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Math Help - "Gradient Theorem" ...not what you think it is

  1. #1
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    "Gradient Theorem" ...not what you think it is

    Hi Everyone.

    I am currently reading a book called "An Introduction to the Finite Element Method" by Reddy (Third Edition).

    In the book he has a chapter on mathematical preliminaries. I am completely stuck on something he wrote entitled "Gradient Theorem" which is completely different from what I learned from Multivariable Calculus. He says the following

    The integral of (The gradient of the scalar function F) dxdy (closed surface) is equal to the line integral around the boundary of (n F) ds

    Where did this come from and may anyone provide the link to online readable text on this?

    Any help would be greatly appreciated. Thanks.
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  2. #2
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    Re: "Gradient Theorem" ...not what you think it is

    Hey rdbateman.

    This is also known as the Fundamental Theorem of Calculus for Line Integrals:

    Gradient theorem - Wikipedia, the free encyclopedia

    It's the same analog that relates normal derivatives and their anti-derivatives (or integrals) but in this case it relates line integrals with th gradient.

    The main analogues in calculus include the normal Fundamental Theorem of Calculus, The Line Integral version (what we are discussing), The Divergence Theorem and Stokes Theorem: They all have similar kinds of statements relating an integral to something else.
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  3. #3
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    Re: "Gradient Theorem" ...not what you think it is

    Perhaps you failed to read my question. What I posted is not the Gradient Theorem. Even in the title of the thread I put "...not what you think it is". What I posted has a double integral in the identity. It isn't stokes theorem and it isnt the divergence theorem, so what is it? Please, no references to Wikipedia.
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    Re: "Gradient Theorem" ...not what you think it is

    "Gradient Theorem" ...not what you think it is-imag0267.jpg
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  5. #5
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    Re: "Gradient Theorem" ...not what you think it is

    Can you relate the above to Greens Theorem?

    Green's theorem - Wikipedia, the free encyclopedia
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    Re: "Gradient Theorem" ...not what you think it is

    Quote Originally Posted by rdbateman View Post
    Please, no references to Wikipedia.
    No. I cannot relate the above to Greens Theorem. I'm going to give you the benefit of the doubt and assume your not being passive aggressive. Please read in full before you post.
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  7. #7
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    Re: "Gradient Theorem" ...not what you think it is

    Here's what his book says:

    \int_\Omega{\nabla{F}}\ dx\ dy=\oint_\Gamma\bold{\hat{n}}F\ ds

    If you break it up into x- and y-components it looks like the Fundamental Theorem of Calculus. So I'm thinking it should fall out of Stoke's Theorem:

    \int_\Omega\ d\omega=\int_{\partial{\Omega}}\omega

    Does anyone know how to get from here to there? Is my approach correct?

    - Hollywood
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