One interpretation of the Cauchy – Riemann equations does not involve complex variables directly.

One use of equations is in mathematical identities, assertions that are true independent of the values of any variables contained within them.

One of the most well-known equations describes Pythagoras ' law relating the length of the sides of a right angle triangle:

One of the simplest equations of state for this purpose is the ideal gas law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures.

One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations.

One of the most important equations in Statistical mechanics ( analogous to in mechanics, or the Schroedinger equation in quantum mechanics ) is the definition of the partition function, which is essentially a weighted sum of all possible states available to a system.

One good example is classical electromagnetism, which encompasses results derived from gauge symmetry ( sometimes called gauge invariance ) in a form of a few equations called Maxwell's equations.

One of the characteristics of Metafont is that all of the shapes of the glyphs are defined with geometrical equations.

One then evolves the quadrature points in time according to the Bohm equations of motion.

It was concluded that magnetic monopoles did not exist: One of Maxwell's equations, now called Gauss's law for magnetism, is the mathematical statement that there are no magnetic monopoles.

In later years McMasters has backed away from this origin, suggesting that there could be multiple sources, and that the earliest he has found was a reference in the 1934 French book ; they had applied the equations of air resistance to insects and found that their flight was impossible, but that " One shouldn't be surprised that the results of the calculations don't square with reality ".

One of the two equations is redundant and can be discarded.

One might naively expect that for angles leading to total internal reflection, the solution would consist of an incident wave and a reflected wave, with no transmitted wave at all, but there is no such solution that obeys Maxwell's equations.

One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations in mathematics, physics and engineering.

One way of solving the field equations is to make an approximation, namely, that far from the source ( s ) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space.

Two types of constitutive equations are given below: One type results from using empirical relationships and includes Newtonian fluids, the Bingham fluid model and the power-law fluid model.

One could build a so-called white-box model based on first principles, e. g. a model for a physical process from the Newton equations, but in many cases such models will be overly complex and possibly even impossible to obtain in reasonable time due to the complex nature of many systems and processes.

One example of a sketch defined by parametric equations is the butterfly curve ( transcendental ) | butterfly curve.

One may then expect to differentiate under the integral in order to verify that the Cauchy – Riemann equations hold, and thus that ƒ defines an analytic function.

One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations.

One needs to modify the double duality of the curvature via scale breaking terms, in order to retain Einstein's equations with an induced cosmological constant of partially topological origin as the unique macroscopic ` background '.

where is a conformable identity matrix One may then solve for by inverting or solving the linear equations.

One suggested by Kerner and Isidori and Krener and Respondek can be applied in a situation when there exists a linearizing transformation ( i. e., a diffeomorphism, like the one used in feedback linearization ) such that in new variables the system equations read

One interesting feature of the Cockcroft and Gault equation is that it shows how dependent the estimation of CCr is based on age.

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:

Some cosmologists and physicists argue that a challenge to the cosmological argument is the nature of time: " One finds that time just disappears from the Wheeler – DeWitt equation " ( Carlo Rovelli ).

One must also assume something about the domains of the functions involved before the equation is fully defined.

One example is the Ramanujan – Nagell equation, 2 < sup > n </ sup > − 7 = x < sup > 2 </ sup >.

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

The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in H. One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense.

One method for calculating the pressure is Bernoulli's equation, which is the mathematical expression of Bernoulli's principle.

One of the first standard results using the class equation is that the center of a non-trivial finite p-group cannot be the trivial subgroup ( proof ).

One further important equation can be found by substituting the Lagrangian into another Euler-Lagrange equation, this time for the field, A < sup > μ </ sup >:

One can then write the line's equation, in point-slope form:

One method to solve the initial value problem ( with the initial values as posed above ) is to take advantage of the property of the wave equation that its solutions obey causality.

One corollary of the Friis equation is that an attenuator prior to the first amplifier will degrade the noise figure due to the amplifier.

One cannot freely switch between coulombs and statcoulombs within a formula or equation, as one would freely switch between centimeters and meters.

One can use the Richardson – Dushman equation directly to determine the work function by temperature variation of the sample, as well.

One complication concerning is that its defining equation is not generally solvable.

One approach to " solving " this equation is to use as an integrating factor, which would eventually yield

One of the fundamental theorems of Galois theory states that an equation is solvable in radicals if and only if it has a solvable Galois group, so the proof of the Abel – Ruffini theorem comes down to computing the Galois group of the general polynomial of the fifth degree.

One of his conjectures states that these L-functions satisfy a certain functional equation generalizing those of other known L-functions.

One degree of freedom occurs when one has an autonomous ordinary differential equation in a single variable, with the resulting one-dimensional system being called a phase line, and the qualitative behaviour of the system being immediately visible from the phase line.

One of the cornerstones of electrostatics is setting-up and solving problems described by the Poisson equation.

One balances a chemical equation by changing the scalar number for each chemical formula.

One form of the equation is

One design uses a differential to add mean ( clock ) time and the equation of time to get solar ( sundial ) time.