In the various subfields of
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, there exist two common usages of the term
flux, both with rigorous mathematical frameworks.
 In the study of transport phenomena
Transport Phenomena is the first textbook that is about transport phenomena. It is specifically designed for chemical engineering students...
(heat transferHeat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phasechange transfer...
, mass transferMass transfer is the net movement of mass from one location, usually meaning a stream, phase, fraction or component, to another. Mass transfer occurs in many processes, such as absorption, evaporation, adsorption, drying, precipitation, membrane filtration, and distillation. Mass transfer is used...
and fluid dynamicsIn physics, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
), flux is defined as flow per unit area, where flow is the movement of some quantity per time. Flux, in this definition, is a vector.
 In the fields of electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
and mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, flux is usually the integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of a vector quantity, flux densityFormal Statement:The flux density is simply defined as the amount of flux passing through a unitarea. Mathematical Statement:The flux density would essentially be the number of field lines passing through a defined unitarea...
, over a finite surface. It is an integral operator that acts on a vector field similarly to the gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
, divergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
and curl operators found in vector analysis. The result of this integration is a scalarIn physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or spacetime translations . This is in contrast to a vector...
quantity called flux. The magnetic fluxMagnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
is thus the integral of the magnetic vector field B over a surface, and the electric flux is defined similarly. Using this definition, the flux of the Poynting vectorIn physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently coinvented the Poynting vector...
over a specified surface is the rate at which electromagnetic energy flows through that surface. Confusingly, the Poynting vector is sometimes called the power flux, which is an example of the first usage of flux, above. It has units of wattThe watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.Definition:...
s per square metreThe square metre or square meter is the SI derived unit of area, with symbol m2 . It is defined as the area of a square whose sides measure exactly one metre...
(W/m^{2}).
One could argue, based on the work of
James Clerk MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
, that the transport definition precedes the more recent way the term is used in electromagnetism. The specific quote from Maxwell is "
In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integralIn mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...
of the flux. It represents the quantity which passes through the surface."
In addition to these few common mathematical definitions, there are many more looser, but equally valid, usages to describe observations from other fields such as biology, the arts, history, and humanities.
Origin of the term
The word
flux comes from
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient ProtoIndoEuropean language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
:
fluxus means "flow", and
fluere is "to flow". As
fluxionFluxion may refer to:* An alternate spelling of fluxon, a quantum of magnetic flux, such as in a superconductor* Fluxion was Isaac Newton's term for the derivative of a "fluent", or continuous function....
, this term was introduced into
differential calculusIn mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
by
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
.
Flux definition and theorems
Flux is surface bombardment rate. There are many fluxes used in the study of transport phenomena. Each type of flux has its own distinct unit of measurement along with distinct physical constants. Seven of the most common forms of flux from the transport literature are defined as:
 Momentum flux, the rate of transfer of momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
across a unit area (N·s·m^{−2}·s^{−1}). (Newton's law of viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
,)
 Heat flux
Heat flux or thermal flux is the rate of heat energy transfer through a given surface. The SI derived unit of heat rate is joule per second, or watt. Heat flux is the heat rate per unit area. In SI units, heat flux is measured in W/m2]. Heat rate is a scalar quantity, while heat flux is a vectorial...
, the rate of heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
flow across a unit area (J·m^{−2}·s^{−1}). (Fourier's law of conductionIn heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....
) (This definition of heat flux fits Maxwell's original definition.)
 Diffusion flux, the rate of movement of molecules across a unit area (mol·m^{−2}·s^{−1}). (Fick's law of diffusion
Fick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855. Fick's first law :...
)
 Volumetric flux
In fluid dynamics, the volumetric flux is the rate of volume flow across a unit area . The density of a particular property in a fluid's volume, multiplied with the volumetric flux of the fluid, thus defines the advective flux of that property...
, the rate of volumeVolume is the quantity of threedimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
flow across a unit area (m^{3}·m^{−2}·s^{−1}). (Darcy's law of groundwater flowDarcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...
)
 Mass flux
Mass flux is the rate of mass flow across a unit area .See also:*Flux*Fick's law*Darcy's law...
, the rate of massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
flow across a unit area (kg·m^{−2}·s^{−1}). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)
 Radiative flux
Radiative flux, or radiative flux density, is the amount of power radiated through a given area, in the form of photons or other elementary particles, typically measured in W/m2. It is used in astronomy to determine the magnitude and spectral class of a star...
, the amount of energy moving in the form of photons at a certain distance from the source per steradianThe steradian is the SI unit of solid angle. It is used to describe twodimensional angular spans in threedimensional space, analogous to the way in which the radian describes angles in a plane...
per second (J·m^{−2}·s^{−1}). Used in astronomy to determine the magnitude and spectral class of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the infrared spectrum.
 Energy flux
Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:# Rate of energy transfer per unit area...
, the rate of transfer of energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
through a unit area (J·m^{−2}·s^{−1}). The radiative flux and heat flux are specific cases of energy flux.
 Particle flux, the rate of transfer of particles through a unit area ([number of particles] m^{2}·s^{1})
These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the
divergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For
incompressible flowIn fluid mechanics or more generally continuum mechanics, incompressible flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid...
, the divergence of the volume flux is zero.
Chemical diffusion
Chemical molar flux of a component A in an isothermal,
isobaricIsobaric may refer to:*in thermodynamics, an isobaric process, i.e. one that is carried out at constant pressure;...
systemSystem is a set of interacting or interdependent components forming an integrated whole....
is defined in abovementioned
Fick's first lawFick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in the year 1855. Fick's first law :...
as:
where:
 is the diffusion coefficient (m^{2}/s) of component A diffusing through component B,
 is the concentration (mol
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon12 , the isotope of carbon with atomic weight 12. This corresponds to a value...
/m^{3}) of species A.
This flux has units of mol·m
^{−2}·s
^{−1}, and fits Maxwell's original definition of flux.
Note:
("
nablarightthumb256pxThe [[harp]], the instrument after which the nabla symbol is namedNabla is the symbol \nabla . The name comes from the Greek word for a Hebrew harp, which had a similar shape. Related words also exist in Aramaic and Hebrew. The symbol was first used by William Rowan Hamilton in...
") denotes the
delIn vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a onedimensional domain, it denotes its standard derivative as defined in calculus...
operator.
For dilute gases, kinetic molecular theory relates the diffusion coefficient
D to the particle density
n =
N/
V, the molecular mass
M, the collision
cross sectionA cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the BeerLambert law....
, and the
absolute temperatureThermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
T by
where the second factor is the
mean free pathIn physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.Derivation:...
and the square root (with
Boltzmann's constant k) is the mean velocity of the particles.
In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.
Quantum mechanics
In
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, particles of mass m in the state
have a probability density defined as
So the probability of finding a particle in a unit of volume, say
, is
Then the number of particles passing through a perpendicular unit of area per unit time is
This is sometimes referred to as the
flux density.
Flux definition and theorems
An example of the first definition of flux is the magnitude of a river's current, that is, the amount of water that flows through a crosssection of the river each second. The amount of sunlight that lands on a patch of ground each second is also a kind of flux.
To better understand the concept of flux in Electromagnetism, imagine a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind speed is high, then the flux through the net is large. If the net is made bigger, then the flux would be larger even though the wind speed is the same. For the most air to move through the net, the opening of the net must be facing the direction the wind is blowing. If the net opening is parallel to the wind, then no wind will be moving through the net. Perhaps the best way to think of flux abstractly is "How much stuff goes through your thing", where the stuff is a field and the thing is the virtual surface.
As a mathematical concept, flux is represented by the surface integral of a vector field,
where:
 E is a vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
of Electric Force,
 dA is the vector area of the surface S, directed as the surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a nonflat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
,
 is the resulting flux.
The surface has to be
orientableIn mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the righthand rule to define a "clockwise" direction of loops in the...
, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.
The surface normal is directed accordingly, usually by the
righthand ruleIn mathematics and physics, the righthand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century....
.
Conversely, one can consider the flux the more fundamental quantity and call the vector field the
flux density.
Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive
divergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
(sources) and end at areas of negative divergence (sinks).
See also the image at right: the number of red arrows passing through a unit area is the flux density, the
curveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals.
If the surface encloses a 3D region, usually the surface is oriented such that the
influx is counted positive; the opposite is the
outflux.
The
divergence theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the
divergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
).
If the surface is not closed, it has an oriented curve as boundary.
Stokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
states that the flux of the curl of a vector field is the
line integralIn mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
of the vector field over this boundary. This path integral is also called
circulation, especially in fluid dynamics. Thus the curl is the circulation density.
We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.
Maxwell's equations
The flux of
electricIn physics, an electric field surrounds electrically charged particles and timevarying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and
magnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
lines is frequently discussed in
electrostaticsElectrostatics is the branch of physics that deals with the phenomena and properties of stationary or slowmoving electric charges....
. This is because
Maxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
in integral form involve integrals like above for electric and magnetic fields.
For instance,
Gauss's lawIn physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
states that the flux of the electric field out of a closed surface is proportional to the
electric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
enclosed in the surface (regardless of how that charge is distributed). The constant of proportionality is the reciprocal of the
permittivityIn electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
of free space.
Its integral form is:

where:
 is the electric field,
 is the area of a differential square on the surface A with an outward facing surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a nonflat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
defining its direction,
 is the charge enclosed by the surface,
 is the permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
of free space
 is the integral over the surface A.
Either
or
is called the
electric flux.
If one considers the flux of the electric field vector,
E, for a tube near a point charge in the field the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any crosssectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q/ε
_{0}.
In free space the electric displacement vector
D = ε
_{0} E so for any bounding surface the flux of
D = q, the charge within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow".
Faraday's law of inductionFaraday's law of induction dates from the 1830s, and is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators...
in integral form is:
where:
 is an infinitesimal element (differential
In mathematics, the term differential has several meanings.Basic notions:* In calculus, the differential represents a change in the linearization of a function....
) of the closed curve C (i.e. a vector with magnitude equal to the length of the infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infiniteth" item in a series.In common speech, an...
line element, and direction given by the tangent to the curve C, with the sign determined by the integration direction).
The
magnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
is denoted by
. Its flux is called the
magnetic fluxMagnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...
. The timerate of change of the magnetic flux through a loop of wire is minus the
electromotive forceIn physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...
created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change. This is the basis for
inductorAn inductor is a passive twoterminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s and many electric generators.
Poynting vector
The flux of the
Poynting vectorIn physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently coinvented the Poynting vector...
through a surface is the electromagnetic
powerIn physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...
, or
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
per unit
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, passing through that surface. This is commonly used in analysis of
electromagnetic radiationElectromagnetic radiation is a form of energy that exhibits wavelike behavior as it travels through space...
, but has application to other electromagnetic systems as well.
Biology
In general,
flux in
biologyBiology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
relates to movement of a substance between compartments. There are several cases where the concept of flux is important.
 The movement of molecules across a membrane: in this case, flux is defined by the rate of diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
or transport of a substance across a permeable membraneA biological membrane or biomembrane is an enclosing or separatingmembrane that acts as a selective barrier, within or around a cell. It consists of a lipid bilayer with embedded proteins that may constitute close to 50% of membrane content...
. Except in the case of active transport, net flux is directly proportional to the concentrationIn chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...
difference across the membrane, the surface areaSurface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...
of the membrane, and the membrane permeabilityA semipermeable membrane, also termed a selectively permeable membrane, a partially permeable membrane or a differentially permeable membrane, is a membrane that will allow certain molecules or ions to pass through it by diffusion and occasionally specialized "facilitated diffusion".The rate of...
constant.
 In ecology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
, flux is often considered at the ecosystemAn ecosystem is a biological environment consisting of all the organisms living in a particular area, as well as all the nonliving , physical components of the environment with which the organisms interact, such as air, soil, water and sunlight....
level  for instance, accurate determination of carbon fluxes using techniques like eddy covarianceThe eddy covariance technique is a key atmospheric flux measurement technique to measure and calculate vertical turbulent fluxes within atmospheric boundary layers...
(at a regional and global level) is essential for modeling the causes and consequences of global warmingGlobal warming refers to the rising average temperature of Earth's atmosphere and oceans and its projected continuation. In the last 100 years, Earth's average surface temperature increased by about with about two thirds of the increase occurring over just the last three decades...
.
 Metabolic flux refers to the rate of flow of metabolites along a metabolic pathway
In biochemistry, metabolic pathways are series of chemical reactions occurring within a cell. In each pathway, a principal chemical is modified by a series of chemical reactions. Enzymes catalyze these reactions, and often require dietary minerals, vitamins, and other cofactors in order to function...
, or even through a single enzymeEnzymes are proteins that catalyze chemical reactions. In enzymatic reactions, the molecules at the beginning of the process, called substrates, are converted into different molecules, called products. Almost all chemical reactions in a biological cell need enzymes in order to occur at rates...
. A calculation may also be made of carbon (or other elements, e.g. nitrogen) flux. It is dependent on a number of factors, including: enzyme concentration; the concentration of precursor, product, and intermediate metabolites; posttranslational modification of enzymes; and the presence of metabolic activators or repressors. Metabolic control analysisMetabolic control analysis is a mathematical framework for describingmetabolic, signaling and genetic pathways. MCA quantifies how variables,such as fluxes and species concentrations, depend on network parameters....
and flux balance analysisFlux balance analysis is a mathematical method for analysing metabolism. It does not require knowledge of metabolite concentration or details of the enzyme kinetics of the system...
provide frameworks for understanding metabolic fluxes and their constraints.