ma215-080-f09 outline 2009-11-11

Last time we introduced Line Integrals
Game: Let C be the curve from (0,0) to (1,0) to (1,1), and F = <2xy² + 2x, 2x²y>. Find the integral of F.dr on C.
Alternately, we can note that F is conservative. Why? Recall that conservative means F is a Gradient Vector Field, which means that F = gradf for some f. Thus (with F = <P,Q>),
In fact, if F lives on an open (no incl. boundary), simply connected (no holes or separate pieces) region, and P and Q are sufficiently nice, we can also say
Game: Find the potential function f for F, and use the FTC for line integrals to find the integral from the first game.
Key Points
1. If F = <P,Q> lives on an appropriate domain, and if P_y = Q_x, then F = gradf for some f.
2. The FTC for Line Integrals
3. Therefore, if F = gradf, the integral on any path from one point to another is the same (the integrals are path independent.
  And, in fact, for fectof fields F that live on open connected regions and are continuous, path independence only occurs if F is conservative.
Game: Let C be given by r = <t, t^{2}>, for 0<=t<=
2. If F = <2xy, x² + 3y>, find the integral
3. If F = <x² y, x y²>, similar.

2009-11-11: The FTC for Line Integrals