Mathematic Methods HW Solutions 19

Mathematic Methods HW Solutions 19 - x 3-3 xy 2 ) / 3! e x...

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Chapter 4 1.1 ∂u/∂x = 2 xy 2 / ( x 2 + y 2 ) 2 , ∂u/∂y = - 2 x 2 y/ ( x 2 + y 2 ) 2 1.2 ∂s/∂t = ut u - 1 , ∂s/∂u = t u ln t 1.3 ∂z/∂u = u/ ( u 2 + v 2 + w 2 ) 1.4 At (0 , 0), both = 0; at ( - 2 / 3 , 2 / 3), both = - 4 1.5 At (0 , 0), both = 0; at (1 / 4 , ± 1 / 2), 2 w/∂x 2 = 6, 2 w/∂y 2 = 2 1.7 2 x 1.8 - 2 x 1.9 2 x (1 + 2 tan 2 θ ) 1.10 4 y 1.11 2 y 1.12 2 y (cot 2 θ + 2) 1.13 4 r 2 tan θ 1.14 - 2 r 2 cot θ 1.15 r 2 sin 2 θ 1.16 2 r (1 + sin 2 θ ) 1.17 4 r 1.18 2 r 1.19 0 1.20 8 y sec 2 θ 1.21 - 4 x csc 2 θ 1.22 0 1.23 2 r sin 2 θ 1.24 0 1.7 p - 2 y 4 /x 3 1.8 p - 2 r 4 /x 3 1.9 p 2 x tan 2 θ sec 2 θ 1.10 p 2 y + 4 y 3 /x 2 1.11 p 2 yr 4 / ( r 2 - y 2 ) 2 1.12 p 2 y sec 2 θ 1.13 p 2 x 2 sec 2 θ tan θ (sec 2 θ + tan 2 θ ) 1.14 p 2 y 2 sec 2 θ tan θ 1.15 p 2 r 2 tan θ sec 2 θ 1.16 p 2 r tan 2 θ 1.17 p 4 r 3 /x 2 - 2 r 1.18 p - 2 ry 4 / ( r 2 - y 2 ) 2 1.19 p - 8 r 3 y 3 / ( r 2 - y 2 ) 3 1.20 p 4 x tan θ sec 2 θ (tan 2 θ + sec 2 θ ) 1.21 p 4 y sec 2 θ tan θ 1.22 p - 8 r 3 /x 3 1.23 p 4 r tan θ sec 2 θ 1.24 p - 8 y 3 /x 3 2.1 y + y 3 / 6 - x 2 y/ 2 + x 4 y/ 24 - x 2 y 3 / 12 + y 5 / 120 + ··· 2.2 1 - ( x 2 + 2 xy + y 2 ) / 2 + ( x 4 + 4 x 3 y + 6 x 2 y 2 + 4 xy 3 + y 4 ) / 24 + ··· 2.3 x - x 2 / 2 - xy + x 3 / 3 + x 2 y/ 2 + xy 2 ··· 2.4 1 + xy + x 2 y 2 / 2 + x 3 y 3 / 3! + x 4 y 4 / 4! ··· 2.5 1 + 1 2 xy - 1 8 x 2 y 2 + 1 16 x 3 y 3 - 5 128 x 4 y 4 ··· 2.6 1 + x + y + ( x 2 + 2 xy + y 2 ) / 2 ··· 2.8 e x cos y = 1 + x + ( x 2 - y 2 ) / 2 + (
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Unformatted text preview: x 3-3 xy 2 ) / 3! e x sin y = y + xy + (3 x 2 y-y 3 ) / 3! 4.2 2 . 5 10-13 4.3 14 . 8 4.4 12 . 2 4.5 14 . 96 4.6 9% 4.7 15% 4.8 5% 4.10 4 . 28 nt 4.11 3 . 95 4.12 2 . 01 4.13 5 / 3 4.14 0 . 005 4.15 8 10 23 5.1 e-y sinh t + z sin t 5.2 w = 1, dw/dp = 0 5.3 2 r ( q 2-p 2 ) 5.4 (4 ut + 2 v sin t ) / ( u 2-v 2 ) 5.6 5( x + y ) 4 (1 + 10 cos10 x ) 5.7 (1-2 b-e 2 a ) cos( a-b ) 6.1 dv/dp =-v/ ( ap ), d 2 v/dp 2 = v (1 + a ) / ( a 2 p 2 ) 6.2 y p = 1, y pp = 0 6.3 y p = 4(ln 2-1) / (2 ln2-1) 19...
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