mat237t2 - 1. Let f : R 3 R 3 be the spherical coordinate...

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1 . Let be the spherical coordinate map, 3 3 : R R f ) cos , sin sin , cos sin ( ) , , ( ) , , ( ϕ θ r r r r z y x = = f . Thus r is the distance to the origin, is the angle from the positive z-axis, and is the longitude. (a) [3 marks] What is the condition on the point ) , , ( 0 0 0 r for f to be locally invertible about this point? What is the corresponding condition on ) , , ( ) , , ( 0 0 0 0 0 0 r z y x f = ? (b) [5 marks] Determine whether the integral dV z y x x z y x ∫∫∫ < + + + + 1 2 / 3 2 2 2 5 1 2 2 2 ) ( converges. 2
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2. Suppose that the surface S is parametrized by the function , where 3 2 : R R f . Let the curve C be the locus (the solution set) of two equations ) , 2 , ( ) , ( v u v u uv v u + = f 0 4 and , 0 6 2 2 2 1 2 2 2 = + + = + + z y x z y x . (a) [6 marks] Under what angle does the curve C intersects the surface S at the point (1,1, 2)? (b) [3 marks] Is there a neighborhood N of ) 0 , 0 ( ) , ( 0 0 = v u in 2 R such that the set S = is the graph of a function ( } ) , ( : ) , ( { N v u v u f 1 C ) , ( y x f z = or or ) , ( z x f y = )? Why or why not? Formulate the theorem that you have used. ) , ( z y f x = 3
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3 . The mass m of the solid V 3 R having the mass density
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.

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mat237t2 - 1. Let f : R 3 R 3 be the spherical coordinate...

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