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Raising and lowering indices, the Glossary

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type.^[1]

28 relations: Basis (linear algebra), Cartesian coordinate system, Covariance and contravariance of vectors, Dot product, Dual space, Einstein notation, Electromagnetic tensor, Falling and rising factorials, Field (mathematics), Four-vector, Gaussian units, Identity matrix, Inner product space, International System of Units, Kronecker delta, Linear form, Mathematical physics, Mathematics, Metric signature, Metric tensor, Minkowski space, Mixed tensor, Musical isomorphism, Orthonormal basis, Ricci calculus, Sylvester's law of inertia, Tensor, Vector space.

Basis (linear algebra)

In mathematics, a set of vectors in a vector space is called a basis (bases) if every element of may be written in a unique way as a finite linear combination of elements of.

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In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system.

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Covariance and contravariance of vectors

In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. Raising and lowering indices and covariance and contravariance of vectors are tensors.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product means literally "product with a scalar as a result". Raising and lowering indices and dot product are tensors.

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Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

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Einstein notation

In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. Raising and lowering indices and Einstein notation are tensors.

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Electromagnetic tensor

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime.

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Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

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Four-vector

In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an object with four components, which transform in a specific way under Lorentz transformations.

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Gaussian units

Gaussian units constitute a metric system of physical units.

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Identity matrix

In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.

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Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product.

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International System of Units

The International System of Units, internationally known by the abbreviation SI (from French Système international d'unités), is the modern form of the metric system and the world's most widely used system of measurement.

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Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.

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Linear form

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of scalars (often, the real numbers or the complex numbers).

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Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Metric signature

In mathematics, the signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis.

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Metric tensor

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. Raising and lowering indices and metric tensor are tensors.

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Minkowski space

In physics, Minkowski space (or Minkowski spacetime) is the main mathematical description of spacetime in the absence of gravitation.

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Mixed tensor

In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript (contravariant). Raising and lowering indices and mixed tensor are tensors.

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Musical isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a Riemannian or pseudo-Riemannian manifold induced by its metric tensor.

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Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

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Ricci calculus

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. Raising and lowering indices and Ricci calculus are tensors.

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References

[1] https://en.wikipedia.org/wiki/Raising_and_lowering_indices

Also known as Index gymnastics, Index lowering, Index raising, Lowering an index, Lowering indices, Raising an index, Raising and lowering of indices, Raising indices, Raising or lowering indices.

Raising and lowering indices, the Glossary

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References