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A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
More abstractly, in the presence of a Riemannian metric, vector fields correspond to differential -forms. The conservative vector fields correspond to the exact 1 {\displaystyle 1} -forms , that is, to the forms which are the exterior derivative d ϕ {\displaystyle d\phi } of a function (scalar field) ϕ {\displaystyle \phi } on U ...
Given a module M over a ring R, a linear form on M is a linear map from M to R, where the latter is considered as a module over itself. The space of linear forms is always denoted Hom k (V, k), whether k is a field or not. It is a right module if V is a left module. The existence of "enough" linear forms on a module is equivalent to ...
In the geometrical setting, a non-zero element of the top exterior power () (which is a one-dimensional vector space) is sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity). The name orientation form comes from the fact that a choice of preferred top element determines an orientation of the ...
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v)
Note that linear functionals (multilinear 1-forms over ) are trivially alternating, so that () = =, while, by convention, 0-forms are defined to be scalars: () = =. The determinant on n × n {\displaystyle n\times n} matrices, viewed as an n {\displaystyle n} argument function of the column vectors, is an important example of an alternating ...
Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).