elliptical coordinates

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1: 28.32 Mathematical Applications
§28.32(i) EllipticalCoordinates and an Integral Relationship
If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. …
2: 31.17 Physical Applications
Introduce elliptic coordinates $z_{1}$ and $z_{2}$ on $S_{2}$. Then …
3: 28.33 Physical Applications
• Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

• Physical problems involving Mathieu functions include vibrational problems in elliptical coordinates; see (28.32.1). …In elliptical coordinates (28.32.2) becomes (28.32.3). …
5: 23.20 Mathematical Applications
23.20.2 $C:y^{2}z=x^{3}+axz^{2}+bz^{3},$
6: 29.18 Mathematical Applications
29.18.6 $\frac{{\mathrm{d}}^{2}u_{2}}{{\mathrm{d}\beta}^{2}}+(h-\nu(\nu+1)k^{2}{% \operatorname{sn}}^{2}\left(\beta,k\right))u_{2}=0,$
29.18.7 $\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2}}+(h-\nu(\nu+1)k^{2}{% \operatorname{sn}}^{2}\left(\gamma,k\right))u_{3}=0,$
7: 22.19 Physical Applications
22.19.6 $x(t)=a\operatorname{cn}\left(t\sqrt{1+2\eta},k\right).$
22.19.7 $x(t)=a\operatorname{sn}\left(t\sqrt{1-\eta},k\right).$
22.19.8 $x(t)=a\operatorname{dn}\left(t\sqrt{\eta},k\right).$
22.19.9 $x(t)=a\operatorname{cn}\left(t\sqrt{2\eta-1},k\right),$
9: 36.5 Stokes Sets
For $z\neq 0$, the Stokes set is expressed in terms of scaled coordinates
Elliptic Umbilic Stokes Set (Codimension three)
With coordinates Figure 36.5.5: Elliptic umbilic catastrophe with z = constant . … Magnify
10: Bibliography P
• F. A. Paxton and J. E. Rollin (1959) Tables of the Incomplete Elliptic Integrals of the First and Third Kind. Technical report Curtiss-Wright Corp., Research Division, Quehanna, PA.
• S. Pratt (2007) Comoving coordinate system for relativistic hydrodynamics. Phy. Rev. C 75, pp. (024907–1)–(024907–10).
• W. H. Press and S. A. Teukolsky (1990) Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.