A7_soln - Math 237 Assignment 7 Solutions 1. Consider the...

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Unformatted text preview: Math 237 Assignment 7 Solutions 1. Consider the following maps T : R 2 → R 2 . Find the image under T of the rectangle D = { ( x, y ) ∈ R 2 | 1 ≤ x ≤ 3 , 2 ≤ y ≤ 3 } . a) T ( x, y ) = ( x + y, x − y ). Solution: We have u = x + y and v = x − y , so For the line x = 1, 2 ≤ y ≤ 3 we get u = 1 + y so 2 ≤ y ≤ 3 ⇒ 3 ≤ u ≤ 4 and v = 1 − y = 1 − ( u − 1) = 2 − u . For the line x = 3, 2 ≤ y ≤ 3 we get u = 3 + y so 2 ≤ y ≤ 3 ⇒ 5 ≤ u ≤ 6 and v = 3 − y = 3 − ( u − 3) = 6 − u . For the line y = 2, 1 ≤ x ≤ 3 we get u = x + 2 so 1 ≤ x ≤ 3 ⇒ 3 ≤ u ≤ 5 and v = x − 2 = ( u − 2) − 2 = u − 4. For the line y = 3, 1 ≤ x ≤ 3 we get u = x + 3 so 1 ≤ x ≤ 3 ⇒ 4 ≤ u ≤ 6 and v = x − 3 = ( u − 3) − 3 = u − 6. 2 b) T ( x, y ) = ( x cos( πy/ 3) , x sin( πy/ 3)). Solution: We have u = x cos πy/ 3 and v = x sin πy/ 3. Thus, we observe that u 2 + v 2 = x 2 For the line x = 1, 2 ≤ y ≤ 3 we have u 2 + v 2 = 1 and when y = 2 we get u = cos 2 π 3 = − 1 2 and v = sin 2 π 3 = √ 3 2 and when y = 3 we get u = − 1 and v = 0. Hence this is the arc of the circle u 2 + v 2 = 1 from ( − 1 2 , √ 3 2 ) to ( − 1 , 0). For the line x = 3, 2 ≤ y ≤ 3 we have u 2 + v 2 = 4 and when y = 2 we get u = 3 cos 2 π 3 = − 3 2 and v = 3 sin 2 π 3 = 3 √ 3 2 and when...
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This note was uploaded on 10/21/2010 for the course MATHEMATIC stats taught by Professor Various during the Spring '10 term at Waterloo.

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A7_soln - Math 237 Assignment 7 Solutions 1. Consider the...

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