Divergence

A vector field is a vector-valued function. 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 scalar

∇⋅F = $\displaystyle {\frac{{\partial p}}{{\partial x}}}$$\displaystyle \left(\vphantom{ a,b,c}\right.$a, b, c$\displaystyle \left.\vphantom{ a,b,c}\right)$ + $\displaystyle {\frac{{\partial q}}{{\partial
y}}}$$\displaystyle \left(\vphantom{ a,b,c}\right.$a, b, c$\displaystyle \left.\vphantom{ a,b,c}\right)$ + $\displaystyle {\frac{{\partial r}}{{\partial z}}}$$\displaystyle \left(\vphantom{ a,b,c}\right.$a, b, c$\displaystyle \left.\vphantom{ a,b,c}\right)$

is the divergence of F at the point $\left(\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right)$. The dot product notation is used because the symbol ∇ can be thought of as the vector operator ∇ = $\left(\vphantom{
\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{%
\partial z}}\right.$${\frac{{\partial }}{{\partial x}}}$,${\frac{{\partial }}{{\partial y}}}$,${\frac{{\partial }}{{%
\partial z}}}$$\left.\vphantom{
\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{%
\partial z}}\right)$. 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.

For the following example, use Define + New Definition to define the following vector fields.

F = [yz, 2xz, xy]                 G = (xz, 2yz, z2)          
H =  $\displaystyle \left[\vphantom{
\begin{array}{ccc}
yz & 2xz & xy
\end{array}}\right.$$\displaystyle \begin{array}{ccc}
yz & 2xz & xy
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{ccc}
yz & 2xz & xy
\end{array}}\right]$        J = $\displaystyle \left(\vphantom{
\begin{array}{c}
x^{2} \\
xy \\
2xz
\end{array}}\right.$$\displaystyle \begin{array}{c}
x^{2} \\
xy \\
2xz
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{c}
x^{2} \\
xy \\
2xz
\end{array}}\right)$  

where F and G are represented as 3-tuples, H is represented as a ×3 matrix, and J as a 3×1 matrix. Compute divergence with Evaluate.

$\blacktriangleright$ Evaluate

∇⋅F =   0

∇⋅G =  5z

∇⋅(xz, 2yz, z2) =  5z

∇⋅(xy, x, 0) =  y

∇⋅H = 0

∇⋅J = 5x

∇⋅$\left(\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right)$ = 0

∇⋅$\left[\vphantom{ ax,bxy,cz^{2}}\right.$ax, bxy, cz2$\left.\vphantom{ ax,bxy,cz^{2}}\right]$ = a + bx + 2cz