How Does the Divergence Theorem Apply in Vector Calculus and PDE?

[Tony11235]

Suppose $$D \subset \Re^3$$ is a bounded, smooth domain with boundary $$\partial D$$ having outer unit normal $$n = (n_1, n_2, n_3)$$. Suppose $$f: \Re^3 \rightarrow \Re$$ is a given smooth function. Use the divergence theorem to prove that

$$\int_{D} f_{y}(x, y, z)dxdydz = \int_{\partial D} f(x, y, z)n_2(x, y, z)dS$$

I think I see how they might be equal but I don't know where to start as far as proving it.

 

[arildno]

Hint:
Consider the vector function:
$$F(x,y,z)=(0,f(x,y,z),0)$$

 

[Tony11235]

While we're on it, I have another similar question. Say F:R^3 -> R^3 is a C^1 function, verifty that

$$\int_{\partial D} \nabla \times F \cdot n dS = 0$$

in two ways, first using Stokes theorem, then using the Divergence theorem.

By the way, I'm currently in vector calculus and at the same time first semester PDE. We don't cover the divergence theorem and such until way later into the semester. But our PDE book requires that you have some minor knowlegde of these theorems. And that's what our professor is having us do right now, especially for those of us that are currently in vector calculus.