Question

\({\bf{F}}(x,y,z) = \left( {{x^3} + {y^3}} \right){\bf{i}} + \left( {{y^3} + {z^3}} \right){\bf{j}} + \left( {{z^3} + {x^3}} \right){\bf{k}}\) \(S\)is the sphere with centre the origin and radius \({\bf{2}}\).

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.

In this example the field is\((0,\,0,\, - y) = - k\). It assigns the vector that is the negative value of \(y\) coordinate of every multiplies by vector \((0,0,1)\).

Use the Geogebra to draw the result. Therefore, the sketch is given above.