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Math Help - vector force fields

  1. #1
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    vector force fields

    By attempting to find a V such that F =  - \nabla V in each case, find which of the following vector force fields are conservative, and obtain an appropriate potential energy function V when it exists.
    (i) F (x, y, z) = 3x^2yz i  + (x^3z - cos y) j  - x^3y k
    (ii) F (x, y, z) = (2xyz^2 - 6xyze^{x^2yz}) i  + (x^2z^2 - 3x^2ze^{x^2yz}) j  + (2x^2yz - 3x^2ye^{x^2yz}) k.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    1. to determine which field is conservative Calculuate curlF
    If CurlF = 0 then F is conservative

    2. To find V
    1. Integrate the x component wrt x and add a fn h(y,z) instead of an integration constant
    2. Differentiate the result wrt y and set it equal to the y component

    Integrate this with respect to find h(y,z) and add on a fn g( z)

    3. Differentiate this wrt to z snd set equal to the z- component

    Integrat this and add it alll together

    I'll illustrate with a simple example

    F = yzi + (xz+2y)j +(xy+2z)k

    V= xyz +h(y,z)

    dV/dy = xz + dh/dy = xz + y

    so dh/dy = y so h= y^2 + g(z)


    V= xyz +y^2 +g(z)

    dV/dz = xy + dg/dz = xy +2z

    dg/dz = 2z so g = z^2

    V= xyz + y^2 +z^2
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  3. #3
    MHF Contributor Calculus26's Avatar
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    You want F = -del V (must be a physics problem mathematicians tend to use F = del V)

    so must multiply result by a negative.
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  4. #4
    MHF Contributor Calculus26's Avatar
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    In all honesty sometimes simple inspection does the trick

    For ii V = - x^2yz^2 +3e^(x^2yz)
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