Problem 1

Let be the part of the sphere lying above the cone . Complete the following parameterizations of .

(a). Find the Range of for where

(b). Find the Upper Bound of for

Problem 2

Let be the part of the cone lying below the plane . Complete the following parameterizations of .

(a). Find , , and for where

(b). Find and for where and

Problem 3

The vector field

is known to be conservative.

(a). Compute the Curl of

(b). Find the Values of , , and

(c). Find a Function and such that

(d). Evaluate the line Integral where is the line Segment from to

Problem 4

Consider the line integral

as well as that

(a). Using Greens Theorem Evaluate the Integral where is the Arc of the Parabola from to

Define to be the line from to and to be the region bounded by

(b). Using Greens Theorem Evaluate the Integral where is the Arc of the Parabola from to

The vector field is undefined at but we can use greens theorem by subtracting an area including the point.

Define to be the unit circle centred at the origin define to be the line from to we can then define the surface to be the region bounded by minus the region bounded by .

(c). Is the Vector Field Conservative? provide a Reason for Your Answer Based on Your Answers to the Previous Parts of This Question

No the vector field is not conservative if it was the from (a) would have the same value as the one from b as they start and end at the same point.