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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.
In this article, vectors are represented in boldface to distinguish them from scalars. [nb 1] [1] A vector space over a field F is a non-empty set V together with a binary operation and a binary function that satisfy the eight axioms listed below. In this context, the elements of V are commonly called vectors, and the elements of F are called ...
For example, the complex numbers C form a two-dimensional vector space over the real numbers R. Likewise, the real numbers R form a vector space over the rational numbers Q which has (uncountably) infinite dimension, if a Hamel basis exists. [b] If V is a vector space over F it may also be regarded as vector space over K. The dimensions are ...
A scalar in physics and other areas of science is also a scalar in mathematics, as an element of a mathematical field used to define a vector space.For example, the magnitude (or length) of an electric field vector is calculated as the square root of its absolute square (the inner product of the electric field with itself); so, the inner product's result is an element of the mathematical field ...
It is common to call these tuples vectors, even in contexts where vector-space operations do not apply. More generally, when some data can be represented naturally by vectors, they are often called vectors even when addition and scalar multiplication of vectors are not valid operations on these data. [disputed – discuss] Here are some examples.
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. [1] [2] The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.
Ordinary vectors are sometimes called true vectors or polar vectors to distinguish them from pseudovectors. Pseudovectors occur most frequently as the cross product of two ordinary vectors. One example of a pseudovector is angular velocity. Driving in a car, and looking forward, each of the wheels has an angular velocity vector pointing to the ...
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .