surface area

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1: 19.33 Triaxial Ellipsoids

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§19.33(i) Surface Area

►The surface area of an ellipsoid with semiaxes $a,b,c$, and volume $V=4\pi abc/3$ is given by …

2: 5.19 Mathematical Applications

… ►The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by …

3: 1.6 Vectors and Vector-Valued Functions

… ►The area $A(S)$ of a parametrized smooth surface is given by … ►

1.6.52 $$A(S)=2\pi {\int }_a^b\left|f(x)\right|\sqrt{1+{(f^{\prime }(x))}^2}dx,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $\left|x\right|$: absolute value of $x$

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http://dlmf.nist.gov/1.6.E52

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See also:

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

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1.6.53 $$A(S)=2\pi {\int }_a^b\left|x\right|\sqrt{1+{(f^{\prime }(x))}^2}dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.6.E53

Encodings:

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Annotations for §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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4: 21.9 Integrable Equations

… ►Riemann theta functions arise in the study of integrable differential equations that have applications in many areas, including fluid mechanics (Ablowitz and Segur (1981, Chapter 4)), magnetic monopoles (Ercolani and Sinha (1989)), and string theory (Deligne et al. (1999, Part 3)). … ►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)): …Here $x$ and $y$ are spatial variables, $t$ is time, and $u(x,y,t)$ is the elevation of the surface wave. …These parameters, including $𝛀$, are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition $u(x,y,0)$ (Deconinck and Segur (1998)). … ►Furthermore, the solutions of the KP equation solve the Schottky problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)). …