• Support PF! Buy your school textbooks, materials and every day products Here!

Vector Calculus II: Divergence

  • #1

Homework Statement



A smooth vector field F has divF(1,2,3) = 5. Estimate the flux of F out of a small sphere of radius 0.01 centered at the point (1,2,3).

Homework Equations



Cartesian Coordinate Definition of Divergence: If F= F1i + F2j +F3k, then divF=dF1/dx + dF2/dy + dF3/dz

The Attempt at a Solution



I graphed the problem in hopes that a method to solve this would come to me, but it didn't. Any hints to get me started on this problem? Thank you for any time and effort.
 

Answers and Replies

  • #2
gabbagabbahey
Homework Helper
Gold Member
5,002
6
Well, what is the definition of flux through a surface?
 
  • #3
Flux is the amount of "something" passing through a surface. Hmmm. If divergence = flux/volume, can I just take the volume of the sphere with radius 0.01 and multiply it by the flux divergence of 5? That sounds right to me.
 
  • #4
Yup. I got the right answer. Thank you for sparking something so simply. Have a good day.
 
  • #5
gabbagabbahey
Homework Helper
Gold Member
5,002
6
You got the right answer, but your method isn't really correct. The definiton of flux involves an integral...what is it?
 
  • #6
The integral over the surface S of Vector field F dotted with dA
 
  • #7
gabbagabbahey
Homework Helper
Gold Member
5,002
6
Right, and what does the divergence theorem tell you that will equal when your surface is closed, like the boundary of a sphere?
 
  • #8
We are learning the divergence theorem tomorrow :\ I'll get back to you on that one. Lol.
 

Related Threads on Vector Calculus II: Divergence

Replies
3
Views
5K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
3
Views
6K
  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
0
Views
2K
Replies
5
Views
29K
  • Last Post
Replies
9
Views
1K
Replies
6
Views
5K
Top