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The ordinary differential equation (1) (Byerly 1959, p. 255). The solution is denoted E_m^p(x) and is known as an ellipsoidal harmonic of the first kind, or Lamé function. ...

In toroidal coordinates, Laplace's equation becomes (1) Attempt separation of variables by plugging in the trial solution f(u,v,phi)=sqrt(coshu-cosv)U(u)V(v)Psi(psi), (2) ...

The parabolic cylinder differential equation is the second-order ordinary differential equation y^('')+(nu+1/2-1/4z^2)y=0 (1) whose solution is given by ...

There are two different definitions of "polar vector." In elementary math, the term "polar vector" is used to refer to a representation of a vector as a vector magnitude ...

The spherical curve obtained when moving along the surface of a sphere with constant speed, while maintaining a constant angular velocity with respect to a fixed diameter ...

A spherical sector is a solid of revolution enclosed by two radii from the center of a sphere. The spherical sector may either be "open" and have a conical hole (left figure; ...

A tesseral harmonic is a spherical harmonic of the form cos; sin(mphi)P_l^m(costheta). These harmonics are so named because the curves on which they vanish are l-m parallels ...

A system of curvilinear coordinates for which several different notations are commonly used. In this work (u,v,phi) is used, whereas Arfken (1970) uses (xi,eta,phi) and Moon ...

Toroidal functions are a class of functions also called ring functions that appear in systems having toroidal symmetry. Toroidal functions can be expressed in terms of the ...

A vector Laplacian can be defined for a vector A by del ^2A=del (del ·A)-del x(del xA), (1) where the notation ✡ is sometimes used to distinguish the vector Laplacian from ...