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There are three types of so-called fundamental forms. The most important are the first and second (since the third can be expressed in terms of these). The fundamental forms ...

The osculating circle of a curve C at a given point P is the circle that has the same tangent as C at point P as well as the same curvature. Just as the tangent line is the ...

Let x:U->R^3 be a regular patch, where U is an open subset of R^2. Then (partiale)/(partialv)-(partialf)/(partialu) = eGamma_(12)^1+f(Gamma_(12)^2-Gamma_(11)^1)-gGamma_(11)^2 ...

A ruled surface is a surface that can be swept out by moving a line in space. It therefore has a parameterization of the form x(u,v)=b(u)+vdelta(u), (1) where b is called the ...

The Schrödinger equation describes the motion of particles in nonrelativistic quantum mechanics, and was first written down by Erwin Schrödinger. The time-dependent ...

Underdamped simple harmonic motion is a special case of damped simple harmonic motion x^..+betax^.+omega_0^2x=0 (1) in which beta^2-4omega_0^2<0. (2) Since we have ...

There are six Painlevé transcendents, corresponding to second-order ordinary differential equations whose only movable singularities are ordinary poles and which cannot be ...

A bivector, also called a 2-vector, is an antisymmetric tensor of second rank (a.k.a. 2-form). For a bivector X^->, X^->=X_(ab)omega^a ^ omega^b, where ^ is the wedge product ...

For a smooth harmonic map u:M->N, where del is the gradient, Ric is the Ricci curvature tensor, and Riem is the Riemann tensor.

The antisymmetric parts of the Christoffel symbol of the second kind Gamma^lambda_(munu).