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Math Help - Can anyone solve this Calculus 3 rate of change problem?

  1. #1
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    Can anyone solve this Calculus 3 rate of change problem?

    The temperature at a point (x,y) on a flat metal plate is given by T(x,y)=20/(1+x^4+y^2) where T is measured in degrees Celsius and x,y in meters. Find the rate of change of temperature with respect to the distance at the point (1,2) in the direction of the unit vector

    u=<4/5,3/5>
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Can anyone solve this Calculus 3 rate of change problem?

    dT(1,2)/dx = -20/9
    dT(1,2)/dy = -20/9

    dT={-20/9, -20/9} as vector
    u={4/5, 3/5} unit vector

    dot product dT.u = -28/9

    Attached Thumbnails Attached Thumbnails Can anyone solve this Calculus 3 rate of change problem?-temperature-t-x-y-.png  
    Last edited by MaxJasper; October 16th 2012 at 10:23 PM.
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  3. #3
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    Re: Can anyone solve this Calculus 3 rate of change problem?

    Hey, MaxJasper, what do you use to create those pretty pictures?

    To allstar2 (mostly just re-hashing what MaxJasper wrote) - the thing you're looking for is called a directional derivative, and it is equal to \nabla{T}\cdot{u}, u being a unit vector in the direction you want to take the derivative. If you write it out, you get \frac{4}{5}\frac{dT}{dx}+\frac{3}{5}\frac{dT}{dy}, which is what you'd write if x and y were functions of some other variable t and you applied the chain rule.

    - Hollywood
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