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# ps4 - g and their determinants(b Check explicitly that J f...

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University of Hong Kong Fall 2011 Math 1813 Mathematical Methods for Actuarial Science Instructor: Siye Wu Problem Set 4 (due at 17:00, Friday, 11 November) I. Consider the cylinder Z = { ( x, y, z ) T R 3 : x 2 + y 2 = 4 } , the sphere S = { ( x, y, z ) T R 3 : x 2 + y 2 + z 2 = 9 } and the cone C = { ( x, y, z ) T R 3 : x 2 + y 2 = z 2 } . Describe each of the surfaces Z, S, C in both cylindrical polar and spherical polar coordinates. II. Let f : R 2 R 2 be deFned by f ( x, y ) = (cos( xy ) , sin( x + y )) T . Suppose x and y are in turn functions of s and t . Write ( x, y ) T = g ( s, t ) and let F ( s, t ) = f ( g ( s, t )) be the composite function. Suppose that g (2 , 3) = (0 , π ) T and J g (2 , 3) = ( 3 7 - 2 0 ) . (a) Compute the Jacobian matrix J F at the point (2 , 3) T . (b) ±ind the approximate value of F (2 . 01 , 2 . 98) by linear approximation. III. Let f be given by f ( x, y, z ) = ( r x 2 + y 2 + z 2 , tan - 1 x 2 + y 2 z , tan - 1 y x ) T . Its inverse is given by g ( r, θ, φ ) = ( r sin θ cos φ, r sin θ sin φ, r cos θ ) T , r g 0 . (a) Compute the Jacobian matrices J f , J
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Unformatted text preview: g and their determinants. (b) Check explicitly that J f and J g are inverses of each other (when ( x, y, z ) and ( r, θ, φ ) are related by f, g ) and so are their determinants. IV. Let Q ( x, y, z ) = 2 xz + y 2 and A be the corresponding matrix of the quadratic form Q . (a) ±ind all the extremum points of Q subject to the constraint x 2 + y 2 + z 2 = 1. (b) ±ind all the Lagrange multipliers in (a). (c) Show that the set of eigenvalues of the matrix A given above is the same as the set of Lagrange multipliers in (c), counting multiplicities. V. Use the method of Lagrange multipliers to prove that a b 2 c 3 l √ 3 36 ( a 2 + b 2 + c 2 ) 3 for any real numbers a, b, c ....
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