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1: 1.6 Vectors and Vector-Valued Functions
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1.6.54 ∬ S f ⁑ ( x , y , z ) ⁒ d S = ∬ D f ⁑ ( 𝚽 ⁑ ( u , v ) ) ⁒ β€– 𝐓 u × π“ v β€– ⁒ d u ⁒ d v .
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1.6.55 ∬ S 𝐅 d 𝐒 = ∬ D 𝐅 ( 𝐓 u × π“ v ) ⁒ d u ⁒ d v ,
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1.6.56 ∬ 𝚽 1 ⁑ ( D 1 ) 𝐅 d 𝐒 = ∬ 𝚽 2 ⁑ ( D 2 ) 𝐅 d 𝐒 ;
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1.6.57 ∬ S ( × π… ) d 𝐒 = S 𝐅 d 𝐬 ,
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1.6.58 ∭ V ( 𝐅 ) ⁒ d V = ∬ S 𝐅 d 𝐒 ,
2: 1.5 Calculus of Two or More Variables
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§1.5(v) Multiple Integrals
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Double Integrals
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Infinite Double Integrals
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Change of Variables
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3: Bibliography Q
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  • W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
  • 4: 18.37 Classical OP’s in Two or More Variables
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    18.37.2 ∬ x 2 + y 2 < 1 R m , n ( Ξ± ) ⁑ ( x + i ⁒ y ) ⁒ R j , β„“ ( Ξ± ) ⁑ ( x i ⁒ y ) ⁒ ( 1 x 2 y 2 ) Ξ± ⁒ d x ⁒ d y = 0 , m j and/or n β„“ .
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    18.37.4 ∬ x 2 + y 2 < 1 R m , n ( α ) ⁑ ( x + i ⁒ y ) ⁒ ( x i ⁒ y ) m j ⁒ ( x + i ⁒ y ) n j ⁒ ( 1 x 2 y 2 ) α ⁒ d x ⁒ d y = 0 , j = 1 , 2 , , min ⁑ ( m , n ) ;
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    18.37.8 ∬ 0 < y < x < 1 P m , n Ξ± , Ξ² , Ξ³ ⁑ ( x , y ) ⁒ P j , β„“ Ξ± , Ξ² , Ξ³ ⁑ ( x , y ) ⁒ ( 1 x ) Ξ± ⁒ ( x y ) Ξ² ⁒ y Ξ³ ⁒ d x ⁒ d y = 0 , m j and/or n β„“ .
    5: 16.15 Integral Representations and Integrals
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    16.15.3 F 3 ⁑ ( Ξ± , Ξ± ; Ξ² , Ξ² ; Ξ³ ; x , y ) = Ξ“ ⁑ ( Ξ³ ) Ξ“ ⁑ ( Ξ² ) ⁒ Ξ“ ⁑ ( Ξ² ) ⁒ Ξ“ ⁑ ( Ξ³ Ξ² Ξ² ) ⁒ ∬ Ξ” u Ξ² 1 ⁒ v Ξ² 1 ⁒ ( 1 u v ) Ξ³ Ξ² Ξ² 1 ( 1 u ⁒ x ) Ξ± ⁒ ( 1 v ⁒ y ) Ξ± ⁒ d u ⁒ d v , ⁑ ( Ξ³ Ξ² Ξ² ) > 0 , ⁑ Ξ² > 0 , ⁑ Ξ² > 0 ,
    β–ΊFor these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). …
    6: Bibliography U
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  • F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.
  • 7: 35.10 Methods of Computation
    β–ΊOther methods include numerical quadrature applied to double and multiple integral representations. …
    8: 3.5 Quadrature
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    3.5.47 1 Ο€ ⁒ h 2 ⁒ ∬ D f ⁒ ( x , y ) ⁒ d x ⁒ d y = j = 1 n w j ⁒ f ⁒ ( x j , y j ) + R ,
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    3.5.48 1 4 ⁒ h 2 ⁒ ∬ S f ⁒ ( x , y ) ⁒ d x ⁒ d y = j = 1 n w j ⁒ f ⁒ ( x j , y j ) + R .
    9: 10.71 Integrals
    β–ΊFor infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631). …
    10: 25.6 Integer Arguments
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    25.6.7 ΢ ⁑ ( 2 ) = 0 1 0 1 1 1 x ⁒ y ⁒ d x ⁒ d y .