# surface area

♦
4 matching pages ♦

(0.001 seconds)

## 4 matching pages

##### 1: 19.33 Triaxial Ellipsoids

…
►

###### §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

##### 3: 1.6 Vectors and Vector-Valued Functions

…
►The

*area*$A(S)$ of a parametrized smooth surface is given by … ►
1.6.52
$$A(S)=2\pi {\int}_{a}^{b}|f(x)|\sqrt{1+{({f}^{\prime}(x))}^{2}}dx,$$

…
►
1.6.53
$$A(S)=2\pi {\int}_{a}^{b}|x|\sqrt{1+{({f}^{\prime}(x))}^{2}}dx.$$

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

…
►
1.6.60
$${\iiint}_{V}(f{\nabla}^{2}g-g{\nabla}^{2}f)dV={\iint}_{S}\left(f\frac{\partial g}{\partial n}-g\frac{\partial f}{\partial n}\right)dA,$$

…
##### 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 $\mathbf{\Omega}$, 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)). …