16.5 Ex 26 - 28

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

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SECTION 16.5 Change of Variables (ET Section 15.5) 1003 25. With 8 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 (8) 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 ZZ 8( R ) 1 dA = R 1 | Jac (8) | dud v = R 1 | 2 u /v | v = Z 4 1 Z 4 1 2 u /v v = Z 4 1 2 udu · Z 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 8 0 ( u = ( φ ( u ,v), ψ ( u ,v)) is amap 8( u = ( a + ( u b + ( u where a , b are constants. Observe that the map 8 0 in Figure 16 maps R 0 to the parallelogram P 0 and the translate 8 1 ( u = ( 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 8 2 and 8 3 of the mapping 8 0 in Figure 16 that map the unit square R 0 to the parallelograms P 2 and P 3 . The parallelogram P 2 is obtained by translating P 0 two units upward and two units to the left. Therefore
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This homework help was uploaded on 04/13/2008 for the course MATH 32B taught by Professor Rogawski during the Winter '08 term at UCLA.

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

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