double integrals

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1: 1.6 Vectors and Vector-Valued Functions

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1.6.54 $${\iint }_{S}f(x,y,z)dS={\iint }_{D}f(𝚽(u,v))\Vert {𝐓}_u\times {𝐓}_v\Vert dudv.$$

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Symbols:

$dx$: differential of $x$ , $\Vert 𝐯\Vert$: vector norm ($l$²), $S$: parameterized surface, $𝚽(x,y,z)$: parameterization and $D$: open set in the plane

Permalink:

http://dlmf.nist.gov/1.6.E54

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See also:

Annotations for §1.6(v), §1.6 and Ch.1

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1.6.55 $${\iint }_S𝐅\cdot d𝐒={\iint }_D𝐅\cdot ({𝐓}_u\times {𝐓}_v)dudv,$$

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Symbols:

$dx$: differential of $x$ , $S$: parameterized surface and $D$: open set in the plane

Permalink:

http://dlmf.nist.gov/1.6.E55

Encodings:

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See also:

Annotations for §1.6(v), §1.6 and Ch.1

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1.6.56 $${\iint }_{{𝚽}_1(D_1)}𝐅\cdot d𝐒={\iint }_{{𝚽}_2(D_2)}𝐅\cdot d𝐒;$$

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Symbols:

$dx$: differential of $x$ , $𝚽(x,y,z)$: parameterization and $D$: open set in the plane

Permalink:

http://dlmf.nist.gov/1.6.E56

Encodings:

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See also:

Annotations for §1.6(v), §1.6 and Ch.1

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1.6.57 $${\iint }_S(\nabla \times 𝐅)\cdot d𝐒={\int }_{\partial S}𝐅\cdot d𝐬,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\partial x$: partial differential of $x$ and $S$: parameterized surface

Referenced by:

§1.6(v), Erratum (V1.2.1) for Subsection 1.6(v)

Permalink:

http://dlmf.nist.gov/1.6.E57

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Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

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1.6.58 $${\iiint }_V(\nabla \cdot 𝐅)dV={\iint }_S𝐅\cdot d𝐒,$$

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Symbols:

$dx$: differential of $x$ , $S$: parameterized surface and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E58

Encodings:

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See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

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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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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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4: 18.37 Classical OP’s in Two or More Variables

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18.37.2 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $\mathrm{i}$: imaginary unit, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E2

Encodings:

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See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.4 $j=1,2,\mathrm{\dots },\min (m,n)$;

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Symbols:

$dx$: differential of $x$ , $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $\mathrm{i}$: imaginary unit, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E4

Encodings:

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See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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18.37.8 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$: triangle polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E8

Encodings:

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See also:

Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

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5: 37.8 Jacobi Polynomials Associated with Root System $B{C}_2$

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37.8.3

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Symbols:

$dx$: differential of $x$ , $(a,b)$: open interval, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Proved:

Sprinkhuizen-Kuyper (1976, Theorem 8.2); Rewrite the Theorem there in terms of $x,y$, where $u=x+y$, $v=xy$.(proved)

Permalink:

http://dlmf.nist.gov/37.8.E3

Encodings:

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See also:

Annotations for §37.8, Part 37.II and Ch.37

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37.8.4 $(k,n)\ne (j,m)$.

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Symbols:

$dx$: differential of $x$ , $(a,b)$: open interval, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Source:

Koornwinder (1974b, (3.14)); Rewrite the formula there in terms of $x,y$, where $u=x+y$, $v=xy$.

Permalink:

http://dlmf.nist.gov/37.8.E4

Encodings:

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See also:

Annotations for §37.8, §37.8, Part 37.II and Ch.37

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6: 16.15 Integral Representations and Integrals

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16.15.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)\mathrm{\Gamma }\left(\gamma -\beta -{\beta }^{\prime }\right)}{\iint }_{\mathrm{\Delta }}\frac{u^{\beta -1}{v}^{{\beta }^{\prime }-1}{(1-u-v)}^{\gamma -\beta -{\beta }^{\prime }-1}}{{(1-ux)}^{\alpha }{(1-vy)}^{{\alpha }^{\prime }}}dudv,$$ $\mathrm{\Re }\left(\gamma -\beta -{\beta }^{\prime }\right)>0$, $\mathrm{\Re }\beta >0$, $\mathrm{\Re }{\beta }^{\prime }>0$,

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Symbols:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Notes:

Erdélyi et al. (1953a, 5.8.1(3)) contains a typo.

Referenced by:

Erratum (V1.0.14) for Equation (16.15.3), Erratum (V1.0.14) for Equation (16.15.3)

Permalink:

http://dlmf.nist.gov/16.15.E3

Encodings:

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Clarification (effective with 1.0.14):

A stray “$f$” was removed; it appeared after the comma at the end of this equation.

See also:

Annotations for §16.15 and Ch.16

… ►For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8). …

7: Bibliography U

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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.

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External Links:

ISSN 0305-0041, Document, MathReview (F. I. Ibragimov), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

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8: 35.10 Methods of Computation

… ►Other methods include numerical quadrature applied to double and multiple integral representations. …

9: 3.5 Quadrature

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3.5.47 $$\frac{1}{\pi h^2}{\iint }_{D}f(x,y)dxdy=\sum _{j=1}^n{w}_{j}f(x_j,y_j)+R,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $(a,b)$: open interval, $D$: disk, $R$: remainder and $w_k$: weights

A&S Ref:

25.4.61

Permalink:

http://dlmf.nist.gov/3.5.E47

Encodings:

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See also:

Annotations for §3.5(x), §3.5 and Ch.3

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3.5.48 $$\frac{1}{4h^2}{\iint }_{S}f(x,y)dxdy=\sum _{j=1}^n{w}_{j}f(x_j,y_j)+R.$$

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Symbols:

$dx$: differential of $x$ , $(a,b)$: open interval, $R$: remainder and $w_k$: weights

A&S Ref:

25.4.62

Permalink:

http://dlmf.nist.gov/3.5.E48

Encodings:

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See also:

Annotations for §3.5(x), §3.5 and Ch.3

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10: 10.71 Integrals

… ►For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631). …