ma192sampleprelim - 1(20 True/False No reasons need to be...

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1. (20) True/False. No reasons need to be given. For any non-zero real a and b , the vectors a i + b j and b i - a j are orthogonal. Proj ¯ w ¯ v always has the same direction as ¯ w . The gradient f is normal to the surface z = f ( x, y ) at every point on the surface. The domain D of the function f ( x, y ) = 1 / ( x 2 - y ) is a closed region. The function f ( x, y ) = sin y + xe y has no local or absolute extrema. The punctured plane { ( x, y ) | ( x, y ) 6 = (0 , 0) } is a connected region but not simply connected. H - y d x + x d y x 2 + y 2 = 0 around every closed curve C containing the origin. H - y d x + x d y x 2 + y 2 = 0 around every closed curve C not containing the origin. If f ( x, y, z ) is harmonic everywhere (i.e. f xx + f yy + f zz = 0), then the outward flux of f across a smooth, closed, oriented surface S is always 0. Let ¯ F be a vector field with continuous first partials everywhere. Let S 1 and S 2 be surfaces with unit normals n 1 and n 2 respectively and having the same boundary C . Suppose C is oriented counter-
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