solve surface integral of a scalar function using divergence theorem

Question

Use the Divergence Theorem to evaluate:

$$\iint_S (2x+2y+z^2) dS$$

where $S$ is the sphere $$x^2+y^2+z^2=1.$$

Most of the examples on the book that needs to be solved using divergence theorem are given in the form of a vector field. This one, however, is a scalar function. We know that if we want to use divergence theorem we need a vector field, take the divergence, and then integrate over the volume. I think this one need to somehow convert the scalar function $$2x+2y+z^2$$ into a vector field and then use divergence theorem. I don't know how to do that.

Answer 1

Note that

• $\vec n=(x,y,z)$
• $\vec F \cdot \vec n=2x+2y+z^2 \implies \vec F=(2,2,z)$

thus

• $\nabla \cdot \vec F=1$

and then

$$\iint_S (2x+2y+z^2) dS=\iint_S \vec F \cdot \vec n\, dS=\iiint_V 1\cdot dV $$