A model for the height of Observatory Hill is given by 1 z = f(x, y)...

 

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A model for the height of Observatory Hill is given by 1 z = f(x, y) = (1+4x²+6xy+12y2)3/2 i) Show that 47² +6ry + 12y² can be written as the matrix expression 4x² +6ry + 12y² = (x, y)S $(3). where S is a symmetric matrix, and obtain an explicit form for S. ii) Obtain the eigenvalues and eigenvectors of S, and hence the diagonal represent S = 0¹ DO, specifying the elements of D and O. iii) By introducing the new variables ( )= 0 () show that 4x² + 6xy+12y² = 3u²+ 13v². iv) By making a further change of variable, Р U= cos √3 cos and V= Si √13 show that the volume of Observatory Hill can be written as V v=dz = [_dz_ dyf(x, y) = √5 do 1 de (1 + 0²3/2 Sot 1 2m and evaluate the integral. sin o A model for the height of Observatory Hill is given by 1 z = f(x, y) = (1+4x²+6xy+12y2)3/2 i) Show that 47² +6ry + 12y² can be written as the matrix expression 4x² +6ry + 12y² = (x, y)S $(3). where S is a symmetric matrix, and obtain an explicit form for S. ii) Obtain the eigenvalues and eigenvectors of S, and hence the diagonal represent S = 0¹ DO, specifying the elements of D and O. iii) By introducing the new variables ( )= 0 () show that 4x² + 6xy+12y² = 3u²+ 13v². iv) By making a further change of variable, Р U= cos √3 cos and V= Si √13 show that the volume of Observatory Hill can be written as V v=dz = [_dz_ dyf(x, y) = √5 do 1 de (1 + 0²3/2 Sot 1 2m and evaluate the integral. sin o

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1 LHS 42 6 xy 1292 ier s 4 3 S 21 RHS x Y S 3 x 233 22 3 4 s 12 4x 3y 3x12y 3 4... View the full answer