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Transformation matrix. In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . [citation needed] Note that has rows and columns, whereas the transformation is from to .
The direct-quadrature-zero (DQZ or DQ0[1] or DQO, [2] sometimes lowercase) transformation or zero-direct-quadrature[3] (0DQ or ODQ, sometimes lowercase) transformation is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix in an effort to simplify analysis. The DQZ transform is the product of ...
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b ...
The divergence of a vector field F(x) at a point x0 is defined as the limit of the ratio of the surface integral of F out of the closed surface of a volume V enclosing x0 to the volume of V, as V shrinks to zero. where |V| is the volume of V, S(V) is the boundary of V, and is the outward unit normal to that surface.
Conjugate transpose. In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , [1] , [2] or (often in physics) .
Kronecker product. In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a ...
Definition. For a vector field defined on a domain , a Helmholtz decomposition is a pair of vector fields and such that: Here, is a scalar potential, is its gradient, and is the divergence of the vector field . The irrotational vector field is called a gradient field and is called a solenoidal field or rotation field.
Dot product. In mathematics, the dot product or scalar product[note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or ...