triple integrals

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1: 1.5 Calculus of Two or More Variables

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Triple Integrals

… ►In case of triple integrals the $(x,y,z)$ sets are of the form … ►

1.5.43 $${\iiint }_{D}f(x,y,z)dxdydz={\iiint }_{D^{\ast }}f(x(u,v,w),y(u,v,w),z(u,v,w))\left|\frac{\partial (x,y,z)}{\partial (u,v,w)}\right|dudvdw.$$

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Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $w$: variable and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.5.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(vi), §1.5(vi), §1.5 and Ch.1

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2: 1.6 Vectors and Vector-Valued Functions

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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:

pMML, png, TeX

See also:

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

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1.6.59 $${\iiint }_V(f{\nabla }^{2}g+\nabla f\cdot \nabla g)dV={\iint }_{S}f\frac{\partial g}{\partial n}dA,$$

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Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E59

Encodings:

pMML, png, TeX

See also:

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

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1.6.60 $${\iiint }_V(f{\nabla }^{2}g-g{\nabla }^{2}f)dV={\iint }_S\left(f\frac{\partial g}{\partial n}-g\frac{\partial f}{\partial n}\right)dA,$$

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Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E60

Encodings:

pMML, png, TeX

See also:

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

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3: 22.4 Periods, Poles, and Zeros

… ►This half-period will be plus or minus a member of the triple $K,\mathrm{i}{K}^{\prime },K+\mathrm{i}{K}^{\prime }$; the other two members of this triple are quarter periods of $\mathrm{p}\mathrm{q}(z,k)$. …

4: 10.43 Integrals

… ►For infinite integrals of triple products of modified and unmodified Bessel functions, see Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

5: 7.13 Zeros

… ►At $z=0$, $C\left(z\right)$ has a simple zero and $S\left(z\right)$ has a triple zero. …

6: 10.22 Integrals

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Triple Products

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7: Bibliography R

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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External Links:

ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

Referenced by:

§34.6.

Encodings:

BibTeX

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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Referenced by:

§7.25(vi).

Encodings:

BibTeX

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