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In general, there is no unique matrix solution A to the matrix equation y=Ax. Even in the case of y parallel to x, there are still multiple matrices that perform this ...

The plane spanned by the three points x(t), x(t+h_1), and x(t+h_2) on a curve as h_1,h_2->0. Let z be a point on the osculating plane, then [(z-x),x^',x^('')]=0, where ...

A topological space X in which each subset of X of the "first category" has an empty interior. A topological space which is homeomorphic to a complete metric space is a Baire ...

The tangent space at a point p in an abstract manifold M can be described without the use of embeddings or coordinate charts. The elements of the tangent space are called ...

A discrete space is simply a topological space equipped with the discrete topology. A discrete space is always a metric space, namely the metric space with the same ...

Minkowski space is a four-dimensional space possessing a Minkowski metric, i.e., a metric tensor having the form dtau^2=-(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2. Alternatively ...

A complex vector bundle is a vector bundle pi:E->M whose fiber bundles pi^(-1)(m) are a copy of C^k. pi is a holomorphic vector bundle if it is a holomorphic map between ...

A connection on a vector bundle pi:E->M is a way to "differentiate" bundle sections, in a way that is analogous to the exterior derivative df of a function f. In particular, ...

A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions: 1. The empty set ...

The partial differential equation del ^2A=-del xE, where del ^2 is the vector Laplacian.