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Thread: 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; Jun 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 $\displaystyle \nabla f\cdot v$. From that it follows that if the rate of change in the direction of vector v is 0, $\displaystyle \nabla f\cdot v= 0$ which says that v is perpendicular to $\displaystyle \nabla f$.

    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, $\displaystyle \nabla f\cdot v= 0$ for v tangent to the curve so $\displaystyle \nabla f$ is perpendicular to the curve.
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