A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity.
From Noether's theorem, every differentiable symmetry leads to a conservation law.
Conservation laws are fundamental to our understanding of the physical world, in that they describe which processes can or cannot occur in nature.
In general, the total quantity of the property governed by that law remains unchanged during physical processes.
With respect to classical physics, conservation laws include conservation of energy, mass (or matter), linear momentum, angular momentum, and electric charge.
With respect to symmetries and invariance principles, three special conservation laws have been described, associated with inversion or reversal of space, time, and charge.
Conservation laws are considered to be fundamental laws of nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and engineering.
Most conservation laws are exact, or absolute, in the sense that they apply to all possible processes.
One particularly important result concerning conservation laws is Noether's theorem, which states that there is a one-to-one correspondence between each one of them and a differentiable symmetry of the Universe.
For example, the conservation of energy follows from the uniformity of time and the conservation of angular momentum arises from the isotropy of space,[2][5][6] i.e. because there is no preferred direction of space.
Accordingly, the conserved quantity, CPT parity, can usually not be meaningfully calculated or determined.
These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.
The total amount of some conserved quantity in the universe could remain unchanged if an equal amount were to appear at one point A and simultaneously disappear from another separate point B.
[7][8] Due to special relativity, if the appearance of the energy at A and disappearance of the energy at B are simultaneous in one inertial reference frame, they will not be simultaneous in other inertial reference frames moving with respect to the first.
A stronger form of conservation law requires that, for the amount of a conserved quantity at a point to change, there must be a flow, or flux of the quantity into or out of the point.
A local conservation law is expressed mathematically by a continuity equation, which states that the change in the quantity in a volume is equal to the total net "flux" of the quantity through the surface of the volume.
In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation.
If we assume that the motion u of the charge is a continuous function of position and time, then
In one space dimension this can be put into the form of a homogeneous first-order quasilinear hyperbolic equation:[9]: 43
where the dependent variable y is called the density of a conserved quantity, and A(y) is called the current Jacobian, and the subscript notation for partial derivatives has been employed.
The dependent variable y is called a nonconserved quantity, and the inhomogeneous term s(y,x,t) is the-source, or dissipation.
the conservation equation can be put into the current density form:
In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:
where the conserved quantity is y(r,t), ⋅ denotes the scalar product, ∇ is the nabla operator, here indicating a gradient, and a(y) is a vector of current coefficients, analogously corresponding to the divergence of a vector current density associated to the conserved quantity j(y):
where: It can be shown that the conserved (vector) quantity and the current density matrix for these equations are respectively:
[9]: 62–63 By integrating in any space-time domain the current density form in 1-D space:
In a similar fashion, for the scalar multidimensional space, the integral form is:
where the line integration is performed along the boundary of the domain, in an anticlockwise manner.
[9]: 62–63 Moreover, by defining a test function φ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition.
In the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.