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A ruled surface M is said to be a binormal developable of a curve y if M can be parameterized by x(u,v)=y(u)+vB^^(u), where B is the binormal vector.

Every continuous map f:S^n->R^n must identify a pair of antipodal points.

The function Pi_(a,b)(x)=H(x-a)-H(x-b) which is equal to 1 for a<=x<=b and 0 otherwise. Here H(x) is the Heaviside step function. The special case Pi_(-1/2,1/2)(x) gives the ...

For a curve with first fundamental form ds^2=Edu^2+2Fdudv+Gdv^2, (1) the Gaussian curvature is K=(M_1-M_2)/((EG-F^2)^2), (2) where M_1 = |-1/2E_(vv)+F_(uv)-1/2G_(uu) 1/2E_u ...

If Omega_1 and Omega_2 are bounded domains, partialOmega_1, partialOmega_2 are Jordan curves, and phi:Omega_1->Omega_2 is a conformal mapping, then phi (respectively, ...

A topological space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose ...

The space of continuously differentiable functions is denoted C^1, and corresponds to the k=1 case of a C-k function.

The surface given by the parametric equations x = e^(bv)cosv+e^(av)cosucosv (1) y = e^(bv)sinv+e^(av)cosusinv (2) z = e^(av)sinu. (3) For a=b=1, the coefficients of the first ...

A standard form of the linear programming problem of maximizing a linear function over a convex polyhedron is to maximize c·x subject to mx<=b and x>=0, where m is a given ...

The crossed trough is the surface z=x^2y^2. (1) The coefficients of its first fundamental form are E = 1+4x^2y^4 (2) F = 4x^3y^3 (3) G = 1+4x^4y^2 (4) and of the second ...