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Math Help - Vector Calculus

  1. #1
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    Vector Calculus

    This is really starting to irritate me as I can't seem to get anywhere with it:

    I need to solve somthing similar to a trigonometric identity but using the definition of grad,div and curl to verify:

    grad(UV) = U grad(V) + Vgrad(U)

    and..

    grad(F.G) = F(curl G) + G(curl F) + (f.grad) G + (G.grad)F

    I have no idea where to begin, I have the definitions but I fail to see how they can help me, can anybody can help me?
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  2. #2
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    The first looks to be a simple application of the product rule...
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  3. #3
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    If you know the definitions, why not use them to do what is shown on both sides? For example, you should know, from the definition, that grad(UV)= \frac{\partial UV}{\partial x}\vec{i}+ \frac{\partial UV}{\partial y}\vec{j}+ \frac{\partial UV}{\partial z}\vec{k}. Now use the product rule as Prove It suggested.

    And be careful how you write the second one. You have one "f" where you mean "F" but more importantly you have written " F.grad" and " G.grad" where I think you really mean F\cdot\nabla and G\cdot\nabla.
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  4. #4
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    Quote Originally Posted by Prove It View Post
    The first looks to be a simple application of the product rule...
    Maybe I forgot to write in my post, It's obviously going to be the product rule. Yo ucould say that where its normally "f" it's now UV

    Quote Originally Posted by HallsofIvy View Post
    And be careful how you write the second one. You have one "f" where you mean "F" but more importantly you have written " F.grad" and " G.grad" where I think you really mean F\cdot\nabla and G\cdot\nabla.
    That is what the question reads in my tutorial exercises.
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  5. #5
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    Okay, I have done the first question.. finally.

    After checking my tutorial exercise sheets, I was wrong and its:

    grad(F.G) = F(curl G) + G(curl F) + (F.grad)G + (G.grad)F

    Still have no idea what to do, I'm thinking I need to prove the identity for the i component and generalize for j and k component?
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  6. #6
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    Do you mean that
     \nabla(F\cdot G) = F(\nabla \times G) + G(\nabla \times F)+ (F\cdot \nabla G) + (G\cdot \nabla F) , where  F and G are scalar functions?

    if so, we note that for the i component of the LHS is  \frac{\partial}{\partial x}F_iG_i = (\frac{\partial}{\partial x}F_i)G_i + (\frac{\partial}{\partial x}G_i)F_i. Where F_i denotes the i component of  F

    Now using the formulas for curl and gradient, we may calculate the RHS and simplify it in order to get it to look like the LHS.

    Note the ith component of \nabla \times F is  \frac{\partial}{\partial y}F_k  - \frac{\partial}{\partial z}F_j
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  7. #7
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    Yes I mean that, but that makes absolutely no sense to me.. at the moment I'm plugging in the "definitions" that I know into the RHS.
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