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Unformatted text preview: Lecture 22. Change of variables See Chapter 8, section 4. Suppose H is a function from a domain the x- y-plane to the u- v-plane. We can express the dependence of ( u, v ) on ( x, y ) by writing: ( u, v ) = ( u ( x, y ) , v ( x, y )) = H ( x, y ) . Here, u = u ( x, y ) and v = v ( x, y ) are two real-valued functions, each of two variables. We say H is invertible if there is a function G defined on a domain in the u- v-plane such that: G ( H ( x, y )) = ( x, y ) and H ( G ( u, v )) = ( u, v ) . If there is such a function, we can view x and y as functions of u and v : ( x, y ) = ( x ( u, v ) , y ( u, v )) = G ( u, v ) . Examples. 1. Suppose H ( x, y ) = ( x + y, x y ), so u = x + y , v = x y . Then the inverse of H is G ( u, v ) = ( u + v 2 , u v 2 ), so x = u + v 2 , y = u v 2 . 2. Suppose H ( x, y ) = ( x + y, y x ), so u = x + y, v = y x ( x negationslash = 0) . We can solve for x and y by noting that y = xv , so u = x + xv = x (1 + v ), so x = u 1 + v , y = uv 1 + v ( u negationslash = 0 , v negationslash = 1) . Thus, if we restrict H to the portion of the x- y-plane where x negationslash = 0 and y negationslash = x , then it has an inverse G ( u, v ) = ( u 1+ v , u v 1+ v ), defined on the set where u negationslash = 0 and v negationslash = 1....
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This note was uploaded on 11/29/2011 for the course MATH 3355 taught by Professor Britt during the Spring '08 term at LSU.
- Spring '08