Hi, Calc D'ers.
16.3 - Fundamental Theorem of Line Integrals
The key here is that if the vector field F is conservative, we have a
shortcut that we can use--the Fundamental Thm. of Line Integrals. (It
doesn't matter if the curve C is closed or not, this thm will work as long
as F is conservative. However, if F is conservative and C is closed then
the work will be zero no matter which way you solve it.) The steps to
solving one of these problems is as follows (sorry, I can't do boldface on
the web-based e-mail):
- Check for conservative (2D: del Q / del x = del P / del y) and (3D: curl
F = 0)
- If not conservative, you must do it the "long way" as in section 16.2.
- If F is conservative, then find the potential function f(x,y) for 2D or
f(x,y,z) for 3D.
- If you are given a set of parametric equations for the curve C and a
range for the parameter t, then you need to figure out the (x,y) location
for the corresponding endpoints of t. If you are just given the (x,y)
location for the start and end points, you are already done with this
- Plug in the coordinates found above into the potential function to
calculate the potential at each of the end points of the curve.
- Subtract the value of the potentials (potential at end point - potential
at initial point)
This sounds complicated, but once you do it a couple times, you'll get the
idea and see that it's just the same thing over and over again.
16.4 - Green's Theorem
The key here is that if you are trying to find work over a CLOSED curve C,
then you can use Green's Thm. It doesn't matter if the vector field is
conservative or not, you can use Green's Thm. either way. (However, if the
vector field IS conservative, then you'll get zero for your answer
anyway.) To solve a problem using Green's Theorem, graph the curve and see
what region it encloses. These problems are all 2D. Set up a double
integral over that region. Use the techniques developed in Chapter 15 to
figure out the limits of integration. Remember that the integrand (the
function you are integrating) is (del Q / del x) - (del P / del y). If you
were to solve the same problem without using Green's Thm., then you would
have to have parametric equations for the entire boundary. In many cases
(including the first couple problems where you need to solve it both
ways), this can mean solving several different line integrals (one for
each part of the boundary) and then adding the results together.
I hope these hints help. Tell your friends they are here on the e-mail.
You know how to reach me if you need me.