Sources and sinks
In the physical sciences, engineering and mathematics, sources and sinks is an analogy used to describe properties of vector fields.
It generalizes the idea of fluid sources and sinks (like the faucet and drain of a bathtub) across different scientific disciplines.
These terms describe points, regions, or entities where a vector field originates or terminates.
The analogy sometimes includes swirls and saddles for points that are neither of the two.
In the case of electric fields the idea of flow is replaced by field lines and the sources and sinks are electric charges.
is a function that returns a vector and is defined for each point (with coordinates
The integral version of the continuity equation is given by the divergence theorem.
These concepts originate from sources and sinks in fluid dynamics, where the flow is conserved per the continuity equation related to conservation of mass, given by where
This equation implies that any emerging or disappearing amount of flow in a given volume must have a source or a sink, respectively.
Sources represent locations where fluid is added to the region of space, and sinks represent points of removal of fluid.
[1] Note that for incompressible flow or time-independent systems,
is directly related to the divergence as For this kind of flow, solenoidal vector fields (no divergence) have no source or sinks.
[2][3] And when both divergence and curl are zero, the point is sometimes called a saddle.
[3] In electrodynamics, the current density behaves similar to hydrodynamics as it also follows a continuity equation due to the charge conservation: where this time
, respectively (for example, the source and sink can represent the two poles of an electrical battery in a closed circuit).
, a source is a point where electric field lines emanate, such as a positive charge (
[6] This happens because electric fields follow Gauss's law given by where
[9] In thermodynamics, the source and sinks correspond to two types of thermal reservoirs, where energy is supplied or extracted, such as heat flux sources or heat sinks.
In thermal conduction this is described by the heat equation.
[10] The terms are also used in non-equilibrium thermodynamics by introducing the idea of sources and sinks of entropy flux.
[11] In chaos theory and complex system, the idea of sources and sinks is used to describes repellors and attractors, respectively.
[12][13] This terminology is also used in complex analysis, as complex number can be desrcibed as vectors in the complex plane.Sources and sinks are associated with zeros and poles of meromorphic function, representing inflows and outflows in a harmonic function.
A complex function is defined to a source or a sink if it has a pole of order 1.
[14] In topology, the terminology of sources and sinks is used when discussing a vector field over a compact differentiable manifold.
In this context the index of a vector field is +1 if it is a source or a sink, if the value is -1 it is called a saddle point.
[15] Other areas where this terminology is used include source–sink dynamics in ecology and current source density analysis in neuroscience.
