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Homework6

# Homework6 - ables to cylindrical coordinates for Z 1-1 Z...

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1 Mathematics 1c: Homework Set 6 Due: Monday, May 17 at 10am. 1. (10 Points) Section 6.1, Exercise 6 Let D * be the parallelogram with vertices ( - 1 , 3) , (0 , 0) , (2 , - 1) and (1 , 2) and D be the rectangle D = [0 , 1] × [0 , 1] . Find a transformation T such that D is the image set of D * under T . 2. (10 Points) Section 6.2, Exercise 6 Deﬁne T ( u,v ) = ( u 2 - v 2 , 2 uv ) . Let D * be the set of ( u,v ) with u 2 + v 2 1 ,u 0 ,v 0 . Find T ( D * ) = D and evaluate ZZ D dxdy. 3. (10 Points) Section 6.2, Exercise 8 Calculate ZZ R dxdy x + y , where R is the region bounded by x = 0 ,y = 0 ,x + y = 1 , and x + y = 4 by using the mapping T ( u,v ) = ( u - uv,uv ) . 4. (10 Points) Section 6.3, Exercise 4 Find the center of mass of the region between y = 0 and y = x 2 , where 0 x 1 / 2 . 5. (10 Points) Section 6.4, Exercise 8 Show that the integral Z 1 0 Z a 0 x p a 2 - y 2 dy dx exists, and compute its value. (You may assume that a is a positive constant). 6. (10 Points) Review Exercise 4b for Chaper 6 Perform a change of vari-

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Unformatted text preview: ables to cylindrical coordinates for Z 1-1 Z √ 1-y 2-√ 1-y 2 Z √ 4-x 2-y 2-√ 4-x 2-y 2 xyz dz dxdy. 7. (10 Points) Section 7.1, Exercise 4(a) Evaluate the path integral of f ( x,y,z ) = x cos z along the path c : t 7→ t i + t 2 j ,t ∈ [0 , 1]. 8. (10 Points) Section 7.2, Exercise 2 Evaluate each of the following integrals: (a) Z c xdy-y dx, c ( t ) = (cos t, sin t ) , ≤ t ≤ 2 π (b) Z c xdx + y dy, c ( t ) = (cos πt, sin πt ) , ≤ t ≤ 2 2 (c) Z c yz dx + xz dy + xy dz , where c consists of straight-line segments join-ing (1 , , 0) to (0 , 1 , 0) to (0 , , 1) (d) Z c x 2 dx-xy dy + dz , where c is the parabola z = x 2 ,y = 0 from (-1 , , 1) to (1 , , 1) ....
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Homework6 - ables to cylindrical coordinates for Z 1-1 Z...

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