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

divergence(curl(field,vars),vars)

ans =
0

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

ans =
6

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

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')