Homework

• Section 14.2: 1, 3, 9, 12, 16*, 21, 25, 30*, 47, 49 (Do these before the Midterm)
• Section 14.3: 11, 16*, 19, 25, 31, 40*, 43, 46 (Do these before the midterm)
• Section 14.4: 1, 7, 11, 16*, 21, 26* (Do these after the midterm)

YOU MUST WORK ALL PROBLEMS!! However, you only need to hand in the problems marked by an asterisk. These are due Tuesday, November 21 (next class after the midterm exam).

The Midterm will cover up through section 14.3. The quiz on November 21 will cover section all three sections.

Overview:

• Section 14.2: there are two types of line integrals we will look at:
  1. Line integrals of a real-valued functions of several variables. (Think of finding the mass of a piece of wire.)
  2. Line integrals of vector fields. (Think of finding the total work done by moving a particle along some path in a gravitational field).
• Section 14.3: This section establishes some very nice results about line integrals for conservative vector fields:
  1. Fundamental Theorem of Line Integrals. This basically says that you can evaluate a line integral by calculating the values of the potential function at the endpoints of the curve, and subtracting. (Note: this is very similar to how you do definite integrals for for real-valued functions).
  2. Independence of Path The line integral along two different paths connecting the same two endpoints will be the same! So, for example, the work done by moving an object from point A to point B in a gravitational field is the same no matter what path you move along!!
• Section 14.4: Green's theorem relates the line integral over a simple closed curve to a certain double integral over the region enclosed by that curve. This is a very convenient theorem because calculating line integrals can be a lot of work, whereas the double integral is often much easier to work with. (NOTE: Green's theorem does not require that the vector field be conservative).
• Calculating line integrals: Putting it all together, here is an outline of how to calculate a line integral of a vector field F:
  1. First, check to see if the vector field is conservative? If it is, you can do one of the following (whichever appears easier):
     1. Find a potential function f for F and use the fundamental theorem of line integrals, OR
     2. Look for a simpler curve to evaluate the line integral over.
  2. If F is not conservative then do one of the following (whichever appears easier):
     1. Use Green's theorem to convert the line integral to a double integral, OR
     2. Use the brute force approach (find a piecewise smooth parameterization of the curve C, rewrite the variables in terms of the parameter, and integrate).

Key Skills :

• Find a piecewise smooth parameterization of a given path. (14.2:1,3)
• Evaluate a line integral of a real-valued function over a given path. (14.2: 9,12,16)
• Evaluate a line integral of a vector field over a given path. (14.2: 21, 25, 47, 49)
• Find the work done by a force on a particle moving in a vector field.
• Evaluate a line integral of a conservative vector field over a given path by finding an easier path for the integration. (14.3: 11,15, 19)
• Use the fundamental theorem of line integrals to calculate the line integral of a conservative vector field. (14.3: 25, 31)
• Use Green's theorem to evaluate a line integral of a conservative vector field by calculating a related double integral. (14.4: 7, 11, 16)
• Use Green's theorem to calculate the work done by a force on a particle moving along a path in a vector field. (14.4: 21)
• Use Green's theorem to calculate the area of a region (by evaluating a line integral). (14.4:26)