Problem 1
Let and let have continuous first partial derivatives.
Prove that using the following methods
(a). Use the Fundamental Theorem of Calculus
(b). Use Green's Theorem
Problem 2
Let be a finite region in the -plane whose boundary consists of a single, piecewise smooth, simple closed curve oriented anticlockwise. ("Simple" means that the curve does not intersect itself.) Use Green's theorem to show that
where , is the outward unit normal to , and is the arc length along .
The get the outward pointing vector we need to rotate by -90º
Problem 3
Integrate anticlockwise around:
(a). The Circle
(b). The Boundary of the Square with Vertices , , , and
(c). The Boundary of the Region ,
Consider two semi circles one of radius and the other of radius we saw that via (a) that these path integrals do not depend on their radius and would be and for both traveling in the counter clockwise direction. Via our boundary though because it's a semi circle one needs to be reversed so the circle portions contribute 0 to the integral. Additionally to connect the two semi circles we need two horizontal lines on the x-axis from to and the positive equivalents, we can see via the integral that these also contribute 0 so the total integral is 0
Problem 4
Show that
for all . Discuss the connection between this result and the results of Problem 3.
The field is conservative for every simply connected region not including the origin, therefore a closed path integral over a simply connected region not including the origin is always zero and one containing the origin is always 1 as it's just the path integral of a circle encompassing the origin.
Problem 5
Evaluate where and is the boundary of the square in the -plane having one vertex at the origin and diagonally opposite vertex at the point , oriented anticlockwise.
Problem 6
Evaluate
where is the anticlockwise boundary of the trapezoid with vertices , , , and .
Problem 7
Evaluate
anticlockwise around the boundary of the half-disk .
Problem 8
Let be the anticlockwise boundary of the rectangle with vertices , , , and . Evaluate
Problem 9
Consider the closed region enclosed by the curves and . Let be its boundary, oriented anticlockwise. Determine the Value of
Problem 10
Let . Let be the boundary of the triangle with vertices , , and , oriented anticlockwise. Compute .
Problem 11
Suppose the curve is the boundary of the region enclosed between the curves and . Determine the value of the line integral
where is traversed anticlockwise.
Problem 12
Let
Find , where is the boundary of the triangle , , , oriented anticlockwise.
Problem 13
Set .
(a). Use Green's Theorem to Evaluate where is the Arc of the Parabola from to
Let be the line from to and to be the region enclosed by the counterclockwise boundary .
(b). Use Green's Theorem to Evaluate where is the Arc of the Parabola from to
Let by the circle of radius centred at the origin and let be the region .
(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 field is not conservative as it's not defined over a simply connected domain alternatively, line integrals are not path independent over .
Problem 14
Suppose the curve is the boundary of the region enclosed between the curves and . Determine the value of the line integral
where is traversed anticlockwise.
Problem 15
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 . Explain Briefly
is a circle of radius 1 centred at and does not enclose the origin so
(b). Find for . Explain Briefly
is a circle of radius 3 centred at and does enclose the origin so
(c). Find for . Explain Briefly
is a circle of radius 3 centred at and does enclose the origin so
Problem 16
Consider the vector field , where
(a). Compute and Simplify
(b). Compute the Integral Directly Using a Parametrization, where is the Circle of Radius , Centred at the Origin, and Oriented Anticlockwise
(c). Is Conservative? Carefully Explain how Your Answer Fits with the Results You Got in the First Two Parts
No it is not as it's not path independent as was shown in (b). Additionally it is also not defined at meaning it is not defined on a simply connected domain.
(d). Use Green's Theorem to Compute where is the Triangle with Vertices , , Oriented Anticlockwise
The region does not include the origin so as is irrotational over the region.
(e). Use Green's Theorem to Compute where is the Triangle with Vertices , , Oriented Anticlockwise
The region includes the origin so
Problem 17
(a). Evaluate where is the Unit Circle , Oriented Anticlockwise
(b). Evaluate where is now the part of the Unit Circle with , Still Oriented Anticlockwise
Problem 18
Evaluate the line integral
where is the arc of the curve for , traversed in the direction of increasing .
Problem 19
Use Green's theorem to establish that if is a simple closed curve in the plane, then the area enclosed by is given by
Use this to calculate the area inside the curve .
Problem 20
Let and be vector fields. Find a number such that for each circle in the plane
Problem 21
Let , .
(a). Compute where is the Unit Circle in the -plane, Positively Oriented
(b). Use (a) and Green's Theorem to Find where is the Ellipse , Positively Oriented
Problem 22
Let be the circle and let be the circle . Let . Find the integrals and .
Problem 23
Let be the region in the first quadrant of the -plane bounded by the coordinate axes and the curve . Let be the boundary of , oriented anticlockwise.
(a). Evaluate
(b). Evaluate , where
Problem 24
Let be the curve defined by the intersection of the surfaces and .
(a). Show that is a Simple Closed Curve
(b). Evaluate where
(i).
(ii).
Problem 25
Find a smooth, simple, closed, anticlockwise oriented curve in the -plane for which the value of the line integral
is a maximum among all smooth, simple, closed, anticlockwise oriented curves.
The optimal curve is .