Question

Consider F and C below.

F(x, y, z) = (2xz + y²)i + 2xy j + (x² + 9z²)k
C: x = t², y = t + 2, z = 3t - 1, 0 ≤ t ≤ 1

Find a function f such that F = ∇f.
f(x,y,z) = ___

Answer 1

The potential function f(x, y, z) satisfying F = ∇f, given F(x, y, z) = (2xz + y²)i + 2xy j + (x² + 9z²)k, is f(x, y, z) = x²z + y²x + 3z³.

Step-by-step explanation:

The student is asking to find a potential function f such that the vector field F is the gradient of f, where F(x, y, z) = (2xz + y²)i + 2xy j + (x² + 9z²)k. To find such a function, we want to integrate the components of F with respect to their corresponding variables, x, y, and z, ensuring that the partial derivatives of f are consistent with the components of F.

By integrating F's x-component with respect to x, we get f(x, y, z) = x²z + h(y, z), where h is a function of y and z only. Integrating F's y-component with respect to y, we add y²x to the potential function and determine h(y, z) must only be a function of z, which we can denote as g(z). Lastly, integrating F's z-component with respect to z, we add 3z³ to the potential function.

Hence, the potential function f(x, y, z) that satisfies F = ∇f is f(x, y, z) = x²z + y²x + 3z³.