Tutorial 3: Line Integrals & Greenβs Theorem
Question PDF
Class Recording
Whiteboard
Slide version: Slides
Line Integrals
- If , evaluate from to along the following paths C:
a.
b. The straight line from to then to , and then to .
c. The straight line joining and .
Solution
Question (a)
Question (b)
From to ,
From to ,
From to ,
Question (c)
- Find the work done in moving a particle in a force field given by along the curve from to .
Solution
- Determine the work done in moving a particle once around a circle in the -plane, if the circle has center at the origin and radius of 3 and if those field is given by .
Solution
In the plane ,
The work done is,
Parametric equation of the circle: , , where varies from to . β
The line integral,
Note: We choose counterclockwise direction, which is known as positive direction. If traversing from counterclockwise (negative) direction, the value of integral would be .
Greenβs Theorem
- Verify Greenβs theorem in the plane for where is the closed curve of the region bounded by and in the positive direction in traversing . and intersect at and .
Solution
Greenβs theorem:
Along , where the line integral equals
Along from to , the line integral equals
L.H.S =
L.H.S = R.H.S = .
- Evaluate where is the triangle of the adjoining figure by using Greenβs theorem in the plane.
Solution
, , ,
Note: Although there exist lines parallel to the coordinate axes (coincident with the coordinate axes in this case) which meet in an infinite number of points, Greenβs theorem in the plane still holds. In general the theorem is valid when is composed of a finite number of straight line segments.
- Calculate where is the circle of radius 2 centered on the origin.
Solution
We let and to get
- Compute the line integral of along the path shown below against a grid of unit-sized squares. Use Greenβs theorem to relate this to a line integral over the vertical path joining to .
Solution
Let be the line segment going from to . Then, we can now apply Greenβs theorem to combination of and . Let be the region bounded by these two paths. Then, by Greenβs theorem, since we are oriented correctly,
because the area of the region is made of exactly 4 unit squares. The boundary of is and :
The line integral along is easier: parametrizing by for , we get
Putting it together,
- Suppose the line integral where is the triangular path that the points , , and in a counterclockwise manner. Use Greenβs theorem to write this line integral as a double integral with the appropriate limits of integration. Then, evaluate the double integral.
Solution
Let with and .
By Greenβs theorem,
- Use Greenβs theorem to evaluate where is the curve from to and the line segment from to .
Solution
First observe that this curve is oriented in the wrong direction.
However, as we observed before, we have , so we need to find the line integral and then simply negate the answer.
Using Greens Theorem, we have
- A particle starts at and moves along the -axis to . Then it moves along the upper part of the circle back to . Compute the work done on this particle by the force field .
Solution
- Let and let be the curve that is the top half of the circle , traversed counterclockwise from to , and the line segment from to .
Evaluate the line integral .
Solution
We consider the integrals over the semicircle, denoted by , and the line segment, denoted by , seperately. We then have,
For the semicircle, we use the parametric equations
This yields
For the line segment, we use the parametric equations
This yields
We conclude
In evaluating these integrals, we have taken advantage of the rule,