(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

80π/3 or about 83.775

step-by-step explanation:

since a cone is 1/3 of a cylinder, we need to find the base's area. since area = πr^2, we get 16π. then, multiplying this by 5, we get 80π. dividing by 3, since the cone's volume is 1/3 of that of a cylinder's, we get 80π/3.

hope this !

question 1 is 8, question 2 is 9, question 3 is 24

step-by-step explanation:

now, the numbers they give you at the top are the key. from there, you just have to plug and chug! let's try 1, 2, and 3!

key: a=3, b=5, c=6

1. a+5

remember, a=3, so we plug that in to get 3+5=8

2. 15-c

c=6, so we plug that in to get 15-6=9

3. 4b

b=6, so we plug it in to get 4*6=24

time to try the rest on your own! if you put the answers below in a comment, i'll check them for you!