solve surface integral of a scalar function using divergence theorem
By Emma Valentine •
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.
$\endgroup$ 61 Answer
$\begingroup$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
$$\iint_S (2x+2y+z^2) dS=\iint_S \vec F \cdot \vec n\, dS=\iiint_V 1\cdot dV $$
$\endgroup$ 2
