ma215-080-f09 outline 09-11-13

Last time we saw that if F is conservative (we can find a potential function f such that F = gradf), then the FTC for Line Integrals holds. Note that when we evaluate f at the endpoints of the curve, we are evaluating it at (x,y) (or (x,y,z)) points, not at a time t.
This is the first of four theorems we learn in calc III that relate an integral of a function's derivative (of some sort) to (fewer integrals of) the function, on the boundary of the region of integration.
Game: Let C1 be r = <t,t²>, for 0 <= t <= 1, and C2 be the line segment from (1,0,1) to (3,2,4).
1. Parameterize C2.
2. Show that F = -y/(x+1)² i + 1/(x+1) j is conservative.
3. Find the integral of F over C
5. Show that G = <y(z+1), xz + x, xy> is conservative.
6. Find the integral of G over C2.
Key Point
1. When evaluating the line integral of F.dr over a curve C, first check to see if F is conservative.
2. For F = <P,Q>, we can usually just check if P_y = Q_x.
3. For F = <P,Q,R>, we get a similar test in section 17.5.
4. In general, we can also just look for a potential function.
Next, we get Green's Theorem
Game: If F = <xy, 3x + y> and C is the circle of radius 2 centered at (0,0),
1. Find dQ/dx and dP/dy.
2. Use Green's Theorem to find the integral of F over C.
Key Point Green's Theorem lets us evaluate whichever of the line or area integral is easier (usually, this is the latter).

09-11-13: The FTC for Line Integrals, Green's Theorem