Alternatively, we may use just the last two equations given for the compositional property of the solvent in each of the preceding sections, together with the relationships given below, to derive the remainder of properties in that set:

The preceding section explains how to derive Fowler-Nordheim-type equations.

He considered, at every instant, the actual motion of a stratum as composed of a motion which it had in the preceding instant and of a motion which it had lost ; and the laws of equilibrium between the motions lost furnished him with equations representing the motion of the fluid.

We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

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

The equations in this section are given in SI units.

the section on evolution equations, above ), the result is an analogue of the Schrödinger equation: the Wheeler – DeWitt equation, which some argue is ill-defined.

The differential cross section can be derived from the equations of motion for a particle interacting with a central potential.

The equations in this section either do not use axioms of quantum mechanics or use relations for which there exists a direct correspondence in classical mechanics.

Therefore the equations in this section are valid for electromagnets for which:

A derivation of these equations will be given in the following section.

The derivation includes the first two equations given in the Practical formula for dry air section above.

* How vacuum tubes really work with a section on thermionic emission, with equations, john-a-harper. com.

For example, eliminating z between the two equations x < sup > 2 </ sup > + y < sup > 2 </ sup > − z < sup > 2 </ sup > = 0 and x + 2y + 3z − 1 = 0, which defines an intersection of a cone and a plane in three dimensions, we obtain the conic section 8x < sup > 2 </ sup > + 5y < sup > 2 </ sup > − 4xy + 2x + 4y − 1 = 0, which in this case is an ellipse.

In this section, we won't go all the way back to Maxwell's equations, but will start instead with the homogeneous Helmholtz equation ( valid in source-free media ), which is one level of refinement up from Maxwell's equations ( Scott ).

The darker gray areas correspond to undersampling with the maximum value of n in the equations of this section.

The equations are almost universally solved by means of an iterative, fixed-point type algorithm ( see the following section for more details ).

* See also Dynamical systems and differential equations section below.

When the cylinder to be studied has a r / t ratio of less than 10 ( often cited as 20 ) the thin-walled cylinder equations no longer hold since stresses vary significantly between inside and outside surfaces and shear stress through the cross section can no longer be neglected.

: Writing Einstein's constant depends on how the stress-energy tensor is defined, so the Einstein field equations are always invariant ( see details in the section " About the two possible writings " further ).

In terms of the definition above, this means that its local equations coincide with the equations of the vanishing locus of a global section.

This section will help you better understand, work with, and solve equations when they have addition and / or subtraction in them.

This section will help you understand, work with, and solve equations of a slightly more complex nature-equations involving the use of multiplication and / or division.

For more details, see the section on stiff equations and multistep methods.

If interstellar colonization is possible, then this assumption is invalid, and the equations of population dynamics would apply instead.

The above collection of equations apply only for a boost in the x-direction.

The same equations of electromagnetic theory apply at all frequencies.

The momentum and energy equations also apply to the motions of objects that begin together and then move apart.

To “ run ” a model, scientists divide the planet into a 3-dimensional grid, apply the basic equations, and evaluate the results.

This occurs because of the presence of a " transonic regime " around M = 1 where approximations of the Navier-Stokes equations used for subsonic design actually no longer apply, the simplest of many reasons being that the flow locally begins to exceed M = 1 even when the freestream Mach number is below this value.

These equations apply to a particle moving linearly, in three dimensions in a straight line, with constant acceleration.

When magnetostatics does not apply, the Biot – Savart law should be replaced by Jefimenko's equations.

Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field.

To apply the first recursion theorem, the recursion equations must first be recast as a recursive operator.

To “ run ” a model, scientists divide the planet into a 3-dimensional grid, apply the basic equations, and evaluate the results.

For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity ( may have to apply the 2nd Fundamental Theorem of Calculus to do this rigorously ).

However, the equations can be further generalized to then be extended to apply to quantum mechanics as well as to classical mechanics, through the deformation of the Poisson algebra over p and q to the algebra of Moyal brackets.

Strictly, Fowler-Nordheim equations apply only to field emission from bulk metals and ( with suitable modification ) to other bulk crystalline solids, but they are often used – as a rough approximation – to describe field emission from other materials.

Strictly, these equations apply only to CFE from bulk metals.

Strictly, Fowler-Nordheim-type equations apply only to emission from the conduction band of bulk crystalline solids.

However, empirical equations of form ( 42 ) should apply to all materials ( though, conceivably, modification might be needed for very sharp emitters ).

Such statements, however, fall in the same category as statements such as " physics assumes that nature applies formulas such as Einstein's E = mc < sup > 2 </ sup > or Newton's F = ma " and " DST models assume that dynamic systems apply differential equations ".

It is a common misconception that the RANS equations do not apply to flows with a time-varying mean flow because these equations are ' time-averaged '.

The following techniques apply to one-dimensional autonomous differential equations.

The same equations apply to other wave based sensors, such as radar and the human ear.

To find the molar specific heat capacity of the gas involved, the following equations apply for any general gas that is calorically perfect.