20091111: 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 = <2xy^{2} + 2x, 2x^{2}y>. 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
 If F = <P,Q> lives on an appropriate domain, and if P_{y} = Q_{x}, then F = gradf for some f.
 The FTC for Line Integrals
 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<=

 If F = <2xy, x^{2} + 3y>, find the integral
 If F = <x^{2} y, x y^{2}>, similar.
ma215080f09 lecture outline 20091111
Created: Wed Nov 11 12:47:05 EST 2009
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©2009 Gavin LaRose