath}
(\nabla f)_x = f_x, \qquad (\nabla f)_y = f_y, \qquad (\nabla f)_z = f_z\end{displaymath}

\begin{displaymath}
(\nabla \times \vec{F})_x = \frac{\partial F_3}{\partial y} ...
 ...rac{\partial F_2}{\partial x} -
\frac{\partial F_1}{\partial y}\end{displaymath}

\begin{displaymath}
\nabla^2 f = f_{xx} + f_{yy} + f_{zz}\end{displaymath}

Cylindrical Coordinates

\begin{displaymath}
f = f(r,\theta,z), \vec{F} = (F_r, F_\theta, F_z)\end{displaymath}

\scalebox {0.5}{\includegraphics{cylind.ps}}

\begin{displaymath}
\nabla \cdot \vec{F} = \frac{1}{r}\frac{\partial}{\partial r...
 ...al F_\theta}{\partial \theta} + \frac{\partial
F_z}{\partial z}\end{displaymath}

\begin{displaymath}
(\nabla f)_r = \frac{\partial f}{\partial r}, \qquad (\nabla...
 ...al \theta}, \qquad (\nabla f)_z =
\frac{\partial f}{\partial z}\end{displaymath}

\begin{displaymath}
(\nabla \times \vec{F})_r = \frac{1}{r}\frac{\partial F_z}{\...
 ...ac{\partial F_r}{\partial z} -
\frac{\partial F_z}{\partial r},\end{displaymath}

\begin{displaymath}
(\nabla \times \vec{F})_z = \frac{1}{r}\frac{\partial}{\part...
 ...x}(rF_\theta) -
\frac{1}{r}\frac{\partial F_r}{\partial \theta}\end{displaymath}

\begin{displaymath}
\nabla^2 f = \frac{1}{r}\frac{\partial}{\partial r}\left( r ...
 ...ial^2 f}{\partial \theta^2}
+ \frac{\partial^2 f}{\partial z^2}\end{displaymath}

Spherical Coordinates

\begin{displaymath}
f = f(\rho, \phi, \theta), \vec{F} = (F_\rho, F_\phi, F_\theta)\end{displaymath}

\scalebox {0.5}{\includegraphics{spher.ps}}

\begin{displaymath}
\nabla \cdot \vec{F} = \frac{1}{\rho^2} \frac{\partial}{\par...
 ...ac{1}{\rho \sin \phi} \frac{\partial
F_\theta}{\partial \theta}\end{displaymath}

\begin{displaymath}
(\nabla f)_\rho = \frac{\partial f}{\partial \rho}, \qquad
(...
 ...a = \frac{1}{\rho \sin \phi} \frac{\partial f}{\partial
\theta}\end{displaymath}

\begin{displaymath}
(\nabla \times \vec{F})_\rho = \frac{1}{\rho \sin \phi} \lef...
 ...hi F_\theta) - \frac{\partial
F_\phi}{\partial \theta} \right],\end{displaymath}

\begin{displaymath}
(\nabla \times \vec{F})_\phi = \frac{1}{\rho \sin \phi} \fra...
 ... \frac{1}{\rho} \frac{\partial}{\partial \rho}
(\rho F_\theta),\end{displaymath}

\begin{displaymath}
(\nabla \times \vec{F})_\theta = \frac{1}{\rho} \left[
\frac...
 ...o} (\rho F_\phi) - \frac{\partial
F_\rho}{\partial \phi}\right]\end{displaymath}

\begin{displaymath}
\nabla^2 f = \frac{1}{\rho^2}\frac{\partial}{\partial \rho}\...
 ...ac{1}{\rho^2 \sin^2\phi} \frac{\partial^2 f}{\partial
\theta^2}\end{displaymath}

Formulae Involving The Del Operator

If $\vec{A}$ and $\vec{B}$ are smooth vector fields and $\phi$ and $\psi$are smooth scalar functions of position (x,y,z) then

1.
$\nabla(\phi + \psi) = \nabla \phi + \nabla \psi$
2.
$\nabla \cdot (\vec{A} + \vec{B}) = \nabla \cdot \vec{A} + \nabla
\cdot \vec{B}$
3.
$\nabla \times (\vec{A} + \vec{B}) = \nabla \times \vec{A} + \nabla
\times \vec{B}$
4.
$\nabla \cdot (\phi \vec{A}) = (\nabla \phi)\cdot \vec{A} +
\phi(\nabla \cdot \vec{A})$
5.
$\nabla \times (\phi\vec{A}) = (\nabla \phi)\times \vec{A} +
\phi(\nabla \times \vec{A})$
6.
$\nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times
\vec{A}) - \vec{A} \cdot (\nabla \times \vec{B})$
7.
$\nabla \times (\vec{A} \times \vec{B}) = (\vec{B} \cdot
\nabla)\vec{A} - \vec{B...
 ... \cdot \vec{A}) - (\vec{A} \cdot
\nabla)\vec{B} + \vec{A}(\nabla \cdot \vec{B})$
8.
$\nabla (\vec{A} \cdot \vec{B}) = (\vec{B} \cdot \nabla)\vec{A} +
(\vec{A} \cdot...
 ...\vec{B} \times (\nabla \times \vec{A}) +
\vec{A} \times (\nabla \times \vec{B})$
9.
$\nabla \cdot \nabla \phi = \phi_{xx} + \phi_{yy} + \phi_{zz}$, in Cartesian coordinates
10.
$\nabla \times (\nabla \phi) = 0$
11.
$\nabla \cdot (\nabla \times \vec{A}) = 0$
12.
$\nabla \times (\nabla \times \vec{A}) = \nabla (\nabla \cdot
\vec{A}) - \nabla^2 \vec{A}$


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Copyright © 1998 by Kris H. Green
The Vector Calculus Website at
http://www.math.arizona.edu/~vector