# Vector Calculus II: Divergence

## 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

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gabbagabbahey
Homework Helper
Gold Member
Well, what is the definition of flux through a surface?

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.

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

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

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

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

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