SpSolHw7 - MATH-4600 Homework VII Solutions to Graded...

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Unformatted text preview: MATH-4600 Homework VII Solutions to Graded Problems 1. 6.2 #1 F (x, y) = -x2 y, xy 2 , D = x2 + y 2 4 (a) C1 : x = 4 - t2 , y = t, -2 t 2 C2 : x = - 4 - t2 , y = t, -2 t 2 M dx + N dy = D D -x2 ydx + xy 2 dy -x2 ydx + xy 2 dy C2 = C1 -x2 ydx + xy 2 dy + 4 - t2 t+ 4 - t2 t+ 4 - t2 t2 dt+ C2 = C1 2 -1/2 -1/2 -2 1/2 -2 4 - t2 t- 4 - t2 t- -2 2 4 - t2 t2 dt 4 - t2 t2 dt = 4 - t2 t2 dt+ 2 2 1/2 2 = 4(arcsin( ) t 2 + 4(arcsin( ) t -2 = 4(arcsin(1) + arcsin(-1)) + 4(arcsin(-1) + arcsin(1)) = 8 (b) 2 (Nx - My )dA = D 2 -2 - 4-y 2 (y 2 + x2 )dxdy 4-y 2 x3 = y2 x + 3 -2 2 - 4-y 2 dy 4-y 2 =2 -2 y2 3 1 4 - y 2 + (4 - y 2 ) 2 dy 3 y = 2(4 arcsin( )) 2 2 -2 = 2(4(arcsin(1) + arcsin(-1))) = 2(4) = 8 1 2. 6.2 #2 F (x, y) = x2 - y, x + y 2 , Drectangle = [0, 2] [0, 1] (a) C1 : x = t, y = 1, 0 t 2 C2 : x = 2, y = t, 0 t 1 C3 : x = t, y = 0, 0 t 2 C4 : x = 0, y = t, 0 t 1 0 1 2 0 M dx + N dy = D 2 (t2 - 1)dt + 0 0 (2 + t2 )dt + 0 1 t2 dt + 1 0 1 t2 dt = t3 -t 3 2t + 2 t3 3 + 0 t2 3 2 + 0 t3 3 8 1 8 1 = (0 - 0) - ( - 2) + (2 + ) - (0 + 0) + ( ) - (0) + (0) - ( ) 3 3 3 3 =4 (b) 1 2 1 2 (Nx - My )dA = D 1 0 0 2 (1 + 1)dxdy = 0 1 0 2dxdy = 0 2x|0 dy = = 4y|0 = 4 1 4dy 0 3. 6.2 #7 1 1 (x2 - y 2 )dx + (x2 + y 2 )dy = - C 1 0 1 0 (2x + 2y)dxdy =- 0 x2 + 2xy dy 0 1 0 =- 4. 6.2 #19 (1 + 2y)dy = -(y + y 2 ) = -2 3x2 ydx + x3 dy = - C D d(x3 ) d(3x2 y) - dx dy dxdy =- D (3x2 - 3x2 )dxdy = - D (0)dxdy = 0 2 5. 6.2 #24 Let f (x, y) be such that d2 f d2 f + 2 =0 dx2 dy df df dx - dy = dy dx = D df df d dy d dx - dx dy dxdy C D d2 f d2 f + 2 dx2 dy dxdy = D (0)dxdy = 0 3 ...
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SpSolHw7 - MATH-4600 Homework VII Solutions to Graded...

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