surface%20area
(0.001 seconds)
1—10 of 172 matching pages
1: 21.7 Riemann Surfaces
§21.7 Riemann Surfaces
►§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
… ►For example, Figure 21.7.1 depicts a genus 2 surface. … ►§21.7(iii) Frobenius’ Identity
… ►The genus of this surface is . …2: 21.10 Methods of Computation
…
►
•
►
•
►
•
§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
►In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix originating from a Riemann surface. … ►Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.
3: 1.6 Vectors and Vector-Valued Functions
…
►
…
►
§1.6(v) Surfaces and Integrals over Surfaces
►A parametrized surface is defined by … ► ►The area of a parametrized smooth surface is given by …4: 21.9 Integrable Equations
…
►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 and are spatial variables, is time, and is the elevation of the surface wave.
…These parameters, including , are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition (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)).
Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4).
…
5: Publications
…
►
B. V. Saunders and Q. Wang (2000)
From 2D to 3D: Numerical Grid Generation and the Visualization of Complex Surfaces,
Proceedings of the
7th International Conference on Numerical Grid Generation in Computational Field Simulations,
Whistler, British Columbia, Canada,
September 25-28, 2000.
…
►
B. V. Saunders and Q. Wang (2006)
From B-Spline Mesh Generation to Effective Visualizations for the
NIST Digital Library of Mathematical Functions,
in Curve and Surface Design, Proceedings of the Sixth International
Conference on Curves and Surfaces,
Avignon, France June 29–July 5, 2006,
pp. 235–243.
…
►
B. Saunders and Q. Wang (2010)
Tensor Product B-Spline Mesh Generation for Accurate Surface Visualizations
in the NIST Digital Library of Mathematical Functions,
in Mathematical Methods for Curves and Surfaces, Proceedings of the 2008 International
Conference on Mathematical Methods for Curves and Surfaces (MMCS 2008), Lecture Notes in Computer
Science, Vol. 5862, (M. Dæhlen, M. Floater., T. Lyche, J. L. Merrien, K. Mørken, L. L. Schumaker, eds),
Springer, Berlin, Heidelberg (2010) pp. 385–393.
…
►
B. I. Schneider, B. R. Miller and B. V. Saunders (2018)
NIST’s Digital Library of Mathematial Functions,
Physics Today
71, 2, 48 (2018), pp. 48–53.
6: 19.33 Triaxial Ellipsoids
…
►
§19.33(i) Surface Area
►The surface area of an ellipsoid with semiaxes , and volume is given by ►
19.33.1
…
►
19.33.2
,
…
►If a conducting ellipsoid with semiaxes bears an electric charge , then the equipotential surfaces in the exterior region are confocal ellipsoids:
…
7: Sidebar 21.SB1: Periodic Surface Waves
Sidebar 21.SB1: Periodic Surface Waves
… ►The caption reads “Mosaic of two overhead photographs, showing surface patterns of waves in shallow water”. …8: 5.19 Mathematical Applications
9: 36.5 Stokes Sets
…
►Stokes sets are surfaces (codimension one) in space, across which or acquires an exponentially-small asymptotic contribution (in ), associated with a complex critical point of or .
…
►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles.
…
►
►
►