091113: The FTC for Line Integrals, Green's Theorem
Stewart section 17.317.4
 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,t^{2}>, for 0 <= t <= 1, and C2 be the line segment from (1,0,1) to (3,2,4).
 Parameterize C2.
 Show that F = y/(x+1)^{2} i + 1/(x+1) j is conservative.
 Find the integral of F over C

 Show that G = <y(z+1), xz + x, xy> is conservative.
 Find the integral of G over C2.
 Key Point
 When evaluating the line integral of F.dr over a curve C, first check to see if F is conservative.
 For F = <P,Q>, we can usually just check if P_{y} = Q_{x}.
 For F = <P,Q,R>, we get a similar test in section 17.5.
 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),
 Find dQ/dx and dP/dy.
 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).
ma215080f09 lecture outline 091113
Created: Mon Nov 16 13:07:47 EST 2009
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