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Thread: Urgent! Help.. Multivariable

  1. #1
    Junior Member
    Mar 2006

    Urgent! Help.. Multivariable

    A particle is located at the point (2,-3) on a metal plate whose temperature at (x,y) is

    T(x,y) = 100 - 4x^2 - y^2

    Find the equation of the path (in terms of x and y) of the particle as it continuously moves in the direction of maximum temperature increase

    Please help.. I can't seems to understand and solve the Qn
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  2. #2
    Member Ruun's Avatar
    Mar 2009
    North of Spain
    By definition of Gradient (Gradient - Wikipedia, the free encyclopedia)

    "In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change."

    $\displaystyle \nabla T = \left( \frac{\partial T}{\partial x },\frac{\partial T}{\partial y} \right) $
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  3. #3
    MHF Contributor

    Apr 2005
    And once you have that, your path must always have that vector as tangent vector so you must have $\displaystyle \frac{dx}{dt}= \frac{\partial T}{\partial x}$ and [tex]\frac{dy}{dt}= \frac{\partial T}{\partial y}[/quote].

    Solving those two differential equations will give you x and y as functions of the parameter t.
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