divergence

Divergence of vector field

Syntax

divergence(V,X)

Description

divergence(V,X) returns the divergence of vector field V with respect to the vector X in Cartesian coordinates. Vectors V and X must have the same length.

Examples

Find Divergence of Vector Field

Find the divergence of the vector field V(x,y,z) = (x, 2y², 3z³) with respect to vector X = (x,y,z).

syms x y z
field = [x 2*y^2 3*z^3];
vars = [x y z];
divergence(field,vars)

ans =
9*z^2 + 4*y + 1

Show that the divergence of the curl of the vector field is 0.

divergence(curl(field,vars),vars)

ans =
0

Find the divergence of the gradient of this scalar function. The result is the Laplacian of the scalar function.

syms x y z
f = x^2 + y^2 + z^2;
divergence(gradient(f,vars),vars)

ans =
6

Find Electric Charge Density from Electric Field

Gauss’ Law in differential form states that the divergence of electric field is proportional to the electric charge density.

$$\overrightarrow{\underset{}{\nabla }}\cdot \overrightarrow{\underset{}{E}}(\overrightarrow{\underset{}{r}})=\frac{\rho (\overrightarrow{\underset{}{r}})}{{ϵ}_0}.$$

Find the electric charge density for the electric field $$\overrightarrow{\underset{}{E}}=x^2\underset{}{\overset{ˆ}{i}}+y^2\underset{}{\overset{ˆ}{j}}$$.

syms x y ep0
E = [x^2 y^2];
rho = divergence(E,[x y])*ep0

rho = $${\mathrm{ep}}_0\hspace{0.17em}\left(2\hspace{0.17em}x+2\hspace{0.17em}y\right)$$

Visualize the electric field and electric charge density for -2 < x < 2 and -2 < y < 2 with ep0 = 1. Create a grid of values of x and y using meshgrid. Find the values of electric field and charge density by substituting grid values using subs. Simultaneously substitute the grid values xPlot and yPlot into the charge density rho by using cells arrays as inputs to subs.

rho = subs(rho,ep0,1);
v = -2:0.1:2;
[xPlot,yPlot] = meshgrid(v);
Ex = subs(E(1),x,xPlot);
Ey = subs(E(2),y,yPlot);
rhoPlot = double(subs(rho,{x,y},{xPlot,yPlot}));

Plot the electric field using quiver. Overlay the charge density using contour. The contour lines indicate the values of the charge density.

quiver(xPlot,yPlot,Ex,Ey)
hold on
contour(xPlot,yPlot,rhoPlot,'ShowText','on')
title('Contour Plot of Charge Density Over Electric Field')
xlabel('x')
ylabel('y')

Input Arguments

V — Vector field

symbolic expression | symbolic function | vector of symbolic expressions | vector of symbolic functions

Vector field to find divergence of, specified as a symbolic expression or function, or as a vector of symbolic expressions or functions. V must be the same length as X.

X — Variables with respect to which you find the divergence

symbolic variable | vector of symbolic variables

Variables with respect to which you find the divergence, specified as a symbolic variable or a vector of symbolic variables. X must be the same length as V.

More About

Divergence of Vector Field

The divergence of the vector field V = (V₁,...,V_n) with respect to the vector X = (X₁,...,X_n) in Cartesian coordinates is the sum of partial derivatives of V with respect to X₁,...,X_n.

$$div(\overrightarrow{V})=\nabla \cdot \overrightarrow{V}={\displaystyle \sum _{i=1}^n\frac{\partial V_i}{\partial x_i}}.$$