To describe the space of solutions to Af, one must know something about differential equations ; ;

The alpha particle also has a charge + 2, but the charge is usually not written in nuclear equations, which describe nuclear reactions without considering the electrons.

In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.

In his 1868 paper " On Governors ", J. C. Maxwell ( who discovered the Maxwell electromagnetic field equations ) was able to explain instabilities exhibited by the flyball governor using differential equations to describe the control system.

Maxwell developed a set of equations that could unambiguously describe the interrelationship between electric field, magnetic field, electric charge, and electric current.

Linear equations are so-called, because when they are plotted, they describe a straight line ( hence linear ).

At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional, pseudo-Riemannian manifold representing spacetime, and the energy – momentum contained in that spacetime.

They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.

Suppose the goal is to find and describe the solution ( s ), if any, of the following system of linear equations:

Al-Khwarizmi also used the word algebra (' al-jabr ') to describe the mathematical operations he introduced, such as balancing equations, which helped in several problems.

Unlike the physics of rigid bodies where the equations of motion describe the physical location of the rigid body, fluid dynamics describes the flow as a velocity field, i. e. a solution associates each point in space with a vector that represents the speed and direction of the flow at that point.

The dimensionless Reynolds number is an important parameter in the equations that describe whether flow conditions lead to laminar or turbulent flow.

Maxwell's equations, which simplify to the Biot-Savart law in the case of steady currents, describe the origin and behavior of the fields that govern these forces.

Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields.

) Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges.

) The other two equations describe how the fields ' circulate ' around their respective sources ; the magnetic field ' circulates ' around electric currents and time varying electric field in Ampère's law with Maxwell's correction, while the electric field ' circulates ' around time varying magnetic fields in Faraday's law.

The chaotic nature of the atmosphere, the massive computational power required to solve the equations that describe the atmosphere, error involved in measuring the initial conditions, and an incomplete understanding of atmospheric processes mean that forecasts become less accurate as the difference in current time and the time for which the forecast is being made ( the range of the forecast ) increases.

The four re-formulated Maxwell's equations describe the nature of static and moving electric charges and magnetic dipoles, and the relationship between the two, namely electromagnetic induction.

‪ File: Hendrik Antoon Lorentz. jpg ‬| Hendrik Lorentz ( 1853 – 1928 ): clarified electromagnetic theory of light, shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect, developed concept of local time, derived the transformation equations subsequently used by Albert Einstein to describe space and time .‬‬‬

Maxwell's equations – the foundation of classical electromagnetism – describe light as a wave which moves with a characteristic velocity.

He wrote, “ What is it that breathes fire into the equations and makes a universe for them to describe ?”.

In 1927, the Belgian Roman Catholic priest Georges Lemaître independently derived the Friedmann-Lemaître-Robertson-Walker equations and proposed, on the basis of the recession of spiral nebulae, that the universe began with the " explosion " of a " primeval atom "— which was later called the Big Bang.

They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann-Lemaître-Robertson-Walker metric and a fluid with a given mass density and pressure.

Einstein published his first paper on relativistic cosmology in 1917, in which he added this cosmological constant to his field equations in order to force them to model a static universe.

The equations of motion governing the universe as a whole are derived from general relativity with a small, positive cosmological constant.

According to observations of structures larger than solar systems, as well as Big Bang cosmology interpreted under the Friedmann equations and the FLRW metric, dark matter accounts for 23 % of the mass-energy content of the observable universe.

In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations — the cosmological constant — to reproduce that " observation ".

Maxwell's equations posit that there is electric charge, but not magnetic charge ( also called magnetic monopoles ), in the universe.

In the Well World series by Jack L. Chalker, the computer Obie had the ability to adjust the equations of the universe and any changes made could be retconned so that all records would adjust so that even very odd things could be made logical.

The motivation behind this approach began with the Kaluza-Klein theory in which it was noted that applying general relativity to a five dimensional universe ( with the usual four dimensions plus one small curled-up dimension ) yields the equivalent of the usual general relativity in four dimensions together with Maxwell's equations ( electromagnetism, also in four dimensions ).

General relativity provides a set of ten nonlinear partial differential equations for the spacetime metric ( Einstein's field equations ) that must be solved from the distribution of mass-energy and momentum throughout the universe.

The assumption of a uniform dust makes it easy to solve Einstein's field equations and predict the past and future of the universe on cosmological time scales.

In cosmology, solving for the history of the universe is done by calculating R as a function of time, given k and the value of the cosmological constant Λ, which is a ( small ) parameter in Einstein's field equations.

Hence, according to Einstein's field equations, R grew rapidly from an unimaginably hot, dense state that existed immediately following this singularity ( when R had a small, finite value ); this is the essence of the Big Bang model of the universe.

In 1927 Georges Lemaître showed that static solutions of the Einstein equations, which are possible in the presence of the cosmological constant, are unstable, and therefore the static universe envisioned by Einstein could not exist.

Einstein included the cosmological constant as a term in his field equations for general relativity because he was dissatisfied that otherwise his equations did not allow, apparently, for a static universe: gravity would cause a universe which was initially at dynamic equilibrium to contract.

However, soon after Einstein developed his static theory, observations by Edwin Hubble indicated that the universe appears to be expanding ; this was consistent with a cosmological solution to the original general-relativity equations that had been found by the mathematician Friedmann.

In fact adding the cosmological constant to Einstein's equations does not lead to a static universe at equilibrium because the equilibrium is unstable: if the universe expands slightly, then the expansion releases vacuum energy, which causes yet more expansion.

For example, equations used to develop models of the origin do not in themselves explain how the conditions of the universe that the equations model came to be in the first place.

This push to confine the study of mass behaviour to the measurements of parameters involved in differential equations has led sociology perilously close to the reduction of the word `` mass '' to mean a small group in which certain relations between all pairs of individuals in such a group can be studied.

This process would be repeated manually for each of the equations, which would result in a system of equations with one fewer variable.

Fermat always started with an algebraic equation and then described the geometric curve which satisfied it, while Descartes starts with geometric curves and produces their equations as one of several properties of the curves.

Only a few months later, Karl Schwarzschild found a solution to Einstein field equations, which describes the gravitational field of a point mass and a spherical mass.

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:

However, he realized that his equations permitted the introduction of a constant term which could counteract the attractive force of gravity on the cosmic scale.

Chemical reactions are described with chemical equations, which graphically present the starting materials, end products, and sometimes intermediate products and reaction conditions.

Based on this idea and the atomic theory of John Dalton, Joseph Proust had developed the law of definite proportions, which later resulted in the concepts of stoichiometry and chemical equations.

Koenig's software translated the calculation of mathematical equations into codes which represented musical notation.

As with much algorithmic music, and algorithmic art in general, more depends on the way in which the parameters are mapped to aspects of these equations than on the equations themselves.

In mathematics, the Cauchy – Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable.

which are the Cauchy – Riemann equations ( 2 ) at the point z < sub > 0 </ sub >.

Conversely, if ƒ: C → C is a function which is differentiable when regarded as a function on R < sup > 2 </ sup >, then ƒ is complex differentiable if and only if the Cauchy – Riemann equations hold.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

It is possible to construct a continuous function satisfying the Cauchy – Riemann equations at a point, but which is not analytic at the point ( e. g., ƒ ( z ) =.

which satisfies the Cauchy – Riemann equations everywhere, but fails to be continuous at z = 0.

An exact solution to the scalar-Einstein equations which forms a counter example to many formulations of the

Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds — those that are locally Banach spaces — in which case the differential equations are partial differential equations.