especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical...

covariance or contravariance in Wiktionary, the free dictionary. Covariance and contravariance may refer to: Covariance and contravariance of vectors...

general and are thus contravariant, whereas vectors in general are covariant since they can be pushed forward. See also Covariance and contravariance of vectors...

between vectors and points Banach space Clifford algebra Complex number Coordinate system Covariance and contravariance of vectors Four-vector, a non-Euclidean...

(covariance and contravariance of vectors) in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean...

use of a column vector for input to matrix transformation. Covariance and contravariance of vectors Index notation Vector of ones Single-entry vector Standard...

variance of the object, and one cannot ignore the distinction; see Covariance and contravariance of vectors. In the above example, vectors are represented...

choice of gauge. This article uses tensor index notation and the Minkowski metric sign convention (+ − − −). See also covariance and contravariance of vectors...

positively oriented orthonormal basis of a vector space. If (a1, a2, a3) and (b1, b2, b3) are the coordinates of the vectors a and b in this basis, then their cross...

under a change of coordinate system Covariance and contravariance of functors, properties of functors General covariance or simply covariance (inaccurate...

sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors...

components of a vector can respond in two distinct ways to a change of basis (see Covariance and contravariance of vectors), where the new basis vectors e ^ i {\displaystyle...

a four-vector and is Lorentz covariant. This article uses the summation convention for indices. See covariance and contravariance of vectors for background...

trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary...

Frenet–Serret formulas are a set of ordinary differential equations of first order. The solution is the set of Frenet vectors describing the curve specified...

} The determinant of a square matrix is the same as the determinant of its transpose. The dot product of two column vectors a and b can be computed as...

quantity to a spacelike vector quantity, and we have 4d vectors, or "four-vectors", in Minkowski spacetime. The components of vectors are written using tensor...

A, in the basis B of the space it transforms vector coordinates [u] as [v] = A[u]. As vectors change with the inverse of B (vectors coordinates are contravariant)...

is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with...

notation and covariance and contravariance of vectors for why. The term "component" of a vector is ambiguous: it could refer to: a specific coordinate of the...

matrices or row vectors, and those with a single column are called column matrices or column vectors. A matrix with the same number of rows and columns is...

so it is easy to move vectors from one tangent space to another. Parallel transport of tangent vectors is a way of moving vectors from one tangent space...

involving pairs, triplets, and multivectors that generalize vectors. With multiple combinational possibilities, the space of multivectors expands to 2n...

construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also...

covariant vectors, but more commonly dual vectors or one-forms) are elements of the cotangent space. At p {\displaystyle p} , these two vector spaces may...

represented in the basis by column vectors, and linear functionals are represented in the basis by row vectors, and one can go back and forth by transposing. However...

multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain...

components of v. In some advanced contexts, a row and a column vector have different meaning; see covariance and contravariance of vectors for more. A vector in...

vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of...

defined on tangent vectors at p so that the following linearity condition holds for all tangent vectors Xp and Yp, and all real numbers a and b: α p ( a X p...