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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.
When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space. [4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered.
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 ...
Then F can be regarded as a vector space over K by restricting scalar multiplication to elements in K (vector addition is defined as normal). The dimension of this vector space, if it exists, [a] is called the degree of the extension. For example, the complex numbers C form a two-dimensional vector space over the real numbers R.
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.
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 .
scalar Angular acceleration: ω a: Change in angular velocity per unit time rad/s 2: T −2: Area: A: Extent of a surface m 2: L 2: extensive, bivector or scalar Area density: ρ A: Mass per unit area kg⋅m −2: L −2 M: intensive Capacitance: C: Stored charge per unit electric potential farad (F = C/V) L −2 M −1 T 4 I 2: scalar ...