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Math Help - Directional Derivatives and Gradient

  1. #1
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    Directional Derivatives and Gradient

    Suppose that the temperature at a point (
    x, y) in the plane is given by T(x, y) = xye^(x^2y^2) .

    Explain how the quantity
    −gradient of T(x, y) is related to the flow of heat in the plane.

    Evaluate the integral from 0 to 1 of (gradient of T(x, y)) dot <x, 1> dx and explain what it represents physically.

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  2. #2
    Super Member Random Variable's Avatar
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     - \nabla T(x_{0}, y_{0}) is the direction at the point  (x_{0}, y_{0}) in which the temperature is decreasing the most rapidly.

    And a line integral in a gradient vector field is independent of the path taken. All that matters is the starting point and ending point.
    Last edited by Random Variable; July 7th 2009 at 08:38 AM.
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  3. #3
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    Hi there! Thanks for your help but I am still a bit confused over what the integral represents physically.

    I calculated it to be (1 - e)/(2e^2) which is less than 0 obviously, but I am not sure what that means.

    Thanks again,

    FatherMike
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  4. #4
    Super Member Random Variable's Avatar
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    Since the line integral is path independent, it just represents the difference in temperature between the two points.
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