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1: 1.6 Vectors and Vector-Valued Functions
1.6.54 S f ( x , y , z ) d S = D f ( Φ ( u , v ) ) T u × T v d u d v .
1.6.55 S F d S = D F ( T u × T v ) d u d v ,
1.6.56 Φ 1 ( D 1 ) F d S = Φ 2 ( D 2 ) F d S ;
1.6.57 S ( × F ) d S = S F d s ,
1.6.58 V ( F ) d V = S F d S ,
2: Bibliography Q
  • 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.
  • 3: 18.37 Classical OP’s in Two or More Variables
    18.37.2 x 2 + y 2 < 1 R m , n ( α ) ( x + i y ) R j , ( α ) ( x - i y ) ( 1 - x 2 - y 2 ) α d x d y = 0 , m j and/or n .
    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 ) ;
    18.37.8 0 < y < x < 1 P m , n α , β , γ ( x , y ) P j , α , β , γ ( x , y ) ( 1 - x ) α ( x - y ) β y γ d x d y = 0 , m j and/or n .
    4: 1.5 Calculus of Two or More Variables
    §1.5(v) Multiple Integrals
    Double Integrals
    Infinite Double Integrals
    Change of Variables
    5: 16.15 Integral Representations and Integrals
    16.15.3 F 3 ( α , α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β - β ) Δ u β - 1 v β - 1 ( 1 - u - v ) γ - β - β - 1 ( 1 - u x ) α ( 1 - v y ) α d u d v , ( γ - β - β ) > 0 , β > 0 , β > 0 ,
    For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). …
    6: Bibliography U
  • 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: 3.5 Quadrature
    3.5.47 1 π h 2 D f ( x , y ) d x d y = j = 1 n w j f ( x j , y j ) + R ,
    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 .
    8: 10.71 Integrals
    For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631). …
    9: 35.10 Methods of Computation
    Other methods include numerical quadrature applied to double and multiple integral representations. …
    10: 25.6 Integer Arguments
    25.6.7 ζ ( 2 ) = 0 1 0 1 1 1 - x y d x d y .