Math 192 Final Answers - 1(20 True/False No reasons need to...

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1. (20) True/False. No reasons need to be given. t For any non-zero real a and b , the vectors a i + b j and b i - a j are orthogonal. f Proj ¯ w ¯ v always has the same direction as ¯ w . (It could be pointing in the opposite direction.) f The gradient f is normal to the surface z = f ( x, y ) at every point on the surface. (The surface in question is the level surface of the function z - f ( x, y ) so its normal vector is h- f x , - f y , 1 i f The domain D of the function f ( x, y ) = 1 / ( x 2 - y ) is a closed region. t The function f ( x, y ) = sin y + xe y has no local or absolute extrema. t The punctured plane { ( x, y ) | ( x, y ) 6 = (0 , 0) } is a connected region but not simply connected. f H - y d x + x d y x 2 + y 2 = 0 around every closed curve C containing the origin. (Green’s theorem doesn’t apply because the domain isn’t simply con- nected, can do work integral around a circle centered at origin and get 2 π .) t H - y d x + x d y x 2 + y 2 = 0 around every closed curve C not containing the origin. (Green’s theorem applies, use it) t If f ( x, y, z ) is harmonic everywhere (i.e. f xx + f yy + f zz = 0), then the outward flux of f

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