Green's theorem and divergence integrals

[mit_hacker]

Homework Statement

Can someone please explain to me what the physical meaning of the divergence integrals and curl integral is? In the problems I have come across, they ask us to calculate areas and etc.. using Green's theorem. Which one should I use in that case?

Thank-you very much for your help!

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

I will try to restrain myself! Mathematical concepts do NOT HAVE a specific "physical meaning" because mathematics is not physics. It is true, that the names "divergence" and "curl" come from physics- or more correctly, fluid motion. If $\vec{f}(x,y,z)$ is the velocity vector of a fluid at point (x,y,z), the div f measures the tendency of the flud to "diverge" (move away from) some central point. Similarly, curl f measures the tendency to circulate around some central point.

However, you say "they ask us to calculate area and etc. using Green's theorem".

Well, Green's theorem says
$$\int\int \left[\frac{\partial Q(x,y)}{\partial x}-\frac{\partial P}{\partial y}\right]dxdy= \oint P(x,y)dx+ Q(x,y)dy$$
which doesn't use either "div" or "grad"!

You know, surely, $\int\int dx dy$, taken over a given region in the xy-plane, is the area of that region. To "use Green's theorem" to find the area of a region, you just have to use functions P(x,y) and Q(x,y) such that
$$\frac{\partial Q(x,y)}{\partial x}- \frac{\partial P(x,y)}{\partial y}= 1$$.
One obvious choice is P(x,y)= 0, Q(x,y)= x. Integrating $\int x dy$ around the boundary of a region will, according to Green's theorem, be the same as integrating 1 over the region itself- and so will give you the area of the region.

 

[mit_hacker]

Thanks a ton!

Thank-you for your help. I really appreciate it!