Curl

If F(x, y, z) = $\left(\vphantom{ p(x,y,z),q(x,y,z),r(x,y,z)}\right.$p(x, y, z), q(x, y, z), r(x, y, z)$\left.\vphantom{ p(x,y,z),q(x,y,z),r(x,y,z)}\right)$ is a vector field, then the vector

∇×F = $\displaystyle \left(\vphantom{
\frac{\partial r}{\partial y}-\frac{\partial q...
...rtial x},\frac{\partial q}{\partial x}-%
\frac{\partial p}{\partial y}}\right.$$\displaystyle {\frac{{\partial r}}{{\partial y}}}$ - $\displaystyle {\frac{{\partial q}}{{\partial z}}}$,$\displaystyle {\frac{{\partial p}}{{\partial z}}}$ - $\displaystyle {\frac{{\partial r}}{{\partial x}}}$,$\displaystyle {\frac{{\partial q}}{{\partial x}}}$ - $\displaystyle {\frac{{\partial p}}{{\partial y}}}$$\displaystyle \left.\vphantom{
\frac{\partial r}{\partial y}-\frac{\partial q...
...rtial x},\frac{\partial q}{\partial x}-%
\frac{\partial p}{\partial y}}\right)$

is called thecurl of F. The default is that the field variables are x, y, and z, in that order. If you wish to label the field variables differently, reset the default with Set Basis Variables on the Vector Calculus submenu. The vector field F in the following example is defined as in the previous section.

$\blacktriangleright$ Evaluate

∇×(yz, 2xz, xy) = $\left(\vphantom{ -x,0,z}\right.$ - x, 0, z$\left.\vphantom{ -x,0,z}\right)$ 6pt

∇×F = $\left[\vphantom{ -x,0,z}\right.$ - x, 0, z$\left.\vphantom{ -x,0,z}\right]$ 6pt

∇×$\left(\vphantom{
\begin{array}{ccc}
yz & 2xz & xy
\end{array}
}\right.$$\begin{array}{ccc}
yz & 2xz & xy
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
yz & 2xz & xy
\end{array}
}\right)$ = $\left(\vphantom{
\begin{array}{ccc}
-x & 0 & z
\end{array}
}\right.$$\begin{array}{ccc}
-x & 0 & z
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
-x & 0 & z
\end{array}
}\right)$ 6pt

∇×$\left(\vphantom{
\begin{array}{c}
x^{2} \\
xy \\
2xz
\end{array}
}\right.$$\begin{array}{c}
x^{2} \\
xy \\
2xz
\end{array}$$\left.\vphantom{
\begin{array}{c}
x^{2} \\
xy \\
2xz
\end{array}
}\right)$ = $\left(\vphantom{
\begin{array}{c}
0 \\
-2z \\
y
\end{array}
}\right.$$\begin{array}{c}
0 \\
-2z \\
y
\end{array}$$\left.\vphantom{
\begin{array}{c}
0 \\
-2z \\
y
\end{array}
}\right)$ 6pt

∇×$\left[\vphantom{
\begin{array}{c}
ax^{2} \\
bxy \\
2cxz
\end{array}
}\right.$$\begin{array}{c}
ax^{2} \\
bxy \\
2cxz
\end{array}$$\left.\vphantom{
\begin{array}{c}
ax^{2} \\
bxy \\
2cxz
\end{array}
}\right]$ = $\left[\vphantom{
\begin{array}{c}
0 \\
-2cz \\
by
\end{array}
}\right.$$\begin{array}{c}
0 \\
-2cz \\
by
\end{array}$$\left.\vphantom{
\begin{array}{c}
0 \\
-2cz \\
by
\end{array}
}\right]$