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16.5 Ex 5 - 17

16.5 Ex 5 - 17 - 996 C H A P T E R 16 M U LTI P L E I N T E...

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996 C H A P T E R 16 MULTIPLE INTEGRATION (ET CHAPTER 15) The image of the horizontal line v = c is the set of the points ( x , y ) = ( u , c ) = ( e u , e u + c ) x = e u , y = e c x y = e c x , x > 0 Since u can take any value, x can take any positive value, and hence the image is the ray y = e c x , x > 0. u Φ x c y = e c x y In Exercises 5–12, let ( u , v) = ( 2 u + v, 5 u + 3 v) be a map from the u v -plane to the xy-plane. 5. Show that the image of the horizontal line v = c is the line y = 5 2 x + 1 2 c . What is the image (in slope-intercept form) of the vertical line u = c ? SOLUTION The image of the vertical line u = c is the set of the following points: ( x , y ) = ( c , v) = ( 2 c + v, 5 c + 3 v) x = 2 c + v, y = 5 c + 3 v By the first equation, v = x 2 c . Substituting in the second equation gives y = 5 c + 3 ( x 2 c ) = 5 c + 3 x 6 c = 3 x c Therefore, the image of the line u = c is the line y = 3 x c in the xy -plane. The image of the horizontal line v = c is the set of the following points: ( x , y ) = ( u , c ) = ( 2 u + c , 5 u + 3 c ) x = 2 u + c , y = 5 u + 3 c The first equation implies u = x c 2 . Substituting in the second equation gives y = 5 ( x c ) 2 + 3 c = 5 x 2 + c 2 Therefore, the image of the line v = c is the line y = 5 x 2 + c 2 in the xy -plane. 6. Describe the image of the line through the points ( u , v) = ( 1 , 1 ) and ( u , v) = ( 1 , 1 ) under in slope-intercept form. SOLUTION 1 1 1 1 1 1 Φ x y u u = 1 y = 3 x 1 The line is the vertical line u = 1 in the u v -plane. The image of the line under the linear map ( u , v) = ( 2 u + v, 5 u + 3 v) is the line through the images of the points ( u , v) = ( 1 , 1 ) and ( u
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