Vector Calculus II: Divergence

[Tylerdhamlin]

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.

 

[gabbagabbahey]

Well, what is the definition of flux through a surface?

 

[Tylerdhamlin]

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.

 

[Tylerdhamlin]

Yup. I got the right answer. Thank you for sparking something so simply. Have a good day.

 

[gabbagabbahey]

You got the right answer, but your method isn't really correct. The definiton of flux involves an integral...what is it?

 

[Tylerdhamlin]

The integral over the surface S of Vector field F dotted with dA

 

[gabbagabbahey]

Right, and what does the divergence theorem tell you that will equal when your surface is closed, like the boundary of a sphere?

 

[Tylerdhamlin]

We are learning the divergence theorem tomorrow :\ I'll get back to you on that one. Lol.