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15-5 - Math 200 Spring 2010 Handout#21 Section 15-5 Change...

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Math 200, Spring 2010 Handout #21 Section 15-5 Change of Variables Transformation Φ from the uv -Plane to the xy -Plane A transformation or mapping is a function : X Y Φ from a set X (the domain) to a set Y . Suppose 2 : D Φ is the transformation ( ) ( ) ( ) ( ) , , , , u v u v u v φ ψ Φ = from the uv -plane to the xy -plane. Here and φ ψ are the component functions and ( ) ( ) , , , x u v y u v φ ψ = = . Transformation Φ is a function whose domain and range are both subsets of 2 . If ( ) ( ) 0 0 0 0 , , u v x y Φ = , then the point ( ) 0 0 , x y is called the image of the point ( ) 0 0 , u v . Φ maps a region D in the uv -plane into a region R in the xy -plane called the image of D . If no two points have the same image, Φ is called one-to-one . If Φ is one-to-one, then it has an inverse transformation from the xy -plane to the uv -plane defined by ( ) 1 ( , ) , x y u v Φ = . 1. Described the image of a rectangle [ ] [ ] 1 2 1 2 , , r r θ θ × under the polar coordinate transformation 2 2 : Φ ℝ ℝ defined by ( ) ( ) , cos , sin r r r θ θ θ Φ = . 2. Suppose D is a square bounded by the lines 0, 1, 0, 1 u u v v = = = = . Find the image R of the set D under the transformation ( ) 2 : , 1 x v y u v Φ = = + .
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Linear Maps A map ( ) , u v Φ is called linear if it has the form ( ) ( ) , , u v Au Cv Bu Dv Φ = + + , where A , B , C , and D are constants. The images of the basis vectors 1,0 and 0,1 = = i j are 1,0 , and 0,1 , A B C D = Φ = Φ r = s = . If ( ) , u v Φ is a linear map, then a segment joining two points P and Q in the uv -plane is mapped to the segment joining ( ) ( ) and P Q Φ Φ in the xy -plane.
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