Problem 1

Let

(a). Find

(b). What Does Your Answer to part (a) Tell You about where is the Circle in the -plane, Oriented Clockwise?

The function is a conservative vector field and thus is path independent because starts and ends at the same point our integral evaluates to 0.

(c). If is Any Closed Curve, what Can You Say about

(d). Now Let Be the Half Circle in the -plane with Traversed from to . Find by Using Your Result from (c) and considering plus the line Segment Connecting the Endpoints of

Consider to be and the line from to connecting the endpoints of .

Problem 2

Consider the open cylindrical can given by the equation of the side edges for a height of 5.

(a). Give a Parametric Equation for the Upper Rim Oriented Clockwise

(b). If is Oriented outward and Downward, Find

Problem 3

Let be the capped cylindrical surface which is the union of two surfaces, a cylinder given by , , and a hemispherical cap defined by , . For the vector field compute .