final solution3

final solution3 - THIRD PRACTICE FINAL Math 21a Fall 2007...

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Unformatted text preview: 1/17/2007, THIRD PRACTICE FINAL Math 21a, Fall 2007 Name: MWF 9 Chen-Yu Chi MWF 10 Oliver Knill MWF 10 Corina Tarnita MWF 11 Veronique Godin MWF 11 Stefan Hornet MWF 11 Jay Pottharst MWF 12 Chen-Yu Chi MWF 12 Ming-Tao Chuan TTH 10 Thomas Barnet-Lamb TTH 10 Rehana Patel TTH 11:30 Thomas Barnet-Lamb TTH 11:30 Thomas Lam • Please mark the box to the left which lists your section. • Do not detach pages from this exam packet or unstaple the packet. • Show your work. Answers without reason- ing can not be given credit except for the True/False and multiple choice problems. • Please write neatly. • Do not use notes, books, calculators, comput- ers, or other electronic aids. • Unspecified functions are assumed to be smooth and defined everywhere unless stated otherwise. • You have 180 minutes time to complete your work. 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 Total: 150 Problem 1) True/False questions (20 points) 1) T F For any two nonzero vectors vectorv, vectorw the vector (( vectorv × vectorw ) × vectorv ) × vectorv ) is parallel to vectorw . Solution: Take vectorv = ( 1 , , ) , vectorw = ( , 1 , ) so that vectorv × vectorw = ( , , 1 ) and ( vectorv × vectorw ) × vectorv ) = ( , 1 , ) and (( vectorv × vectorw ) × vectorv ) × vectorv ) = ( , , 1 ) . 2) T F The cross product satisfies the law ( vectoru × vectorv ) × vectorw = vectoru × ( vectorv × vectorw ). Solution: Take vectorv = vectorw , then the right hand side is the zero vector while the left hand side is not zero in general (for example if vectoru = vector i,vectorv = vector j ). 3) T F If the curvature of a smooth curve vector r ( t ) in space is defined and zero for all t , then the curve is part of a line. Solution: One can see that with the formula κ ( t ) = | vector r ′ ( t ) × vector r ′′ ( t ) | / | vector r ′ ( t ) | 3 which shows that the acceleration vector r ′′ ( t ) is in the velocity direction at all times. One can also see it intuitively or with the definition κ ( t ) = vector T ′ ( t ) / | vector T ′ ( t ) | . If curve is not part of a line, then vector T ′ has to change which means that κ is not zero somewhere. 4) T F The curve vector r ( t ) = (1 − t ) A + tB,t ∈ [0 , 1] connects the point A with the point B . Solution: The curve is a parameterization of a line and for t = 0, one has vector r (0) = A and for t = 1 one has vector r (1) = B . 5) T F For every c , the function u ( x,t ) = (2 cos( ct ) + 3 sin( ct )) sin( x ) is a solution to the wave equation u tt = c 2 u xx . Solution: Just differentiate. 6) T F The length of the curve vector r ( t ) = ( t, sin( t )), where t ∈ [0 , 2 π ] is integraltext 2 π radicalBig 1 + cos 2 ( t ) dt ....
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final solution3 - THIRD PRACTICE FINAL Math 21a Fall 2007...

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