SpSolHw9 - MATH-4600 Homework IX Solutions to Graded...

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Unformatted text preview: MATH-4600 Homework IX Solutions to Graded Problems 1. 7.2 #1 - (s, t) = (s, s + t, t): 0 s 1, 0 t 2 x Evaluate x2 + y 2 + z 2 ds - x Ts (s, t) = (1, 1, 0) , Tt = (0, 1, 1) i 1 0 j 1 1 k 0 1 =i+j+k 3 3dsdt = Ts Tt = Ts Tt = 2 1 - x x2 + y 2 + z 2 ds = 0 0 2s2 + 2st + 2t2 1 = 2 3 0 2s3 + ts2 + 2t2 s 3 2t3 2t 2t2 + + 3 4 3 16 4 +2+ 3 3 = dt 0 2 0 = = 2. 7.2 #10 3 3 26 3 3 zdS S Ts Tt = i -3 sin(s) 0 j k 3 cos(s) 0 0 1 Ts Tt = 3 = 3 cos(s)i + 3 sin(s)j 2 4 IL = 0 0 (3t) dsdt = 6t2 2 4 = 48 0 Ts Tt = i j k -r sin(s) r cos(s) 0 cos(s) sin(s) 0 Ts Tt = r 2 3 = -rk IT = 0 0 (4r) drdt = 8r2 2 3 = 36 0 f dS = IL + IT = 48 + 36 = 84 S 1 3. 7.2 #14 (xi + yj)dS S Ts Tt = 4 0 0 2 i -3 sin(s) 0 j k 3 cos(s) 0 0 1 = 3 cos(s)i + 3 sin(s)j (3 cos(s), 3 sin(s), 0) (3 cos(s), 3 sin(s), 0) dsdt 4 2 = 0 0 9dsdt = (4)(2)(9) = 72 4. 7.3 #1 F = xz, yz, x2 + y 2 Verify S , S : x2 + y 2 + 5z = 1, z 0 ( S F ds = F) dS (a) Since z = dz = 0 on S, it follows that xz, yz, x2 + y 2 (dx, dy, dz) = S S 0, 0, x2 + y 2 (dx, dy, 0) = 0 (b) 1 1 - x2 - y 2 5 2 2 dS = (-gx (x, y), -gy (x, y), 1) dA = x, y, 1 dA 5 5 F = (y, -x, 0) , z = g(x, y) = ( F)dS = S R (y, -x, 0) 2 2 x, y, 1 dA = 5 5 16 - y 2 - z 2 , z 0 ( S R 2 2 xy - xy dA = 0 5 5 5. 7.3 #3 F = (x, y, z) , S : x = Verify S F ds = F) dS (a) x = 4 cos(t) sin(s) , y = 4 sin(t) sin(s) , z = 4 cos(s): 0 t 2, 0 s We know that at the boundary s = 2 2. F ds = S 2 0 (4 cos(t), 4 sin(t), 0) (-4 sin(t), 4 cos(t), 0) dt = 2 (-16 sin(t) cos(t) + 16 sin(t) cos(t)) dt = 0 0 (0) = 0 (b) F = (0, 0, 0) (0, 0, 0) dS = S R (0)dA = 0 2 ...
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This note was uploaded on 04/07/2008 for the course MATH 4600 taught by Professor Boudjelhka during the Spring '08 term at Rensselaer Polytechnic Institute.

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SpSolHw9 - MATH-4600 Homework IX Solutions to Graded...

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