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An orientation on an n-dimensional manifold is given by a nowhere vanishing differential n-form. Alternatively, it is an bundle orientation for the tangent bundle. If an ...

Given a plane ax+by+cz+d=0 (1) and a point x_0=(x_0,y_0,z_0), the normal vector to the plane is given by v=[a; b; c], (2) and a vector from the plane to the point is given by ...

For a scalar function f over a surface parameterized by u and v, the surface integral is given by Phi = int_Sfda (1) = int_Sf(u,v)|T_uxT_v|dudv, (2) where T_u and T_v are ...

The French metro metric is an example for disproving apparently intuitive but false properties of metric spaces. The metric consists of a distance function on the plane such ...

Let u_(p) be a unit tangent vector of a regular surface M subset R^3. Then the normal curvature of M in the direction u_(p) is kappa(u_(p))=S(u_(p))·u_(p), (1) where S is the ...

The orthogonal complement of a subspace V of the vector space R^n is the set of vectors which are orthogonal to all elements of V. For example, the orthogonal complement of ...

If X_i for i=1, ..., m has a multivariate normal distribution with mean vector mu=0 and covariance matrix Sigma, and X denotes the m×p matrix composed of the row vectors X_i, ...

The four parameters e_0, e_1, e_2, and e_3 describing a finite rotation about an arbitrary axis. The Euler parameters are defined by e_0 = cos(phi/2) (1) e = [e_1; e_2; e_3] ...

A tube of radius r of a set gamma is the set of points at a distance r from gamma. In particular, if gamma(t) is a regular space curve whose curvature does not vanish, then ...

The plane spanned by the tangent vector T and binormal vector B.