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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 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.
But for >, the point at = is unstable and the point at = is stable. So the bifurcation occurs at r = 0 {\displaystyle r=0} . A typical example (in real life) could be the consumer-producer problem where the consumption is proportional to the (quantity of) resource.
A scalar field is invariant under any Lorentz transformation. [1] The only fundamental scalar quantum field that has been observed in nature is the Higgs field. However, scalar quantum fields feature in the effective field theory descriptions of many physical phenomena. An example is the pion, which is actually a pseudoscalar. [2]
The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. If φ : U ⊆ R n → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point q, then
A simpler example of a 4-derivative wave equation is the scalar 4-derivative wave equation: = The solution for this in a central field of force is: = + + The first two terms are the same as a normal wave equation.
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 ...
which is thus equivalent to the SO(2) model of real scalar fields ,, as can be seen by expanding the complex field in real and imaginary parts. With N {\displaystyle N} real scalar fields, we can have a φ 4 {\displaystyle \varphi ^{4}} model with a global SO(N) symmetry given by the Lagrangian