# Divergence Theorem

Let Q denote the unit cube in $$\Re^3$$ (that is the unite cube with 0<x,y,z<1). Let G(x,y,z) = (y, xe^z+3y, y^3*sinx). Verify the validity of the divergence theorem.

$$\int_{Q} \bigtriangledown} \cdot G dxdydz = \int_{\partial Q} G \cdot n dS$$

I am not sure how to evaluate the right side. Any help would be good.

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arildno
Homework Helper
Gold Member
Dearly Missed
Well, you've got six sides on your surface, right?
Treat the contribution from each side separately.

What would say..the first integral of the six look like? I just need one example.

arildno
Homework Helper
Gold Member
Dearly Missed
Okay, let's look at the side x=1, 0<=y,z<=1.
Here, the normal vector is +i.
G(1,y,z)=(y,e^z+3y,y^3sinx)
Forming the dot product between G and the normal vector yields the integrand "y".
This is easy to integrate over the y,z-square (yielding 1/2 in contribution)

Okay?

Ok that makes sense. So in the end I add the value of each of the six integrals for the total value of the original integral, $$\int_{\partial Q} G \cdot n dS$$?

Hurkyl
Staff Emeritus
Gold Member
By the way, is your book actually using $\partial\Omega$? I'm more used to seeing $d\Omega$ or $\delta\Omega$.

arildno
Ok that makes sense. So in the end I add the value of each of the six integrals for the total value of the original integral, $$\int_{\partial Q} G \cdot n dS$$?