Problem 1

(a). Evaluate where is the Unit Circle , Oriented Counterclockwise

(b). Evaluate where is the Unit Circle where , Oriented Counterclockwise

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

Problem 2

Consider the vector field

(a). Compute and Simplify

(b) Compute the Integral Directly Using a Parameterization, where is the Circle of Radius , Centred at the Origin, and Oriented in the Counterclockwise Direction

(c). Is Conservative? Carefully Explain how Your Answer Fits with the Results You Got in the First Two Parts

is not conservative as the domain of it's curl is not simply connected (it's undefined at )).

(d). Use Green’s Theorem to Compute where is the Triangle with Vertices , , Oriented in the Counterclockwise Direction

(d). Use Green’s Theorem to Compute where is the Triangle with Vertices , , Oriented in the Counterclockwise Direction

Define to be a circle of radius centred around the origin and to be the area bounded by minus the region bounded by .

Problem 3

Let be a smooth plane vector field defined for , and suppose for . In the following for integer and all are positively oriented circles. Suppose where is the circle .

(a). Find for

is a circle of radius 1 centred around the point , define to be the region bounded by .

(b). Find for

is a circle of radius 3 centred around the point , define to be the region bounded by minus the region bounded by .

(b). Find for

is a circle of radius 3 centred around the point , define to be the region bounded by minus the region bounded by .

Problem 4

Consider the closed region enclosed by the curves and . Let be its boundary and suppose that is oriented counter–clockwise. Determine the value of

Problem 5