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Math Help - vector analysis question

  1. #1
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    vector analysis question

    can anybody explain me Green's theorem in plane physically?
    i cant understand that how two scalar functions M and N are representing the same curve?
    does the line integral like general ones give the area bounded by the region regarding the proof?
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  2. #2
    MHF Contributor Calculus26's Avatar
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    See the Green's Thm discussion Line Integrals
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  3. #3
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    the section wasnt of much help sorry. i want to clarify a few things
    1. we cant use stokes, circulation, curls regarding scalar functions.
    2. i wanted to know what the physical interpretation of the theorem. what the lhs & rhs separately mean? does it give the total area ultimately? could we use M and N in the region R not surrounding R?
    i didnt get the meaning anywhere in the part that u supplied.
    pls somebody interpret the theorem physically.
    thanks in advance.
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  4. #4
    MHF Contributor Calculus26's Avatar
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    The line integral is the circulation of the vector field along the boundary

    the left hand side if you will.

    To understand the double integral consider the integrand:

    dg/dx - df/dy this the gives the rotatation of the vector field about each point in the domain.

    In the notes I recommended is a diagram which shows that if you partition
    the region enclosed by the curve into rectangles the rotataions cancel everywhere but along the boundary So the sum of the rotaions i.e. the double integral of dg/dx - df/dy is the same as the circulation along the boundary.

    This is the best I can do
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