Gradient of Surface: Find the Gradient Vector of z=(x^2)*(y^3) at A(1,1)

[magorium]

Homework Statement

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

Homework Equations

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

 

[algebrat]

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

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 believe 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.

 

[magorium]

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.