(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

irrational, then, just means all the numbers that aren't rational. every integer is a rational number, since each integer n can be written in the form n/1. however, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers.

step-by-step explanation:

hey there : ) your answer would be a.

step-by-step explanation: