That is, equations were determined by curves, but curves were not determined by equations.

Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.

The Euler equations were extended to incorporate the effects of viscosity in the first half of the 1800s, resulting in the Navier-Stokes equations.

Their equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste Perrin in 1913.

These equations of state were also previously published by the Soviet physicist Yakov Frenkel in 1928, together with some other remarks on the physics of degenerate matter.

As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found, most importantly in the area of integral equations.

The speed of light and other EMR predicted by Maxwell's equations did not appear unless the equations were modified in a way first suggested by FitzGerald and Lorentz ( see history of special relativity ), or else otherwise it would depend on the speed of observer relative to the " medium " ( called luminiferous aether ) which supposedly " carried " the electromagnetic wave ( in a manner analogous to the way air carries sound waves ).

Radio waves were not detected first from a natural source, but were rather produced deliberately an artificially by the German scientist Heinrich Hertz in 1887, using electrical circuits calculated to produce oscillations in the radio frequency range, following recipies suggested by the equations of James Clerk Maxwell.

Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations.

By 1951, 87 subroutines in the following categories were available for general use: floating point arithmetic ; arithmetic operations on complex numbers ; checking ; division ; exponentiation ; routines relating to functions ; differential equations ; special functions ; power series ; logarithms ; miscellaneous ; print and layout ; quadrature ; read ( input ); nth root ; trigonometric functions ; counting operations ( simulating repeat until loops, while loops and for loops ); vectors ; and matrices.

The matter must, of course, also satisfy whatever additional equations were imposed on its properties.

De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold.

During the Manhattan project, future Nobel laureate Richard Feynman was the supervisor of human computers who understood the use of differential equations which were being solved for the war effort.

Since black holes are exact solutions of Einstein's equations, they were thought not to have any entropy either.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Larmor and Lorentz, who believed the luminiferous ether hypothesis, were seeking the transformation under which Maxwell's equations were invariant when transformed from the ether to a moving frame.

Lorentz and Fitzgerald offered within the framework of Lorentz ether theory a more elegant solution to how the motion of an absolute aether could be undetectable ( length contraction ), but if their equations were correct, the new special theory of relativity ( 1905 ) could generate the same mathematics without referring to an aether at all.

The study of linear algebra and matrices first emerged from the study of determinants, which were used to solve systems of linear equations.

The sequence of equations ( 6 ) can be solved for Af when Af is known, and clearly Af, the maximization being over all admissible Af.

If the differential equations are nonlinear and have a known solution, it may be possible to linearize the nonlinear differential equations at that solution.

Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877, resulting in what is now known as the Routh – Hurwitz theorem.

Impurity distribution in the condensate is described by known equations of zone recrystallization with various numbers of iteration of process – with replacement distribution efficient k of crystallization on separation factor α of distillation.

* The systems studied may only be known approximately — the parameters of the system may not be known precisely or terms may be missing from the equations.

In classical electromagnetism, the electromagnetic field obeys a set of equations known as Maxwell's equations, and the electromagnetic force is given by the Lorentz force law.

After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations.

In 1670, he wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which Newton then supplied.

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

In his PhD thesis project, Paul Dirac discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.

Except for special cases like the Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies.

The equations of motion of the moons, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation.

* Wolfgang M. Schmidt. Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000.

In a private correspondence, Wolfgang Pauli formulated in 1953 a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher dimensional internal space.

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.

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra.

The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

The evidence was compiled by W de Sitter ( 1927 ) who wrote " If we accept this hypothesis, then the ' astronomical time ', given by the earth's rotation, and used in all practical astronomical computations, differs from the ' uniform ' or ' Newtonian ' time, which is defined as the independent variable of the equations of celestial mechanics ".

Furthermore, a mathematician is not someone who merely manipulates formulas, numbers or equations — the diversity of mathematics allows for research concerning how concepts in one area of mathematics can be used in other areas too.

Oliver Heaviside FRS ( ( 18 May 1850 – 3 February 1925 ) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations ( later found to be equivalent to Laplace transforms ), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.

This equation was first studied extensively in ancient India, starting with Brahmagupta, who developed the chakravala method to solve Pell's equation and other quadratic indeterminate equations in his Brahma Sphuta Siddhanta in 628, about a thousand years before Pell's time.

It starred Dustin Hoffman as David Sumner, a timid American mathematician ( his wife at one point attempts to erase Einstein's field equations from his blackboard ) who leaves the chaos of college anti-war protests to live with his young wife Amy ( Susan George ) in her native village in Cornwall, England.

Although implicit in Ampère's force law the force due to a magnetic field on a moving electric charge was not correctly and explicitly stated until 1892 by Hendrik Lorentz who theoretically derived it from Maxwell's equations.

A simplified proof of the second Nash embedding theorem was obtained by who reduced the set of nonlinear partial differential equations to an elliptic system, to which the contraction mapping theorem could be applied.

Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions.

Other important mathematical physicists of the seventeenth century included the Dutchman Christiaan Huygens ( famous for having suggested the wave theory of light ), and the German Johannes Kepler ( Tycho Brahe's assistant who discovered the equations for planetary motion ).

John McMasters recounted an anecdote about an unnamed Swiss aerodynamicist at a dinner party who performed some rough calculations and concluded, presumably in jest, that according to the equations, bumblebees cannot fly.

This idea was elaborated by Hermann Minkowski, who used it to restate the Maxwell equations in four dimensions, showing directly their invariance under the Lorentz transformation.

This fold is called the Beloch fold after Margharita P. Beloch who in 1936 showed using it that origami can be used to solve general cubic equations.