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(For vectors in three-dimensional space, the bivector-valued wedge product has the same magnitude as the vector-valued cross product, but unlike the cross product, which is only defined in three-dimensional Euclidean space, the wedge product is well-defined in an affine space of any dimension.)
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their ...
The cross product with respect to a right-handed coordinate system. 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 .
An affine transformation of the Euclidean plane has the form +, where is a regular matrix (its determinant is not 0) and is an arbitrary vector. If f → 1 , f → 2 {\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}} are the column vectors of the matrix A {\displaystyle A} , the unit hyperbola ( ± cosh ( t ) , sinh ( t ) ) , t ∈ R ...
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
A coordinate vector is commonly organized as a column matrix (also called a column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space. A linear map A from a vector space of dimension n into a vector space of dimension m maps a column vector