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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.
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 ...
Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers; Scalar (physics), a physical quantity that can be described by a single element of a number field such as a real number Lorentz scalar, a quantity in the theory of relativity which is invariant under a Lorentz transformation
A scalar field is a tensor field of order zero, [3] and the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form. The scalar field of ((+)) oscillating as increases. Red represents positive values, purple represents negative values, and sky blue represents ...
An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a ring homomorphism from the ...
The scalar projection is a scalar, equal to the length of the orthogonal projection of on , with a negative sign if the projection has an opposite direction with respect to .
Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral, the ambiguity of V reflects the freedom in the choice of the reference point r 0 .
By contrast, according to the electromagnetism definition, flux is the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather ...