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Math Help - find the gradient and plot on level surface

  1. #1
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    find the gradient and plot on level surface

    f(x,y,z) = z-x+y find \nabla f(0,0,1) plot it on the level surface of f that passes through (0,0,1)


    \nabla f = -\hat{i} + \hat{j} + \hat{k} = <-1,1,1>

    \nabla f (0,0,1) = <-1,1,1>

    and f(x,y,z) is a plane passing through (0,0,1)

    z-x+y=1 find intercepts and draw then plot from (0,0,1) the vector <-1,1,1>

    ??
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  2. #2
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    Re: find the gradient and plot on level surface

    A "level surface" for f(x,y,z) is a surface on which f(x,y,z)= C for some constant C. For f(x)= z- x+ y, you are correct, that will be a plane: z- x+ y= C. And, of course, (0, 0, 1) satisfies 1- 0+ 0= 1 so the level surface containing (0, 0, 1) is the plane z- x+ y= 1. Yes, grad f, at (0, 0, 1) is <-1, 1, 1> which does NOT lie in the plane z- x+ y= 1. So I don't know what this question is asking! Perhaps it means the projection of the vector on that plane.
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    Re: find the gradient and plot on level surface

    yeah I may have to clarify that with my professor. When I sketched it, it made no sense.
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  4. #4
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    Re: find the gradient and plot on level surface

    gradient is orthogonal to the level surface.
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