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Math Help - Can anyone explain in English what is LaPlace and Poisson equations?

  1. #1
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    Can anyone explain in English what is LaPlace and Poisson equations?


    \nabla^2 V = \nabla \cdot \nabla V.

    Let me first break this down in English from my understanding:

    \nabla V is the gradient of a scalar function  V. \nabla V is a vector field at each point P where the vector points to the direction the maximum rate of increase and |\nabla V| is the value of the slope.


     \nabla \cdot \vec{A} at a point P is the divergence of  \vec{A} at point P. If  \nabla \cdot \vec{A} at a point P is not zero, there must be a source or sink because the inflow to point P is not equal to the outflow from point P.


    So what is the meaning of the divergence of a gradient ( \nabla^2 V = \nabla \cdot \nabla V)?

    What is the meaning of Laplace equation where \nabla^2 V = 0?

    What is the meaning of Poisson's equation where \nabla^2 V = some function?

    Please explain to me in English. I know all the formulas already, I just want to put the formulas into context.

    Thanks

    Alan
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  2. #2
    A Plied Mathematician
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    Here's a review of Farlow's Partial Differential Equations for Scientists and Engineers on Amazon:

    This book is a rarity because Farlow actually succeeds in explaining how to model physical problems using PDE's. This is a volume for engineers rather than mathematicians, so expect clarity rather than pages of ugly and worthless abstractions. It's not exhaustive, but, given the price, you wouldn't be justified in demanding a detailed treatment of all the intricacies of a subject as vast as PDE's. As a pedagogical tool, Farlow stresses the physical origin of PDE's , so many problems include units and very insightful diagrams. For example, unlike many other authors, Farlow reveals the intuitive meaning of the LaPlacian, which is a noteworthy distinction reminiscent of the writing of Tristan Needham, the author of Visual Complex Analysis. If your primary interest is real understanding rather than an adeptness at manipulating meaningless symbols, this book contains all the physical motivations necessary to advance your ambitions.
    So it sounds to me like Farlow's book would have the explanation for which you're looking. It's not an expensive book, at any rate. Farlow could probably explain the Laplacian better than I. I would recommend you check that book out.
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  3. #3
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    Thanks

    I actually have the book and I'll read that later.

    Alan
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  4. #4
    A Plied Mathematician
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    You're welcome. Have a good one!
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  5. #5
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    Quote Originally Posted by Ackbeet View Post
    You're welcome. Have a good one!
    I read the first two pages and it really help me a lot. Thanks for all your help here and in the past questions.

    Alan
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  6. #6
    A Plied Mathematician
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    Excellent. You're welcome, again.
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