Divergence theorem again

  • Thread starter Tony11235
  • Start date
  • #1
254
0
Suppose [tex] D \subset \Re^3 [/tex] is a bounded, smooth domain with boundary [tex] \partial D [/tex] having outer unit normal [tex] n = (n_1, n_2, n_3) [/tex]. Suppose [tex] f: \Re^3 \rightarrow \Re [/tex] is a given smooth function. Use the divergence theorem to prove that

[tex] \int_{D} f_{y}(x, y, z)dxdydz = \int_{\partial D} f(x, y, z)n_2(x, y, z)dS[/tex]

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

Answers and Replies

  • #2
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
Hint:
Consider the vector function:
[tex]F(x,y,z)=(0,f(x,y,z),0)[/tex]
 
  • #3
254
0
While we're on it, I have another similar question. Say F:R^3 -> R^3 is a C^1 function, verifty that

[tex]\int_{\partial D} \nabla \times F \cdot n dS = 0 [/tex]

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 thats what our professor is having us do right now, especially for those of us that are currently in vector calculus.
 
Last edited:

Related Threads on Divergence theorem again

  • Last Post
Replies
6
Views
4K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
5
Views
915
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
11
Views
12K
  • Last Post
Replies
4
Views
1K
Replies
1
Views
5K
Replies
9
Views
13K
  • Last Post
Replies
5
Views
1K
Replies
1
Views
3K
Top