Practice Final Solutions

# Practice Final Solutions - Math E-21a Practice Final Exam...

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1 Math E-21a Practice Final Exam solutions – Fall 2009 1 ) True or False (circle one – you need not show your work): a) The dot product of two nonzero vectors gives the cosine of the angle between these two vectors. FALSE - cos  uv u v , so in general the dot product would give the angle only for a pair of unit vectors. b) () ( )   uvw uwv for all vectors u, v, w. FALSE - ( ) ( )          uvw vuw vuw uwv c) The line integral of the vector field (, ) ( ,) x yy x   F along the counterclockwise boundary of any region R in the xy -plane is twice the area of R . TRUE -  1(1 ) 2 2A r e a CB n dR n R R R QP dP d x Q d y d A d A d A R xy        Fr  d) If F is a vector field in R 3 , then curl(2 ) 8 curl( ) FF . FALSE – If ,, PQR F , then 2 2 ,2 ,2 F and (2 ) curl(2 ) , , 2 2 ,2 2 2 2, , 2 c u r l R RQ P R Q P R Q P y zz xx y y z z x x y PR yz zx  F F e) If F is a vector field that is everywhere tangent to a surface S , then the flux S d FS of F through S is zero. TRUE - SS dd S Fn where n is a unit normal vector to the surface at every point of the surface. If F is everywhere tangent to the surface S , then F and n will be perpendicular at all points of the surface, so 0 everywhere and the integrand in the flux integral will be constantly zero. Therefore the flux will be zero. 2 ) Find the distance from the point (3,3,3) S to the plane passing through the points (1,2,3) P , (2, 1,0) Q  , and (1 ,4 , 1 ) R . Solution : For the normal vector to the plane, 1, 3, 3 2,2, 2 12,8, 4 4 3,2, 1 PQ PR   , so we can use 3, 2, 1 n as a normal vector to the plane. The simplest way to find the distance from the point S to the plane is with scalar projection, namely: distance = 2,1,0 3,2, 1 8 941 1 4 PS   n n  . 3 ) Consider the function 22 (, ,) 3 5 7 f xyz x y x z . a) Suppose we were to pass through the point (1,1,1) with unit speed in a direction parallel to the vector (1,2, 2) . At what rate would the values of this function be changing at that moment? That is, what would be the directional derivative of f at the given point and in the given direction?

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Practice Final Solutions - Math E-21a Practice Final Exam...

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