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thatONEguy94

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Anyway, I get grad, div, and curl, but I am confused about these theorems so I will state what I think they are, and hopefully someone will correct me where I'm wrong.

First off Green's Theorem:

∮CLdx+Mdy=∫D∫(∂M∂x−∂L∂y)dydx

so to my understanding, the closed path integral about two functions defining a closed area C is equivalent to the double integral of that area.

Then with the Divergence Theorem:

∫∫∫V(∇⋅F⃗ )dV=∫S∫F⃗ ⋅dS⃗

To my understanding, the triple integral of the divergence of vector F through a three dimensional region is equivalent to the cross product of the vector F and the unit vector normal to the the surface of a the three dimensional region. It is my understanding that The Divergence theorem is Green's Theorem extended to three dimensions, and deals with a closed three dimensional region bounded by a surface that can be collapsed to two dimensions, while Green's theorem deals with a two-dimensional region outlined by two one-dimensional functions. Where I'm confused is that the divergence theorem seems to deal with a flow through the 3 dimensional area while Greens does not.

Finally for Stokes' Theorem:

∮CF⃗ ⋅dr⃗ =∫S∫∇×F⃗ ⋅dS⃗

my understanding is that it deals with a two dimensional surface in 3-dimensional space that is bounded by a 1-dimensional curve (my textbook used the example of a butterfly net). However, I am really confused by this theorem, and I don't even have an idea of what it means.

I know my ideas here are way off, but my textbook is really not of much help, so any clarification anyone could give me would be really helpful!

Thanks.