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11: 1.6 Vectors and Vector-Valued Functions
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1.6.19 = 𝐒 ⁒ x + 𝐣 ⁒ y + 𝐀 ⁒ z .
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1.6.20 grad ⁑ f = f = f x ⁒ 𝐒 + f y ⁒ 𝐣 + f z ⁒ 𝐀 .
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1.6.22 curl ⁑ 𝐅 = × π… = | 𝐒 𝐣 𝐀 x y z F 1 F 2 F 3 | = ( F 3 y F 2 z ) ⁒ 𝐒 + ( F 1 z F 3 x ) ⁒ 𝐣 + ( F 2 x F 1 y ) ⁒ 𝐀 .
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1.6.46 𝐓 u = x u ⁒ ( u 0 , v 0 ) ⁒ 𝐒 + y u ⁒ ( u 0 , v 0 ) ⁒ 𝐣 + z u ⁒ ( u 0 , v 0 ) ⁒ 𝐀
β–Ίwhere g / n = g 𝐧 is the derivative of g normal to the surface outwards from V and 𝐧 is the unit outer normal vector. …
12: 23.21 Physical Applications
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23.21.2 ( η ΢ ) ⁒ ( ΢ ξ ) ⁒ ( ξ η ) ⁒ 2 = ( ΢ η ) ⁒ f ⁑ ( ξ ) ⁒ f ⁑ ( ξ ) ⁒ ξ + ( ξ ΢ ) ⁒ f ⁑ ( η ) ⁒ f ⁑ ( η ) ⁒ η + ( η ξ ) ⁒ f ⁑ ( ΢ ) ⁒ f ⁑ ( ΢ ) ⁒ ΢ ,
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23.21.5 ( ⁑ ( v ) ⁑ ( w ) ) ⁒ ( ⁑ ( w ) ⁑ ( u ) ) ⁒ ( ⁑ ( u ) ⁑ ( v ) ) ⁒ 2 = ( ⁑ ( w ) ⁑ ( v ) ) ⁒ 2 u 2 + ( ⁑ ( u ) ⁑ ( w ) ) ⁒ 2 v 2 + ( ⁑ ( v ) ⁑ ( u ) ) ⁒ 2 w 2 .
13: 28.32 Mathematical Applications
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28.32.3 2 V ξ 2 + 2 V η 2 + 1 2 ⁒ c 2 ⁒ k 2 ⁒ ( cosh ⁑ ( 2 ⁒ ξ ) cos ⁑ ( 2 ⁒ η ) ) ⁒ V = 0 .
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28.32.4 2 K z 2 2 K ΢ 2 = 2 ⁒ q ⁒ ( cos ⁑ ( 2 ⁒ z ) cos ⁑ ( 2 ⁒ ΢ ) ) ⁒ K .
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28.32.5 K ⁑ ( z , ΢ ) ⁒ d u ⁑ ( ΢ ) d ΢ u ⁑ ( ΢ ) ⁒ K ⁑ ( z , ΢ ) ΢
14: 31.10 Integral Equations and Representations
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31.10.4 π’Ÿ z = z ⁒ ( z 1 ) ⁒ ( z a ) ⁒ ( 2 / z 2 ) + ( Ξ³ ⁒ ( z 1 ) ⁒ ( z a ) + Ξ΄ ⁒ z ⁒ ( z a ) + Ο΅ ⁒ z ⁒ ( z 1 ) ) ⁒ ( / z ) + Ξ± ⁒ Ξ² ⁒ z .
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31.10.5 p ⁑ ( t ) ⁒ ( 𝒦 t ⁒ w ⁑ ( t ) 𝒦 ⁒ d w ⁑ ( t ) d t ) | C = 0 ,
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31.10.8 sin 2 ⁑ ΞΈ ⁒ ( 2 𝒦 ΞΈ 2 + ( ( 1 2 ⁒ Ξ³ ) ⁒ tan ⁑ ΞΈ + 2 ⁒ ( Ξ΄ + Ο΅ 1 2 ) ⁒ cot ⁑ ΞΈ ) ⁒ 𝒦 ΞΈ 4 ⁒ Ξ± ⁒ Ξ² ⁒ 𝒦 ) + 2 𝒦 Ο• 2 + ( ( 1 2 ⁒ Ξ΄ ) ⁒ cot ⁑ Ο• ( 1 2 ⁒ Ο΅ ) ⁒ tan ⁑ Ο• ) ⁒ 𝒦 Ο• = 0 .
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31.10.18 2 𝒦 u 2 + 2 𝒦 v 2 + 2 𝒦 w 2 + 2 ⁒ Ξ³ 1 u ⁒ 𝒦 u + 2 ⁒ Ξ΄ 1 v ⁒ 𝒦 v + 2 ⁒ Ο΅ 1 w ⁒ 𝒦 w = 0 .
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31.10.21 2 𝒦 r 2 + 2 ⁒ ( Ξ³ + Ξ΄ + Ο΅ ) 1 r ⁒ 𝒦 r + 1 r 2 ⁒ 2 𝒦 ΞΈ 2 + ( 2 ⁒ ( Ξ΄ + Ο΅ ) 1 ) ⁒ cot ⁑ ΞΈ ( 2 ⁒ Ξ³ 1 ) ⁒ tan ⁑ ΞΈ r 2 ⁒ 𝒦 ΞΈ + 1 r 2 ⁒ sin 2 ⁑ ΞΈ ⁒ 2 𝒦 Ο• 2 + ( 2 ⁒ Ξ΄ 1 ) ⁒ cot ⁑ Ο• ( 2 ⁒ Ο΅ 1 ) ⁒ tan ⁑ Ο• r 2 ⁒ sin 2 ⁑ ΞΈ ⁒ 𝒦 Ο• = 0 .
15: 36.4 Bifurcation Sets
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s ⁑ Φ ( U ) ⁑ ( s j ⁒ ( 𝐱 ) , t j ⁑ ( 𝐱 ) ; 𝐱 ) = 0 ,
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t ⁑ Φ ( U ) ⁑ ( s j ⁒ ( 𝐱 ) , t j ⁑ ( 𝐱 ) ; 𝐱 ) = 0 .
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36.4.4 2 s 2 ⁑ Φ ( U ) ⁑ ( s , t ; 𝐱 ) ⁒ 2 t 2 ⁑ Φ ( U ) ⁑ ( s , t ; 𝐱 ) ( 2 s ⁒ t ⁒ Φ ( U ) ⁑ ( s , t ; 𝐱 ) ) 2 = 0 .
16: 36.12 Uniform Approximation of Integrals
β–ΊAlso, f is real analytic, and K + 2 f / u K + 2 > 0 for all 𝐲 such that all K + 1 critical points coincide. If K + 2 f / u K + 2 < 0 , then we may evaluate the complex conjugate of I for real values of 𝐲 and g , and obtain I by conjugation and analytic continuation. … β–Ί
36.12.2 u ⁑ f ⁑ ( u j ⁑ ( 𝐲 ) ; 𝐲 ) = 0 .
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36.12.10 G n ⁑ ( 𝐲 ) = g ⁑ ( t n ⁑ ( 𝐲 ) , 𝐲 ) ⁒ 2 Φ K ⁑ ( t n ⁑ ( 𝐱 ⁑ ( 𝐲 ) ) ; 𝐱 ⁑ ( 𝐲 ) ) / t 2 2 f ⁑ ( u n ⁑ ( 𝐲 ) ) / u 2 .
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f ± ′′ = 2 u 2 ⁑ f ⁑ ( u ± ⁑ ( y ) , y ) ,
17: 19.4 Derivatives and Differential Equations
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19.4.5 F ⁑ ( Ο• , k ) k = E ⁑ ( Ο• , k ) k 2 ⁒ F ⁑ ( Ο• , k ) k ⁒ k 2 k ⁒ sin ⁑ Ο• ⁒ cos ⁑ Ο• k 2 ⁒ 1 k 2 ⁒ sin 2 ⁑ Ο• ,
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19.4.7 Ξ  ⁑ ( Ο• , Ξ± 2 , k ) k = k k 2 ⁒ ( k 2 Ξ± 2 ) ⁒ ( E ⁑ ( Ο• , k ) k 2 ⁒ Ξ  ⁑ ( Ο• , Ξ± 2 , k ) k 2 ⁒ sin ⁑ Ο• ⁒ cos ⁑ Ο• 1 k 2 ⁒ sin 2 ⁑ Ο• ) .
β–ΊLet D k = / k . …
18: 14.11 Derivatives with Respect to Degree or Order
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14.11.1 Ξ½ ⁑ 𝖯 Ξ½ ΞΌ ⁑ ( x ) = Ο€ ⁒ cot ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) 1 Ο€ ⁒ 𝖠 Ξ½ ΞΌ ⁑ ( x ) ,
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14.11.2 Ξ½ ⁑ 𝖰 Ξ½ ΞΌ ⁑ ( x ) = 1 2 ⁒ Ο€ 2 ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) + Ο€ ⁒ sin ⁑ ( ΞΌ ⁒ Ο€ ) sin ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ sin ⁑ ( ( Ξ½ + ΞΌ ) ⁒ Ο€ ) ⁒ 𝖰 Ξ½ ΞΌ ⁑ ( x ) 1 2 ⁒ cot ⁑ ( ( Ξ½ + ΞΌ ) ⁒ Ο€ ) ⁒ 𝖠 Ξ½ ΞΌ ⁑ ( x ) + 1 2 ⁒ csc ⁑ ( ( Ξ½ + ΞΌ ) ⁒ Ο€ ) ⁒ 𝖠 Ξ½ ΞΌ ⁑ ( x ) ,
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19: 3.4 Differentiation
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§3.4(iii) Partial Derivatives
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3.4.21 u 0 , 0 x = 1 4 ⁒ h ⁒ ( u 1 , 1 u 1 , 1 + u 1 , 1 u 1 , 1 ) + O ⁑ ( h 2 ) .
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3.4.22 2 u 0 , 0 x 2 = 1 h 2 ⁒ ( u 1 , 0 2 ⁒ u 0 , 0 + u 1 , 0 ) + O ⁑ ( h 2 ) ,
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3.4.27 2 u = 2 u x 2 + 2 u y 2 .
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20: 32.4 Isomonodromy Problems
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𝚿 Ξ» = 𝐀 ⁑ ( z , Ξ» ) ⁒ 𝚿 ,
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𝚿 z = 𝐁 ⁑ ( z , λ ) ⁒ 𝚿 ,
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32.4.3 𝐀 z 𝐁 Ξ» + 𝐀 ⁒ 𝐁 𝐁 ⁒ 𝐀 = 0 .