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1: 1.5 Calculus of Two or More Variables
β–ΊThe function f ⁑ ( x , y ) is continuously differentiable if f , f / x , and f / y are continuous, and twice-continuously differentiable if also 2 f / x 2 , 2 f / y 2 , 2 f / x ⁒ y , and 2 f / y ⁒ x are continuous. … β–Ί
1.5.9 v ⁑ f ⁑ ( x ⁑ ( u , v ) , y ⁑ ( u , v ) , z ⁑ ( u , v ) ) = f x ⁒ x v + f y ⁒ y v + f z ⁒ z v .
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1.5.15 2 f = 2 f x 2 + 2 f y 2 + 2 f z 2 = 2 f r 2 + 1 r ⁒ f r + 1 r 2 ⁒ 2 f Ο• 2 + 2 f z 2 .
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1.5.38 ( f , g ) ( x , y ) = | f / x f / y g / x g / y | ,
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1.5.40 ( f , g , h ) ( x , y , z ) = | f / x f / y f / z g / x g / y g / z h / x h / y h / z | ,
2: 36.10 Differential Equations
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36.10.6 l ⁒ n Ψ K x m l ⁒ n = i n ⁒ ( l m ) ⁒ m ⁒ n Ψ K x l m ⁒ n , 1 m K , 1 l K .
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36.10.9 3 ⁒ n Ψ 3 x 3 ⁒ n = ( 1 ) n ⁒ n Ψ 3 z n ,
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36.10.10 3 ⁒ n Ψ 3 y 3 ⁒ n = i n ⁒ 2 ⁒ n Ψ 3 z 2 ⁒ n .
3: 19.18 Derivatives and Differential Equations
β–ΊLet j = / z j , and 𝐞 j be an n -tuple with 1 in the j th place and 0’s elsewhere. … β–Ί
19.18.6 ( x + y + z ) ⁒ R F ⁑ ( x , y , z ) = 1 2 ⁒ x ⁒ y ⁒ z ,
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19.18.14 2 w x 2 = 2 w y 2 + 1 y ⁒ w y .
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19.18.15 2 W t 2 = 2 W x 2 + 2 W y 2 .
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19.18.16 2 u x 2 + 2 u y 2 + 1 y ⁒ u y = 0 ,
4: 10.38 Derivatives with Respect to Order
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10.38.2 K Ξ½ ⁑ ( z ) Ξ½ = 1 2 ⁒ Ο€ ⁒ csc ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ ( I Ξ½ ⁑ ( z ) Ξ½ I Ξ½ ⁑ ( z ) Ξ½ ) Ο€ ⁒ cot ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ K Ξ½ ⁑ ( z ) , Ξ½ β„€ .
β–ΊFor I Ξ½ ⁑ ( z ) / Ξ½ at Ξ½ = n combine (10.38.1), (10.38.2), and (10.38.4). … β–Ί
I ν ⁑ ( z ) ν | ν = 0 = K 0 ⁑ ( z ) ,
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K ν ⁑ ( z ) ν | ν = 0 = 0 .
β–Ί
5: 16.14 Partial Differential Equations
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x ⁒ ( 1 x ) ⁒ 2 F 1 x 2 + y ⁒ ( 1 x ) ⁒ 2 F 1 x ⁒ y + ( γ ( α + β + 1 ) ⁒ x ) ⁒ F 1 x β ⁒ y ⁒ F 1 y α ⁒ β ⁒ F 1 = 0 ,
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x ⁒ ( 1 x ) ⁒ 2 F 2 x 2 x ⁒ y ⁒ 2 F 2 x ⁒ y + ( γ ( α + β + 1 ) ⁒ x ) ⁒ F 2 x β ⁒ y ⁒ F 2 y α ⁒ β ⁒ F 2 = 0 ,
β–Ί
y ⁒ ( 1 y ) ⁒ 2 F 2 y 2 x ⁒ y ⁒ 2 F 2 x ⁒ y + ( γ ( α + β + 1 ) ⁒ y ) ⁒ F 2 y β ⁒ x ⁒ F 2 x α ⁒ β ⁒ F 2 = 0 ,
β–Ί
x ⁒ ( 1 x ) ⁒ 2 F 4 x 2 2 ⁒ x ⁒ y ⁒ 2 F 4 x ⁒ y y 2 ⁒ 2 F 4 y 2 + ( γ ( α + β + 1 ) ⁒ x ) ⁒ F 4 x ( α + β + 1 ) ⁒ y ⁒ F 4 y α ⁒ β ⁒ F 4 = 0 ,
β–Ί
y ⁒ ( 1 y ) ⁒ 2 F 4 y 2 2 ⁒ x ⁒ y ⁒ 2 F 4 x ⁒ y x 2 ⁒ 2 F 4 x 2 + ( γ ( α + β + 1 ) ⁒ y ) ⁒ F 4 y ( α + β + 1 ) ⁒ x ⁒ F 4 x α ⁒ β ⁒ F 4 = 0 .
6: 10.15 Derivatives with Respect to Order
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10.15.1 J ± Ξ½ ⁑ ( z ) Ξ½ = ± J ± Ξ½ ⁑ ( z ) ⁒ ln ⁑ ( 1 2 ⁒ z ) βˆ“ ( 1 2 ⁒ z ) ± Ξ½ ⁒ k = 0 ( 1 ) k ⁒ ψ ⁑ ( k + 1 ± Ξ½ ) Ξ“ ⁑ ( k + 1 ± Ξ½ ) ⁒ ( 1 4 ⁒ z 2 ) k k ! ,
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10.15.2 Y Ξ½ ⁑ ( z ) Ξ½ = cot ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ ( J Ξ½ ⁑ ( z ) Ξ½ Ο€ ⁒ Y Ξ½ ⁑ ( z ) ) csc ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ J Ξ½ ⁑ ( z ) Ξ½ Ο€ ⁒ J Ξ½ ⁑ ( z ) .
β–Ί
10.15.3 J Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = n = Ο€ 2 ⁒ Y n ⁑ ( z ) + n ! 2 ⁒ ( 1 2 ⁒ z ) n ⁒ k = 0 n 1 ( 1 2 ⁒ z ) k ⁒ J k ⁑ ( z ) k ! ⁒ ( n k ) .
β–ΊFor J Ξ½ ⁑ ( z ) / Ξ½ at Ξ½ = n combine (10.2.4) and (10.15.3). … β–Ί
10.15.5 J Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = 0 = Ο€ 2 ⁒ Y 0 ⁑ ( z ) , Y Ξ½ ⁑ ( z ) Ξ½ | Ξ½ = 0 = Ο€ 2 ⁒ J 0 ⁑ ( z ) .
7: 12.17 Physical Applications
β–Ί
12.17.2 2 = 2 x 2 + 2 y 2 + 2 z 2
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12.17.4 1 ξ 2 + η 2 ⁒ ( 2 w ξ 2 + 2 w η 2 ) + 2 w ΢ 2 + k 2 ⁒ w = 0 .
8: 10.73 Physical Applications
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10.73.1 2 V = 1 r ⁒ r ⁑ ( r ⁒ V r ) + 1 r 2 ⁒ 2 V Ο• 2 + 2 V z 2 = 0 ,
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10.73.2 2 ψ = 1 c 2 ⁒ 2 ψ t 2 ,
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10.73.3 4 W + λ 2 ⁒ 2 W t 2 = 0 .
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10.73.4 ( 2 + k 2 ) ⁒ f = 1 ρ 2 ⁒ ρ ⁑ ( ρ 2 ⁒ f ρ ) + 1 ρ 2 ⁒ sin ⁑ ΞΈ ⁒ ΞΈ ⁑ ( sin ⁑ ΞΈ ⁒ f ΞΈ ) + 1 ρ 2 ⁒ sin 2 ⁑ ΞΈ ⁒ 2 f Ο• 2 + k 2 ⁒ f .
9: 20.13 Physical Applications
β–Ί
20.13.1 ΞΈ ⁑ ( z | Ο„ ) / Ο„ = ΞΊ ⁒ 2 ΞΈ ⁑ ( z | Ο„ ) / z 2 ,
β–Ί
20.13.2 θ / t = α ⁒ 2 θ / z 2 ,
10: 30.13 Wave Equation in Prolate Spheroidal Coordinates
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30.13.3 h ξ 2 = ( x ξ ) 2 + ( y ξ ) 2 + ( z ξ ) 2 = c 2 ⁒ ( ξ 2 η 2 ) ξ 2 1 ,
β–Ί β–Ί β–Ί
30.13.6 2 = 1 h ΞΎ ⁒ h Ξ· ⁒ h Ο• ⁒ ( ΞΎ ⁑ ( h Ξ· ⁒ h Ο• h ΞΎ ⁒ ΞΎ ) + Ξ· ⁑ ( h ΞΎ ⁒ h Ο• h Ξ· ⁒ Ξ· ) + Ο• ⁑ ( h ΞΎ ⁒ h Ξ· h Ο• ⁒ Ο• ) ) = 1 c 2 ⁒ ( ΞΎ 2 Ξ· 2 ) ⁒ ( ΞΎ ⁑ ( ( ΞΎ 2 1 ) ⁒ ΞΎ ) + Ξ· ⁑ ( ( 1 Ξ· 2 ) ⁒ Ξ· ) + ΞΎ 2 Ξ· 2 ( ΞΎ 2 1 ) ⁒ ( 1 Ξ· 2 ) ⁒ 2 Ο• 2 ) .