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Math Help - Gradient and Directional Derivative

  1. #1
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    Gradient and Directional Derivative

    A geological map shows that the altitude at point (x,y) is
    A(x,y)=100-x^2-y^2 feet. If water is spilled at (3,4), in which direction will it run off?

    I used the angle of inclination and ended up with tan^-1(10) = 84 degrees but Im not sure if this is the correct method.
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  2. #2
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    Well,  \nabla A(x,y) = -2x \hat{i} - 2y \hat{j}

    The gradient is the direction of the greatest increase. But I think what we want is the direction of the greatest decrease, which is simply  -\nabla A(x,y)

    at the point (3,4) ,  -\nabla A(x,y) = 6 \hat{i} + 8 \hat{j}

    I would just leave the answer like that (or normalize it if you like).
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  3. #3
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    Of course, the vector 6\vec{i}+ 8\vec{j} has angle tan^{-1}\frac{4}{3} with the positive x axis, NOT tan^{-1} 10.

    I might also point out that the contour lines here are circles so lines perpendicular to the circle are radii. The radius through (3, 4) has direction vector 3\vec{i}+ 4\vec{j} which is in the same direction as 6\vec{i}+ 8\vec{j}.
    Last edited by HallsofIvy; June 9th 2009 at 03:17 PM.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Of course, the vector 6\vec{i}+ 8\vec{j} has angle tan^{-1}\frac{4}{3} with the positive x axis, NOT tan^{-1} 10.

    I might also point out that the contour lines here are circles so lines perpendicular to the circle are radii. The radius through (3, 4) has direction vector 3\vec{i}+ 4\vec{j} which is in the same direction as 6\vec{i}+ 8\vec{j}.
    So 6i + 8j would be the direction of the runoff? Or could I also say tan-1(4/3)=53.1degrees is the direction.
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