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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 π a b c / 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 π a b | f ( x ) | 1 + ( f ( x ) ) 2 d x ,
1.6.53 A ( S ) = 2 π a b | x | 1 + ( f ( x ) ) 2 d x .
1.6.59 V ( f 2 g + f g ) d V = S f g n d A ,
1.6.60 V ( f 2 g - g 2 f ) d V = S ( f g n - g f n ) d A ,
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)). …