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Understanding Greene's, Stoke's, and the Divergence theorems

  1. Jul 14, 2014 #1
    Hi everyone, first post. Anyway, I am reviewing my math physics, and I am having trouble understanding the Divergence Theorem, Green's Theorem, and Stokes' Theorem. I was able to satisfactorily pass math physics by only being able to regurgitate them, but soon I will be taking e&m, and it seems that these theorems are very important to it.
    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.
     
  2. jcsd
  3. Jul 14, 2014 #2

    jedishrfu

    Staff: Mentor

  4. Jul 15, 2014 #3
    So after watching Khan Academy's Videos for the three of them (much more helpful than my miserable textbook), I think I understand it a lot better. Would I be correct in saying that Green's theorem allows one to evaluate a line integral in two dimensional space, the Divergence theorem allows you to evaluate a surface integral in three dimensional space, and stokes theorem allows you to evaluate a line integral in three dimensional space. Would I be correct in saying this??
     
  5. Jul 15, 2014 #4
    First, let me say that your "struggle" grasping these concepts is normall. Most students coming out of their first vector calculus course are in a similar position. Taking E&M and other vector based physics courses will help. And a good E&M prof will spend some time reviewing these concepts when the arise in the course.

    That being said, do you remember the fundamental theorem of calculus?

    If [itex]F' \left( x \right) = f\left(x \right) [/itex]

    Then

    [itex]\int_a^b f\left(x \right) dx = F(b)-F(a) [/itex]

    By now you probably use this theorem routinely with out thinking about it, but basically this theorem allows you to evaluate an integral over a region simply by looking at the behavior at the boundary.

    It turns out that Green's Theorem, the Divergence Theorem, and Stokes Theorem are extension extension of the fundamental theorem of calculus to multi-dimension space. And they do exactly the same thing.

    The divergence theorem allows you the evaluate you the integral of the divergence over the volume of a region by considering integral of the flux over the surface (or boundary) of that volume.

    Stokes Theorem allows you to calculate the integral of the curl over an area by performing a line integral along the boundary.
    (Green's theorem is actually just a simple case of Stokes Theorem).
     
  6. Aug 7, 2014 #5

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    learn differential forms, then it says the integral of w over ∂X equals the integral of dw over X,
     
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