# 2009-11-11: The FTC for Line Integrals

Stewart section 17.3

• 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 Py = Qx, 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<=
1. If F = <2xy, x2 + 3y>, find the integral
2. If F = <x2 y, x y2>, similar.
ma215-080-f09 lecture outline 2009-11-11
Created: Wed Nov 11 12:47:05 EST 2009