A sample question with its solution has been provided below.

Please answer the question in the same format and exact steps as the sample questions - both questions are very similar. A typed answer is preferable for clarity. Note the difference in region in Part (c).

Sample:

Let F = -yi+ (x + 6y) j + 2z k and G = [yz2 - cos(x + y z)] i+ [xz2 -2yz cos(x +yz)] j+ [2xyz -y cos(x + yz)] k. (a) Which of these two fields (if any ) are conservative on RS ? Give detailed reasoning. (b) Find potential functions for the fields that are conservative. (c) Calculate the line integrals F . dr and G . dr where C is the arc of the C C curve formed by the intersection of the plane z = 4 and the surface z = 22 4 in the first octant, oriented anti-clockwise when view from above.(b) Let Vo = G. Then =yz2 - cos(x + y?z) ay =xz2 - 2yz cos(x + y"z) az =2xyz - y' cos(x + yz) Integrating, o =xyz" - sin(x + y'z) + Ci(y, z) p =xyz2 - sin(x + yz) + C2(2, z) o =xyz2 - sin(x + yz) + C3(2, y). From the above, we can choose o = xyz2 - sin(x + yz) to be a potential function. (c) The curve C is given by the intersection of the ellipse 20 2 y2 16 = 1 on the plane {z = 4}. So we can parametrist C by r(t) = 4costi + 2sintj + 4k, 4 0