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

The derivations of the above equations for waves in an acoustic medium are given below.

The equations for the conservation of linear momentum for a fluid medium are

The equations for the conservation of momentum may then be written as

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

The equations for the conservation of momentum may then be written as

In terms of components, these three equations for the conservation of momentum in cylindrical coordinates are

The acoustic equations for the conservation of momentum and the conservation of mass are often expressed in time harmonic form ( at fixed frequency ).

Consider, for instance, the following equations:

The ABC was designed for a specific purpose, the solution of systems of simultaneous linear equations.

It could handle systems with up to twenty-nine equations, a difficult problem for the time.

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

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions.

Aerodynamics allows the definition and solution of equations for the conservation of mass, momentum, and energy in air.

The conservation of momentum equations are often called the Navier-Stokes equations, while others use the term for the system that includes conversation of mass, conservation of momentum, and conservation of energy.

The ideal gas law or another equation of state is often used in conjunction with these equations to form a determined system to solve for the unknown variables.

However, the law of mass action is valid only for concerted one-step reactions that proceed through a single transition state and is not valid in general because rate equations do not, in general, follow the stoichiometry of the reaction as Guldberg and Waage had proposed ( see, for example, nucleophilic aliphatic substitution by S < sub > N </ sub > 1 or reaction of hydrogen and bromine to form hydrogen bromide ).

A more complete set of equations for combustion of methane in air is therefore:

This convention is easier to use in chemical equations, replacing the need to write out the mass number for each atom.

When a body is acted upon by external contact forces, internal contact forces are then transmitted from point to point inside the body to balance their action, according to Newton's second law of motion of conservation of linear momentum and angular momentum ( for continuous bodies these laws are called the Euler's equations of motion ).

For atoms with two or more electrons, the governing equations can only be solved with the use of methods of iterative approximation.

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.

As a result, in matter with approximately equal numbers of protons and electrons, proton degeneracy pressure is much smaller than electron degeneracy pressure, and proton degeneracy is usually modeled as a correction to the equations of state of electron-degenerate matter.

During corrosion there are two reactions, oxidation ( equation ), where electrons leave the metal ( and results in the actual loss of metal ) and reduction, where the electrons are used to convert water or oxygen to hydroxides ( equations and ).

This shows clearly that the pre-exponential factor a φ < sup >− 1 </ sup > F < sup > 2 </ sup >, that appears in Fowler-Nordheim-type equations, relates to the effective supply of electrons to the emitter surface, in a free-electron model.

Combining the above two equations, and noting that is the charge on an electron, results in a formula for the Lorentz force experienced by the electrons:

This system of ordinary differential equations relates the number or density of photons and charge carriers ( electrons ) in the device to the injection current and to device and material parameters such as carrier lifetime, photon lifetime, and the optical gain.

Relevant equations of state for pressure may have to include the perfect gas law, radiation pressure, pressure due to degenerate electrons, etc.

He derived the Hartree equations for the distribution of electrons in an atom and proposed the self-consistent field method for their solution.

When the electrons reach dynamic equilibrium, the inertial and the collisional terms of the momentum equations are zero, and the only terms left in the equation are the pressure and electric terms.

Instead of the Boltzmann equation, the following system of equations was proposed for description of charged components of plasma ( electrons and positive ions ):

The essential difference of this system of equations from equations for particles in an external electromagnetic field is that the self-consistent electromagnetic field depends in a complex way on the distribution functions of electrons and ions and.

What made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct ingredients ( the second-quantized Maxwell-Dirac field equations ) for the theoretical description of interacting photons and electrons were well in place, and no major conceptual change was needed analogous to that which was necessitated by a finite and physically sensible account of the radiative behavior of hot objects, as provided by the Planck radiation law.

In contrast to Born – Oppenheimer molecular dynamics wherein the nuclear ( ions ) degree of freedom are propagated using ionic forces which are calculated at each iteration by approximately solving the electronic problem with conventional matrix diagonalization methods, the Car – Parrinello method explicitly introduces the electronic degrees of freedom as ( fictitious ) dynamical variables, writing an extended Lagrangian for the system which leads to a system of coupled equations of motion for both ions and electrons.

ABINIT implements density functional theory by solving the Kohn-Sham equations describing the electrons in a material, expanded in a plane wave basis set and using a self-consistent conjugate gradient method to determine the energy minimum.

The Navier – Stokes equations assume that the fluid being studied is a continuum ( it is infinitely divisible and not composed of particles such as atoms or molecules ), and is not moving at relativistic velocities.

In the most common version, the trajectories of molecules and atoms are determined by numerically solving the Newton's equations of motion for a system of interacting particles, where forces between the particles and potential energy are defined by molecular mechanics force fields.

Rational functions are used to approximate or model more complex equations in science and engineering including ( i ) fields and forces in physics, ( ii ) spectroscopy in analytical chemistry, ( iii ) enzyme kinetics in biochemistry, ( iv ) electronic circuitry, ( v ) aerodynamics, ( vi ) medicine concentrations in vivo, ( vii ) wave functions for atoms and molecules,

For linear molecules with more atoms, rather more work is required, and it is necessary to measure molecules in which more than one isotope of each atom have been substituted ( effectively this gives rise to a set of simultaneous equations that can be solved for the bond lengths ).

These equations suggest a vertically oriented dumbbell of stresses surrounding the dislocation, with compression experienced by the atoms near the " extra " plane, and tension experienced by those atoms near the " missing " plane.

At Hartree ’ s suggestion, Bertha Swirles proceeded to derive equations with exchange for atoms using the Dirac equation in 1935.

Quantum mechanics provides equations based on the Hartree-Fock method and the Roothaan equations that CNDO uses to model atoms and their locations.