M11.F08.examfinalpracticesolns

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
(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
Background image of page 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 15

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online