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Math Help - Differential vector operators

  1. #1
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    Differential vector operators

    1. f(x,y,z) = e^(x^2+y^2)

    Eval the gradient del F and is it perpendicular to the isovalue surfaces?
    What is the value of the gradient in the origin? If r is any 3D vector and s a real parameter, evaluate df(sr)
    d
    s

    2. F(x,y) = -cosx ei -2y ej

    Determine the curl del x F. Is the force conservative. If so find the scalar potential V (x,y) from which F arises.

    By evaluating a line integral, compute the total work
    W done against the force field F when the particle moves from the point A = (-Pi/2,-1) to the point B = (Pi/2,1) along the segment (a Pi/2, a) for -1 <a < 1. Does the work W depend on the path taken to go from A to B? Give an expression forW in terms of the potential V , without evaluating any integral.

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  2. #2
    Member Greengoblin's Avatar
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    For the first part, let u=x^2+y^2 then f(x,y,z)=e^u, and the gradient is \nabla f=\left< \frac{df}{du}\frac{\partial u}{\partial x}, \frac{df}{du}\frac{\partial u}{\partial y},\frac{df}{du}\frac{\partial u}{\partial z}\right> =\left< 2xe^{x^2+y^2}, 2ye^{x^2+y^2},0\right>

    edit: I don't know what an isovalue surface is, but perpendicular vectors have a dot product of 0. To get the value at the origin simply plug (0,0,0) into the gradient. I don't understand the next part of that question, and I'm afraid I can't help with the second.
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