Consider F and C below. F(x, y, z) = y2 sin(z) i + 2xy sin(z) j + xy2 cos(z) k C: r(t) = t2 i + sin(t) j + t k, 0 ≤ t ≤ π (a) Find a function f such that F = ∇f. f(x, y, z) = (b) Use part (a) to evaluate C ∇f · dr along the given curve C.

a)

b)

Step-by-step explanation:

Recall that given a function f(x,y,z) then . To find f, we will assume it exists and then we will find its form by integration.

First assume that F = . This implies that

if we integrate with respect to x we get that

for some function g(y,z). If we take the derivative of this equation with respect to y, we get

This must be equal to the second component of F. Then

This implies that , which means that g depends on z only. So

Taking the derivative with respect to z and making it equal to the third component of F, we get

which implies that which means that g(z) = K, where K is a constant. So

b) To evaluate we can evaluate it by using f. We can calculate the value of f at the initial and final point of C and the subtract them as follows.

Recall that so

Also so

So