a4f08 - dV z y x x x z y x > > + + +...

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ASSIGNMENT #4 1. Prove that the linear mapping maps straight lines into straight lines. 2 2 : R R T Under what condition on T , parallelograms are mapped onto parallelograms? 2. Find a linear mapping T such that the image of the trapezoidal region with vertices ) 2 , 0 ( ), 1 , 0 ( ), 0 , 2 ( ), 0 , 1 ( is again a trapezoidal region } , 2 1 : ) , {( v u v v v u S = . 3. Let be defined by 2 2 2 1 : R S R S G ) , ( ) , ( ) , ( u uv v u y x = = G , where . ] 1 , 0 [ ] 1 , 0 [ 1 × = S (a) Is G one-to-one? If not, can we eliminate some subset (having zero content) of so that on the remainder G is one-to-one? 1 S (b) Determine the image set . 2 S 4. Evaluate where D is the region bounded by the ellipse using the transformation dA y xy x D ) ( 2 2 + ∫∫ 2 2 2 = + y xy x v u y v u x 3 / 2 2 , 3 / 2 2 + = = . 5. Prove, using problem #6 (assignment 2), that for a one-to-one, map , defined by 1 C 2 2 : R R G ) , ( ), , ( v u y y v u x x = = we have ) , ( ) , ( 1 ) , ( ) , ( y x v u v u y x = (similar statement is true in n R ). 6. Find the volume of the region bounded by the surface 1 = + + z y x and the coordinate planes using the transformation defined by . 3 3 : R R G ) , , ( ) , , ( 2 2 2 w v u w v u = G 7. Determine whether the integral
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Unformatted text preview: dV z y x x x z y x > > + + + + , 1 2 / 5 2 2 2 2 3 2 2 2 ) ( converges. 8. Evaluate dA y x e R y x + + 2 2 2 ) ( ) ( 1 . HINT: Take squares as the sequence and apply suitable change of variables. } { j U 9. Evaluate dy y y 1 ln 1 . HINT: Define dy y y x F x = 1 ln 1 ) ( , x > 0. Evaluate (improper integral, modify and apply Th.4.47) and then integrate to get . ) ( x F ) ( x F 10. Verify that the improper integrals dx x 3 3 ) 1 ( 1 and dx x 1 1 2 1 diverge. Show that the first integral has a principal value and the second does not. HINT: set 2 1 = . Knowing: all the definitions, precise formulation of all the Theorems done on the lecture and the proofs of the few Theorems listed below is also part of the Assignment. 3.18, 4.46, 4.47, 4.65, 4.66 ....
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.

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