Assignment 9 Solutions

Assignment 9 Solutions - Math 237 Assignment 9 Solutions 1....

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 237 Assignment 9 Solutions 1. Use T ( x,y ) = ( x + y,- x + y ) to evaluate R R - y ( x + y ) cos( x- y ) dx dy . Solution: We have u = x + y and v =- x + y . The region of integration is 0 x - y and 0 y . Thus, the region is bounded by the lines x = 0, y = 0 and x = - y . Under the mapping T we get: LINE 1: x = 0, 0 y gives v = y = u with 0 u . LINE 2: y = 0, 0 x gives v =- x =- u with- u 0. LINE 3: x + y = , 0 x gives u = . We have u- v = 2 x , hence v = u- 2 x = - 2 x so- v . Drawing rectangle vertically in the region we get- u v u and 0 u . The Jacobian is ( x,y ) ( u,v ) = 1 2- 1 2 1 2 1 2 = 1 2 6 = 0 . Hence, since the transformation has continuous partial derivatives we get by the change of variables theorem Z Z - y ( x + y ) cos( x- y ) dx dy = Z Z u- u u cos(- v ) 1 2 dv du = 1 2 Z - u sin(- v ) u- u du = 1 2 Z 2 u sin u du =- u cos u + sin u = 2. Find a linear transformation that maps x 2 + 4 xy + 5 y 2 = 4 onto a unit circle. Hence show that the area enclosed by the ellipse equals 4 . Solution: Completing the square we get x 2 + 4 xy + 5 y 2 = 4 is ( x + 2 y ) 2 + y 2 = 4. Hence, we let u = x +2 y and v = y . Thus, the inverse transformation is x = u- 2 v , y = v . Which maps the circle of radius 2 onto the ellipse. The Jacobian is ( x,y ) ( u,v ) = 1- 2 1 = 1 6 = 0 . Hence since the mapping has continuous partial derivatives the change of variables theorem gives Area = ZZ D 1 dA = ZZ D uv 1 dA = 4 , since RR D uv 1 dA just calculates the area of circle in the uv-plane. 2 3. Evaluate RRR R x 2 + y dV where R is the region bounded by...
View Full Document

This note was uploaded on 01/04/2012 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

Page1 / 5

Assignment 9 Solutions - Math 237 Assignment 9 Solutions 1....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online