gradient question

Question

Suppose that f(x,y,z) is a function satisfying ∇f=2î+3j+k at (2, -2, -6) and z+x³+y=0.

Let g(x,y)= f(x,y,z(x,y)); find ∇g at (x,y) = (2,-2)

Answer 1

First off, let's consider our expression for g(x,y) a bit. In the problem, it's given in terms of the function z(x,y), but we can find what that is explicitly: since z + x³ + y = 0, z = -x³-y. Thus, g(x,y) = f(x, y, -x³-y).

Next, recall the definition of the gradient. The gradient finds each partial derivative of the function and assigns that value to the corresponding vector component: ∇g = ∂g/∂x•i + ∂g/∂y•j.

The important thing to note is that because z is a function of x and y, it's impossible to consider a differential change in x or y without also considering a corresponding differential change in z. Those changes are given by the partial derivatives ∂z/∂x and ∂z/∂y.

∂g/∂x, then, will have two components: ∂f/∂x (the change caused directly by x) and ∂f/∂z * ∂z/∂x (the change caused by x influencing z). The same goes for y: ∂g/∂y = ∂f/∂y + ∂f/∂z * ∂z/∂y.

Now that we have our expressions set up, let's assign numbers to these partial derivatives. As shown in the given gradient vector ∇f = 2i + 3j + k, ∂f/∂x = 2, ∂f/∂y = 3, and ∂f/∂z = 1 at this point. Based on the function z we found at the beginning, ∂z/∂x = -3x² = -12 and ∂z/∂y = -1.

We're ready to plug everything in. ∂g/∂x = 2 + 1*-12 = -10, and ∂g/∂y = 3 + 1*-1 = 2. These are the two components of our gradient vector ∇g! Thus, ∇g = -10i + 2j at the point (x,y) = (2,-2).