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Gradient of Surface

  1. Jun 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the gradient vector of surface z=(x^2) * (y^3) at A(1,1).

    2. Relevant equations



    3. The attempt at a solution
    I am confused with book's solution.
    Books solution is :

    f(x,y,z) = (x^2) * (y^3) - z

    grad(f) = < 2x(y^2) , 3(y^2)(x^2) , -1 >
    at A(1,1) = <2 , 3 , -1>


    The reason of my confusion is , since z=(x^2) * (y^3) , z looks like a function of x and y.
    if we create a function f(x,y,z) = (x^2) * (y^3) - z , then we see f(x,y,z)=0. So doesn't differentiation of f becomes an implicit differentiation ?

    P.S. I know thinking as implicit doesn't effects the result here but z might have been 2z or z^2
     
  2. jcsd
  3. Jun 22, 2012 #2
    When you head in the direction along the surface, you obey f(x,y,z)=0. When you head towards f(x,y,z)=1, you may do so orthogonally to the surface, which is the direction of the gradient. The gradient conceptually relies on the value of f(x,y,z) varying.

    I guess I'm not sure on what your question about implicit differentiation is. Is it implicit? The gradient I beleive is an explicit derivative.

    Ah, I think I see wehre the confusion lies.

    f(x,y,z)=0 is an implicit function.

    z=x^2*y^3 is an explicit function for the same surface.

    But grad f is an explicit derivative (from the family of curves, f=c). We might've called it an implicit derivative if we had wanted to dig a partial out or something. But we never did, we wanted the explicit derivative.
     
    Last edited: Jun 22, 2012
  4. Jun 23, 2012 #3
    Since f(x,y,z)=c is a level surface , it's gradient is explicit i see. Thanks for the answer. The real problem is , our Calculus teacher said "You will not need level curves and level surfaces so i am not going to teach them" and passed out them completely. (Yeah pretty awkward.) So that's why i am failing on that. Doesn't even know what is a level surface or level curve. He just teached , that's the function , that's it's grad , and that's how you find the directional derivative. So I believe since we haven't told the level surface , level curve , tangent planes , normal vectors etc. he can't ask something like that one.
     
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