Question

\({\bf{F}}(x,y,z) = {x^2}\sin y{\bf{i}} + x\cos y{\bf{j}} - xz\sin y\,{\bf{k}}\), \(S\)is the "fat sphere" \({x^8} + {y^8} + {z^8} = 8\).

Short answer

Sketched the vector field F by drawing following diagram.

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.

Sketch the vector field \(F\)

Let us solve the given problem.

The situation here is similar as in previous task, but this time it assigns the vector \(\left( {0,{\rm{ }}0{\rm{ }},{\rm{ }}x} \right)\) to every point in the space. Here x is the x coordinate of the given point.

Therefore, the sketch is given above.