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Spherical basis

Index Spherical basis

In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. [1]

34 relations: Applied mathematics, Basis (linear algebra), Cartesian coordinate system, Complex conjugate, Complex number, Computer graphics, Coordinate system, Coordinate vector, Cross product, Dot product, Euclidean space, Hermitian adjoint, Imaginary unit, Invertible matrix, Lie algebra, Norm (mathematics), Orthogonal coordinates, Orthogonality, Orthonormal basis, Positive definiteness, Pure mathematics, Quantum mechanics, Rotation group SO(3), Spherical coordinate system, Spherical harmonics, Standard basis, Tensor operator, Transformation matrix, Transpose, Unit vector, Unitary matrix, Unitary operator, Wigner D-matrix, Wigner–Eckart theorem.

Applied mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.

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Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

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Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Computer graphics

Computer graphics are pictures and films created using computers.

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Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

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Coordinate vector

In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis.

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Cross product

In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Hermitian adjoint

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding adjoint operator.

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Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

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Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

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Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

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Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

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Orthogonal coordinates

In mathematics, orthogonal coordinates are defined as a set of d coordinates q.

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Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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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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Positive definiteness

In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive definite.

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Pure mathematics

Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts.

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Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

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Rotation group SO(3)

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.

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Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

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Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere.

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Standard basis

In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.

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Tensor operator

In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors.

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Transformation matrix

In linear algebra, linear transformations can be represented by matrices.

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Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

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Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

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Unitary matrix

In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.

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Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a surjective bounded operator on a Hilbert space preserving the inner product.

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Wigner D-matrix

The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3).

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Wigner–Eckart theorem

The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics.

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Spherical tensor.

References

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

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