The equation is used for the mathematical process of solving the problem.

Using the same figures as for the previous example, the equation becomes: Af

The total volume of the system above the reference height is Af, and H can be eliminated to obtain an equation for the total potential energy of the system in terms of H.

Thomas also presents a simple equation for deriving an index of persistence, which weights not only the number of stems ( ' roots ' ) per meaning, but their relative frequency.

Atomic orbitals can be the hydrogen-like " orbitals " which are exact solutions to the Schrödinger equation for a hydrogen-like " atom " ( i. e., an atom with one electron ).

With de Broglie's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of hydrogen like atom, a Bohr electron " wavelength " could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength ( this historically incorrect Bohr model is still occasionally taught to students ).

In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the eigenstates of the Hamiltonian operator for the energy.

We make several assumptions to derive the momentum balance equation for an acoustic medium.

The equation for the conservation of mass in a fluid volume ( without any mass sources or sinks ) is given by

The equation for the conservation of mass for an acoustic medium can also be derived in a manner similar to that used for the conservation of momentum.

In order to close the system of equations we need an equation of state for the pressure.

The equation for the conservation of mass can similarly be written in cylindrical coordinates as

Dirac realised that his relativistic version of the Schrödinger wave equation for electrons predicted the possibility of antielectrons.

Such temperature changes can be quantified using the ideal gas law, or the hydrostatic equation for atmospheric processes.

The mathematical equation for an ideal gas undergoing a reversible ( i. e., no entropy generation ) adiabatic process is

If it is assumed that all color values are premultiplied by their alpha values (), we can rewrite the equation for output color as:

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation.

These points form a line, and y = x is said to be the equation for this line.

The equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = r < sup > 2 </ sup > is the equation for any circle with a radius of r.

This equation is incorrect for the calculation of drag in most cases.

The propagation of sound waves in a fluid ( such as water ) can be modeled by an equation of motion ( conservation of momentum ) and an equation of continuity ( conservation of mass ).

Because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave.

Therefore, these equations allow self-sustaining " electromagnetic waves " to travel through empty space ( see electromagnetic wave equation ).

Central to the study of electromagnetic waves is the wave equation.

He demonstrated that radio radiation had all the properties of waves ( now called electromagnetic radiation ), and discovered that the electromagnetic equations could be reformulated into a partial differential equation called the wave equation.

The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves.

Solutions of this equation that are initially zero outside some restricted region propagate out from the region at a fixed speed in all spatial directions, as do physical waves from a localized disturbance ; the constant c is identified with the propagation speed of the wave.

The elastic wave equation in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium.

Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials.

which says that the length of this vector is proportional to the rest mass m. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get an equation describing the propagation of waves, constructed from relativistically invariant objects,

It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves.

The cutoff frequency is found with the characteristic equation of the Helmholtz equation for electromagnetic waves, which is derived from the electromagnetic wave equation by setting the longitudinal wave number equal to zero and solving for the frequency.

The equation was simple: wealth brought them happiness, and their united front to the world was their warning that they meant to keep everything they had, let no one in on the secrets.

The radiation loss from the anode surface was computed according to Af where Af is the mean of the fourth powers of the temperatures Af and Af calculated analogously to equation ( 1 ).

Apart from the personal equation, another reason advanced in favor of the meeting was that too often in the past the U.S. appeared to have been dragged reluctantly to the summit.

Three of the four persons present, all foreign students in Tokyo, had been playing a game of judging popular Japanese foods by the In and Out system, an equation in which Zen philosophy was used as the modifier.

) TAI was henceforth a realisation of TT, with the equation TT ( TAI )

The study of altruism was the initial impetus behind George R. Price's development of the Price equation, which is a mathematical equation used to study genetic evolution.

One simple version of adaptive expectations is stated in the following equation, where is the next year's rate of inflation that is currently expected ; is this year's rate of inflation that was expected last year ; and is this year's actual rate of inflation:

In 1930, Stoner derived the internal energy-density equation of state for a Fermi gas, and was then able to treat the mass-radius relationship in a fully relativistic manner, giving a limiting mass of approximately ( for μ < sub > e </ sub >= 2. 5 ) 2. 19 · 10 < sup > 30 </ sup > kg.

The mathematical properties of the catenary curve were first studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.

His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory.

Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

The equation was devised in 1961 by Frank Drake, Emeritus Professor of Astronomy and Astrophysics at the University of California, Santa Cruz.

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

One solution was all he looked for in a quadratic equation.

Using these modelling techniques, Malthus ' population principle of growth was later transformed into a model known as the logistic equation:

At 15, he was reading the original papers of Joseph Louis Lagrange, such as the landmark Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory, and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.

He submitted his memoir on equation theory several times, but it was never published in his lifetime due to various events.

Boyle's Law was perhaps the first expression of an equation of state.

His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich – Kwong equation of state and the Soave modification of Redlich – Kwong.

Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law.

Introduced in 1949 the Redlich – Kwong equation of state was a considerable improvement over other equations of the time.

In 1923, while writing the appendix for the Italian edition of the book The Mathematical Theory of Relativity by A. Kopff, Enrico Fermi pointed out, for the first time, that hidden inside the famous Einstein equation (), there was an enormous amount of nuclear potential energy to be exploited.