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Math Help - stokes proof-help ! for exams coming soon

  1. #1
    n22
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    stokes proof-help ! for exams coming soon

    Hi,
    Can some one guide me towards the right answer?thanks.
    S be a smooth, orientable surface, with unit normal n and boundary
    C. Let C be oriented, with respect to n, as in the statement of Stokesí
    Theorem. Let f and g be continuous functions defined on S, with
    continuous first and second order partial derivatives.
    a)show that ∮(f∇g).dr=∫∫s(∇fx∇g).nds


    b)show that ∮(f∇g+g∇f).dr=0
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  2. #2
    Newbie Phantasma's Avatar
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    Re: stokes proof-help ! for exams coming soon

    Quote Originally Posted by n22 View Post
    Hi,
    Can some one guide me towards the right answer?thanks.
    S be a smooth, orientable surface, with unit normal n and boundary
    C. Let C be oriented, with respect to n, as in the statement of Stokesí
    Theorem. Let f and g be continuous functions defined on S, with
    continuous first and second order partial derivatives.
    a)show that ∮(f∇g).dr=∫∫s(∇fx∇g).nds


    b)show that ∮(f∇g+g∇f).dr=0
    Hint for (a): You want to prove that d(\langle f\nabla{g}, dr \rangle) is equivalent to what follows the integral on the RHS, where d denotes the exterior derivative and the brackets denote the inner product.

    Hint for (b): \nabla(fg)=f\nabla{g}+g\nabla{f}.
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