M11.F08.examfinalpracticesolns

# M11.F08.examfinalpracticesolns - Math 11 Fall 2007 Practice...

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Math 11 Fall 2007 Practice Problem Solutions Here are some problems on the material we covered since the second midterm. This collection of problems is not intended to mimicthe±na lin length, content, or di²culty. The ±nal exam will concentrate on material covered since the second midterm, but there will also be problems on earlier material. 1. True or False: (a) The function ,r ( t )= ,a + t ( , b - )0 t 1 parametrizes the straight line segment from to , b . TRUE. This is a standard way to parametrize a line segment. (b) If the coordinate functions of , F : R 3 R 3 have continuous second partial derivatives, then curl ( div ( , F )) equals zero. FALSE. The divergence of , F is a scalar function, so its curl is not even de±ned. (c) Putting together the two di³erent vector forms of Green’s Theo- rem, we can see that if D is a region satisfying the hypotheses of the theorem, and P and Q are functions satisfying the hypotheses of the theorem, we must have ± ∂D ( P,Q ) · ˆ Tds = ± ( ) · ˆ Nds Here denotes the positively oriented boundary of D ,and ˆ T denotes the unit tangent vector to a curve, and ˆ N the unit nor- mal vector, so the integral on the left is the usual line integra lo f , F =( )a long , and the integral on the right is the integral representing the ´ux of , F ) across . FALSE: The two di³erent vector forms of Green’s Theorem do deal with these two line integrals, but they are equal to di³erent double integrals over D . 1

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(d) For any vector feld , F in R 3 all o± whose coordinate ±unctions have continuous frst and second partial derivatives, we have that div ( curl ( , F )) = 0. TRUE: This is one o± the things Clairaut’s Theorem tells us; another is that, ±or a scalar ±unction f with continuous frst and second partial derivatives, curl ( grad ( f )) = 0. (e) I± the vector feld , F is conservative on the open region D then line integrals o± , F are path-independent on D , regardless o± the shape D . TRUE: The Fundamental Theorem o± Line Integrals tells us this. It is necessary ±or D to have no holes i± we want to use the ±act that curl ( , F )= , 0on D to tell us that , F is conservative on D . (±) I± , F is any vector feld, then curl ( , F ) is a conservative vector feld. FALSE: We know the curl o± a conservative vector feld must be , 0, so i± this were true then we would always have curl ( curl ( , F )) = , 0 ±or any vector feld , F . But this is not true. (Try computing curl ( curl ( xz, yz, 0)).) 2. (a) Find a potential ±unction f ±or the vector feld , F ( x, y )=(2 x +2 y, 2 x y ) . A potential ±unction is just a ±unction f such that , F = f . SOLUTION: There are several ways to go about fnding a po- tential ±unction. The organized antidi²erentiation methodi sa s ±ollows: f = , F we must have f x =2 x y (1) f y x y (2) Starting with Equation 1 and integrating with respect to x (treat- ing y as a constant) we get f x x y 2
f = x 2 +2 xy + C ( y )( 3 ) Note that the “constant” of integration C can actually be a func- tion of y , as we are treating y as a constant. Now we diFerentiate this equation with respect to y to get f y =2 x + C ± ( y ) Comparing this to Equation 2 we see that we must have f y x y f y x

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M11.F08.examfinalpracticesolns - Math 11 Fall 2007 Practice...

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