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Math Help - Gradient, Function 2 variables

  1. #1
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    Gradient, Function 2 variables

    Hi everybody. Given is a function f(x,y). Sought is a point Px/Py on f(x,y)=1, whose tangent is perpendicular to the vektor n=(1,0). Without any reference to the specific function in the solution it is stated, that grad(Px/Py)=\lambda*n=1,0.

    1) why is this generally true?

    (If needed, the specific function is f(x,y)=3{x}^{2} +xln(y)+{e}^{ax} , x1≥0,x2>0,a∈R.)
    Thanks for your help

    Alex

    (sry latex smhw doesnt work with me; \lambda* just means vector n multiplied with a random number
    Last edited by Schdero; June 11th 2011 at 09:12 AM. Reason: latex probs
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  2. #2
    MHF Contributor

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    grad f always points in the direction of fastest increase. Further, the directional derivative, the derivative in the direction of unit vector v, is the dot product \nabla f\cdot v. From that it follows that if the rate of change in the direction of vector v is 0, \nabla f\cdot v= 0 which says that v is perpendicular to \nabla f.

    In particular, along the curve f(x,y)= 1 (or any constant) the value of f does not change so its derivative along that curve (in the direction tangent to the curve) is 0. That is, \nabla f\cdot v= 0 for v tangent to the curve so \nabla f is perpendicular to the curve.
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