Question

Use a computer algebra system to plot the vector field \({\bf{F}}(x,y,z) = \sin x{\cos ^2}y{\bf{i}} + {\sin ^3}y{\cos ^4}z{\bf{j}} + {\sin ^5}z{\cos ^6}x{\bf{k}}\) in the cube cut from the first octant by the planes \(x = \pi /2,\,\,y = \pi /2\), and \(z = \pi /2\). Then compute the flux across the surface of the cube.

Short answer

We conclude that the corresponding graph is \(I\).

Step-by-step solution

Statement of divergence theorem

The divergence theorem is a mathematical expression of the physical truth that, in the absence of matter creation or destruction, the density within a region of space can only be changed by allowing it to flow into or out of the region through its boundary.

Compute the flux across the surface of the cube

Let us solve the given problem.

Graph I matches \({\bf{F}}(x,y,z) = {\bf{i}} + 2{\bf{j}} + z{\bf{k}}\) because :

All vectors have constant \({\bf{i}}\) and \({\bf{j}}\) components

Therefore, the vectors above \(xy\)-plane point upwards while below \(xy\)-plane point downwards and hence, the result is graph \(I\).