In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces ( see below ), or as a system of parametric equations.

< td > This is a linear Diophantine equation ( see the section " Linear Diophantine equations " below ).</ td >

The previous equation can use Briggsian logarithm as shown below:

25 ° C ) and R = 8. 3145 J /( K · mol ), the equation above can be expressed on base — 10 logarithm as shown below:

This result is derived from the general conic equation given below:

The above equation makes use of information on the composition of parent and daughter isotopes at the time the material being tested cooled below its closure temperature.

If the group velocity ( see below ) is wavelength-independent, this equation can be simplified as:

In 2004, when scientists fit the evolution of dark energy with the cosmological data, they found that the equation of state had possibly crossed the cosmological constant boundary ( w =- 1 ) from above to below.

To appreciate this equation ’ s relationship to the earlier ones, below is this same equation expressed using sines and cosines.

In short, the Arrhenius equation gives " the dependence of the rate constant k of chemical reactions on the temperature T ( in absolute temperature kelvins ) and activation energy E < sub > a </ sub >", as shown below:

where P is the pressure ( Pa ), V the volume ( m < sup > 3 </ sup >) of a gas, and k < sub > 1 </ sub > ( measured in joules ) is the constant from this equation — it is not the same as the constants from the other equations below.

He presented a slightly more complex equation than the one above ( see History section below ) to explain his experimental results.

This is done by subtracting the sum of the standard enthalpies of formation of the reactants ( each being multiplied by its respective stoichiometric coefficient, ν ) from the sum of the standard enthalpies of formation of the products ( each also multiplied by its respective stoichiometric coefficient ), as shown in the equation below:

For example, the wave equation ( described below ) can be written as

The form of the Schrödinger equation depends on the physical situation ( see below for special cases ).

The time-independent Schrödinger equation is discussed further below.

Wave – particle duality follows from the Schrödinger equation, as presented below.

The table below essentially simplifies the ideal gas equation for a particular processes, thus making this equation easier to solve using numerical methods.

In Newtonian mechanics, an equation of motion M takes the general form of a 2nd order ordinary differential equation ( ODE ) in the position r ( see below for details ) of the object:

From the period and the measured distance between the pivots, the acceleration of gravity can be calculated with great precision from the periodicity equation for a simple pendulum, ( 1 ) below.

This is the geodesic equation, discussed below.

; Associativity: For all a, b and c in A, the equation ( a • b ) • c = a • ( b • c ) holds.

; Identity element: There exists an element e in A, such that for all elements a in A, the equation holds.

In case of an ideal mix, the particle diffusion equation holds true and the diffusion coefficient D the speed of diffusion in the particle diffusion equation is independent of particle concentration.

This is an approximate equation, but holds with good accuracy.

; Associativity: For all a, b and c in S, the equation ( a • b ) • c = a • ( b • c ) holds.

; Identity element: There exists an element e in S, such that for all elements a in S, the equation holds.

If denotes the state of the system at any one time t, the following Schrödinger equation holds:

If, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for.

It can be shown using the Gibbs – Duhem equation that if Raoult's law holds over the entire concentration range x = 0 – 1 in a binary solution then, for the second component, the same must also hold.

Note that in, the equation holds, and therefore

More generally, while the basic demographic equation holds true by definition, in practice the recording and counting of events ( births, deaths, immigration, emigration ) and the enumeration of the total population size are subject to error.

Furthermore, he added that his assertion holds “ unless the equation is incomplete ”, by which he meant that no coefficient is equal to 0.

Note that this equation only holds when the density operator is taken to be in the Schrödinger picture, even though this equation seems at first look to emulate the Heisenberg equation of motion in the Heisenberg picture, with a crucial sign difference:

More mundanely, an identity in mathematics may be an equation that holds true for all values of a variable.

With the assumptions of the Black – Scholes model, this second order partial differential equation holds for any type of option as long as its price function is twice differentiable with respect to and once with respect to.

Let D, E and F be given on the lines BC, AC and AB so that the equation holds.

Then by the theorem, the equation also holds for D, E and F ′.

* Λ < sub > 1 </ sub >= the set of all λ's such that equation ( 1 ) holds,

* Λ < sub > 2 </ sub >= the set of all λ's such that equation ( 2 ) holds,

* Λ < sub > 3 </ sub >= the set of all λ's such that equation ( 3 ) holds,

* Λ < sub > 4 </ sub >= the set of all λ's such that equation ( 4 ) holds.