(a) show that the vector field f = sin y cos x i + cos y sin x j + z k is conservative, or equivalently satisfies the equations of the (curl) component test. (b) find a scalar potential function f(x, y, z) for the vector field f(x, y, z), and show all steps of your construction of f. (c) evaluate the line integral ∫cf *dr where c is a path from the point a(0, 1, 0) to b(π/2, π/2, 2).

a. If a vector field is conservative, then its curl is 0:

(Note that, generally, if the curl is 0, the field need not be conservative!)

b. If there is a scalar function for which , then

From these equations we find (first by integrating both sides of the first equation with respect to )

So we have

c. By the gradient theorem (i.e. fundamental theorem of calculus), we have for any path beginning at and terminating at

so that

true

step-by-step explanation:

5050 i think

step-by-step explanation: