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16.5 Ex 26 - 28

# 16.5 Ex 26 - 28 - S E C T I O N 16.5 Change of Variables(ET...

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S E C T I O N 16.5 Change of Variables (ET Section 15.5) 1003 25. With as in Example 3, use the Change of Variables Formula to compute the area of the image of [ 1 , 4 ] × [ 1 , 4 ] . SOLUTION Let R represent the rectangle [ 1 , 4 ] × [ 1 , 4 ] . We proceed as follows. Jac ( ) is easily calculated as Jac ( T ) = ∂( x , y ) ∂( u , v) = 1 /v u /v 2 v u = 2 u /v Now, the area is given by the Change of Variables Formula as ( R ) 1 d A = R 1 | Jac ( ) | du d v = R 1 | 2 u /v | du d v = 4 1 4 1 2 u /v du d v = 4 1 2 u du · 4 1 1 v d v = ( 16 1 )( ln 4 ln 1 ) = 15 ln 4 In Exercises 26–28, let R 0 = [ 0 , 1 ] × [ 0 , 1 ] be the unit square. The translate of a map 0 ( u , v) = ( φ ( u , v), ψ ( u , v)) is a map ( u , v) = ( a + φ ( u , v), b + ψ ( u , v)) where a , b are constants. Observe that the map 0 in Figure 16 maps R 0 to the parallelogram P 0 and the translate 1 ( u , v) = ( 2 + 4 u + 2 v, 1 + u + 3 v) maps R 0 to P 1 . R 0 u 1 1 (4, 1) (6, 4) (2, 3) P 0 (6, 2) (2, 1) (8, 5) (4, 4) P 1 x y Φ 0 ( u , ) = (4 u + 2 , u + 3 ) R 0 u 1 1 x y Φ 1 ( u , ) = (2 + 4 u + 2 , 1 + u + 3 ) (3, 2) ( 1, 1) (1, 4) P 3 x y (6, 3) (2, 2) (4, 5) P 2 x y FIGURE 16 26. Find translates 2 and 3 of the mapping 0 in Figure 16 that map the unit square R 0 to the parallelograms P 2 and P 3 .

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16.5 Ex 26 - 28 - S E C T I O N 16.5 Change of Variables(ET...

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