From the mathematical point of view, Euler equations are notably hyperbolic conservation equations in the case without external field (i.e. in the limit of high Froude number). In fact, like any Cauchy equation, the Euler equations originally formulated in convective form (also called usually "Lagrangian form", but this name is not self-explanatory and historically wrong, so it will be avoided) can also be put in the "conservation form" (also called usually "Eulerian form", but also this name is not self-explanatory and is historically wrong, so it will be avoided here). The conservation form emphasizes the mathematical interpretation of the equations as conservation equations through a control volume fixed in space, and is the most important for these equations also from a numerical point of view. The convective form emphasizes changes to the state in a frame of reference moving with the fluid.
The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in Mémoires de l'Academie des Sciences de Berlin in 1757 (in this article Euler actually published only the general form of the continuity equation and the momentum equation; the energy balance equation would be obtained a century later). They were among the first partial differential equations to be written down. At the time Euler published his work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible fluid. An additional equation, which was later to be called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816.
During the second half of the 19th century, it was found that the equation related to the balance of energy must at all times be kept, while the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stress–energy tensor, and energy and momentum were likewise unified into a single concept, the energy–momentum vector.
Incompressible Euler equations with constant and uniform density
In convective form (i.e. the form with the convective operator made explicit in the momentum equation), the incompressible Euler equations in case of density constant in time and uniform in space are:
Incompressible Euler equations with constant and uniform density (convective or Lagrangian form)
In fact for a flow with uniform density the following identity holds:
where is the mechanic pressure. The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field (the order of the equations is not casual, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following). Notably, the continuity equation would be required also in this incompressible case as an additional third equation in case of density varying in time or varying in space. For example, with density uniform but varying in time, the continuity equation to be added to the above set would correspond to:
So the case of constant and uniform density is the only one not requiring the continuity equation as additional equation regardless of the presence or absence of the incompressible constraint. In fact, the case of incompressible Euler equations with constant and uniform density being analyzed is a toy model featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevancy.
The equations above thus represent respectively conservation of mass (1 scalar equation) and momentum (1 vector equation containing scalar components, where is the physical dimension of the space of interest). In 3D for example and the and vectors are explicitly and . Flow velocity and pressure are the so-called physical variables.
These equations may be expressed in subscript notation:
where the and subscripts label the N-dimensional space components. These equations may be more succinctly expressed using Einstein notation:
where the and subscripts label the N-dimensional space components, and ; is the Kroenecker delta. In 3D and the and vectors are explicitly and , and matched indices imply a sum over those indices and and .
In order to make the equations dimensionless, a characteristic length , and a characteristic velocity , need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
Substitution of these inversed relations in Euler equations, defining the Froude number, yields (omitting the * at apix):
Incompressible Euler equations with constant and uniform density (nondimensional form)
Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory.
The conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for computational fluid dynamics simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods called conservative methods.
The free Euler equations are conservative, in the sense they are equivalent to a conservation equation:
or simply in Einstein notation:
where the conservation quantity in this case is a vector, and is a flux matrix. This can be simply proved.
where I is the identity matrix with dimension N and δij its general element, the Kroenecker delta.
Thanks to these vector identities, the incompressible Euler equations with constant and uniform density and without external field can be put in the so-called conservation (or Eulerian) differential form, with vector notation:
or with Einstein notation:
Then incompressible Euler equations with uniform density have conservation variables:
Note that in the second component u is by itself a vector, with length N, so y has length N+1 and F has size N(N+1). In 3D for example y has length 4, I has size 3x3 and F has size 3x4, so the explicit forms are:
At last Euler equations can be recast into the particular equation:
Incompressible Euler equation(s) with constant and uniform density (conservation or Eulerian form)
For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by Riemann's method of characteristics. This involves finding curves in plane of independent variables (i.e., and ) along which partial differential equations (PDE's) degenerate into ordinary differential equations (ODE's). Numerical solutions of the Euler equations rely heavily on the method of characteristics.
The equations above thus represent conservation of mass, momentum, and energy: the energy equation expressed in the variable internal energy allows to understand the link with the incompressible case, but it is not in the simplest form. Mass density, flow velocity and pressure are the so-called convective variables (or physical variables, or lagrangian variables), while mass density, momentum density and total energy density are the so-called conserved variables (also called eulerian, or mathematical variables). If one explicitates the material derivative the equations above are:
Coming back to the incompressible case, it now becomes apparent that the incompressible constraint typical of the former cases actually is a particular form valid for incompressible flows of the energy equation, and not of the mass equation. In particular, the incompressible constraint corresponds to the following very simple energy equation:
Thus for an incompressible inviscid fluid the specific internal energy is constant along the flow lines, also in a time-dependent flow. The pressure in an incompressible flow acts like a Lagrange multiplier, being the multiplier of the incompressible constraint in the energy equation, and consequently in incompressible flows it has no thermodynamic meaning. In fact, thermodynamics is typical of compressible flows and degenerates in incompressible flows.
Basing on the mass conservation equation, one can put this equation in the conservation form:
meaning that for an incompressible inviscid nonconductive flow a continuity equation holds for the internal energy.
Deduction of the form valid for thermodynamic systems
Considering the first equation, variable must be changed from density to specific volume. By definition:
Thus the following identities hold:
Then by substituting these expressions in the mass conservation equation:
And by multiplication:
Note that this equation is the only belonging to general continuum equations, so only this equation have the same form for example also in Navier-Stokes equations.
On the other hand, the pressure in thermodynamics is the opposite of the partial derivative of the specific internal energy with respect to the specific volume:
since the internal energy in thermodynamics is a function of the two variables aforementioned, the pressure gradient contained into the momentum equation should be explicited as:
It is convenient for brevity to switch the notation for the second order derivatives:
Finally, the energy equation:
can be furtherly simplified in convective form by changing variable from specific energy to the specific entropy: in fact the first law of thermodynamics in local form can be written:
by substituting the material derivative of the internal energy, the energy equation becomes:
now the term between parenthesis is identically zero according to the conservation of mass, then the Euler energy equation becomes simply:
For a thermodynamic fluid, the compressible Euler equations are consequently best written as:
Euler equations (convective form, for a thermodynamic system)
is the specific volume
is the flow velocity vector
is the specific entropy
Note that, in the general case and not only in the incompressible case, the energy equation means that for an inviscid thermodynamic fluid the specific entropy is constant along the flow lines, also in a time-dependent flow. Basing on the mass conservation equation, one can put this equation in the conservation form:
meaning that for an inviscid nonconductive flow a continuity equation holds for the entropy.
On the other hand, the two second-order partial derivatives of the specific internal energy in the momentum equation require the specification of the fundamental equation of state of the material considered, i.e. of the specific internal energy as function of the two variables specific volume and specific entropy:
Note that the fundamental equation of state contains all the thermodynamic information about the system (Callen, 1985), exactly like the couple of a thermal equation of state together with a caloric equation of state.
Quasilinear form and characteristic equations
Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. In regions where the state vector y varies smoothly, the equations in conservative form can be put in quasilinear form :
Obviously this Jacobian does not exist in discontinuity regions (e.g. contact discontinuities, shock waves in inviscid nonconductive flows). Note that if the flux Jacobians are not functions of the state vector , the equations reveals linear.
The compressible Euler equations can be decoupled into a set of N+2 wave equations that describes sound in Eulerian continuum if they are expressed in characteristic variables instead of conserved variables.
In fact the tensor A is always diagonalizable. If the eigenvalues (the case of Euler equations) are all real the system is defined hyperbolic, and physically eigenvalues represent the speeds of propagation of information. If they are all distinguished, the system is defined strictly hyperbolic (it will be proved to be the case of one-dimensional Euler equations). Furthermore, note that diagonalisation of compressible Euler equation is easier when the energy equation is expressed in the variable entropy (i.e. with equations for thermodynamic fluids) than in other energy variables. This will become clear by considering the 1D case.
One can finally find the characteristic variables as:
Since A is constant, multiplying the original 1-D equation in flux-Jacobian form with P−1 yields the characteristic equations:
The original equations have been decoupled into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds. The variables wi are called the characteristic variables and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it is:
Then the solution in terms of the original conservative variables is obtained by transforming back:
this computation can be explicited as the linear combination of the eigenvectors:
Now it becomes apparent that the characteristic variables act as weights in the linear combination of the jacobian eigenvectors. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. Each i-th wave has shape wipi and speed of propagation λi. In the following we show a very simple example of this solution procedure.
Waves in 1D inviscid, nonconductive thermodynamic fluid
If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: g = 0) :
If one defines the vector of variables:
recalling that is the specific volume, the flow speed, the specific entropy, the corresponding jacobian matrix is:
This determinant is very simple: the fastest computation starts on the last row, since it has the highest number of zero elements.
Now by computing the determinant 2x2:
by defining the parameter:
or equivalently in mechanical variables, as:
This parameter is always real according to the second law of thermodynamics. In fact the second law of thermodynamics can be expressed by several postulates. The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. the hessian matrix of the specific energy expresseed as function of specific volume and specific entropy:
is defined positive. This statement corresponds to the two conditions:
The first condition is the one ensuring the parameter a is defined real.
The characteristic equation finally results:
That has three real solutions:
Then the matrix has three real eigenvalues all distinguished: the 1D Euler equations are a strictly hyperbolic system.
At this point one should determine the three eigenvectors: each one is obtained by substituting one eigenvalue in the eigenvalue equation and then solving it. By substituting the first eigenvalue λ1 one obtains:
Basing on the third equation that simply has solution s1=0, the system reduces to:
The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector:
The other two eigenvectors can be found with analogous procedure as:
Then the projection matrix can be built:
Finally it becomes apparent that the real parameter a previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. it is the wave speed. It remains to be shown that the sound speed corresponds to the particular case of an isoentropic transformation:
By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form:
in the convective form of Euler momentum equation, one arrives to:
Friedmann deduced this equation for the particular case of a perfect gas and published it in 1922. However, this equation is general for an inviscid nonconductive fluid and no equation of state is implicit in it.
On the other hand, by substituting the enthalpy form of the first law of thermodynamics in the rotational form of Euler momentum equation, one obtains:
The Euler equations are quasilinearhyperbolic equations and their general solutions are waves. Under certain assumptions they can be simplified leading to Burgers equation. Much like the familiar oceanic waves, waves described by the Euler Equations 'break' and so-called shock waves are formed; this is a nonlinear effect and represents the solution becoming multi-valued. Physically this represents a breakdown of the assumptions that led to the formulation of the differential equations, and to extract further information from the equations we must go back to the more fundamental integral form. Then, weak solutions are formulated by working in 'jumps' (discontinuities) into the flow quantities – density, velocity, pressure, entropy – using the Rankine–Hugoniot equations. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by viscosity and by heat transfer. (See Navier–Stokes equations)
To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the local forms (all the above forms are local forms, since the variables being described are typical of one point in the space caonsidered, i.e. they are local variables) of Euler equations through finite difference methods generally too many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some weak forms, like the finite volume one.
Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain:
where in general F is the flux matrix. By integrating this local equation over a fixed volume Vm, it becomes:
Then, basing on the divergence theorem, we can transform this integral in a boundary integral of the flux:
This global form simply states that there is no net flux of a conserved quantity passing through a region in the case steady and without source. In 1D the volume reduces to an interval, its boundary being its extrema, then the divergence theorem reduces to the fundamental theorem of calculus:
In the steady one dimensional case the become simply:
Thanks to the mass difference equation, the energy difference equation can be simplified without any restriction:
where is the specific total enthalpy.
These are the usually expressed in the convective variables:
is the flow speed
is the specific internal energy.
Note that the energy equation is an integral form of the Bernoulli equation in the compressible case. The former mass and momentum equations by substitution lead to the Rayleigh equation:
Since the second member is a constant, the Rayleigh equation always describes a simple line in the pressure volume plane not depending of any equation of state, i.e. the Rayleigh line. By substituition in the Rankine–Hugoniot equations, that can be also made explicit as:
One can also obtain the kinetic equation and to the Hugoniot equation. The analytical passages are not shown here for brevity.
It has been shown that Euler equations are not a complete set of equations, but they require some additional constraints to admit a unique solution: these are the equation of state of the material considered. To be consistent with thermodynamics these equations of state should satisfy the two laws of thermodynamics. On the other hand, by definition non-equilibrium system are described by laws lying outside these laws. In the following we list some very simple equations of state and the corresponding influence on Euler equations.
where is the specific energy, is the specific volume, is the specific entropy, is the molecular mass, here is considered a constant (polytropic process), and can be shown to correspond to the heat capacity ratio. This equation can be shown to be consistent with the usual equations of state employed by thermodynamics.
Demonstration of consistency with the thermodynamics of an ideal gas
By the thermodynamic definition of temperature:
Where the temperature is measured in energy units. At first, note that by combining these two equations one can deduce the ideal gas law:
or, in the usual form:
where: is the number density of the material. On the other hand the ideal gas law is less strict than the original fundamental equation of state considered.
Now consider the molar heat capacity associated to a process x:
according to the first law of thermodynamics:
it can be simply expressed as:
Now inverting the equation for temperature T(e) we deduce that for an ideal polytropic gas the isocoric heat capacity is a constant:
and similarly for an ideal polytropic gas the isobaric heat capacity results constant:
The specific energy is then, by inverting the relation T(e):
The specific enthalpy results by substitution of the latter and of the ideal gas law:
From this equation one can derive the equation for pressure by its thermodynamic definition:
By inverting it one arrives to the mechanical equation of state:
Then for an ideal gas the compressible Euler equations can be simply expressed in the mechanical or primitive variables specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in convective form they result:
Euler equations for an ideal polytropic gas (convective form)
and in one-dimensional quasilinear form they results:
Let be a Frenet–Serret orthonormal basis which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively. Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-handed side of the above equation, the convective derivative of velocity, can be described as follows:
The "Streamline curvature theorem" states that the pressure at the upper surface of an airfoil is lower than the pressure far away and that the pressure at the lower surface is higher than the pressure far away; hence the pressure difference between the upper and lower surfaces of an airfoil generates a lift force.
Let be the distance from the center of curvature of the streamline, then the second equation is written as follows:
Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature. Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem". 
This "theorem" explains clearly why there are such low pressures in the centre of vortices, which consist of concentric circles of streamlines. This also is a way to intuitively explain why airfoils generate lift forces.
In incompressible flow, all potential flow solutions are also solutions of the Euler equations.
A two-dimensional parallel shear-flow.
Solutions to the Euler equations with vorticity are:
parallel shear flows – where the flow is unidirectional, and the flow velocity only varies in the cross-flow directions, e.g. in a Cartesian coordinate system the flow is for instance in the -direction – with the only non-zero velocity component being only dependent on and and not on 
Two solutions of the three-dimensional Euler equations with cylindrical symmetry have been presented by Gibbon, Moore and Stuart in 2003. These two solutions have infinite energy; they blow up everywhere in space in finite time.
^Friedmann A. An essay on hydrodynamics of compressible fluid (Опыт гидромеханики сжимаемой жидкости), Petrograd, 1922, 516 p., reprinted in 1934 under the editorship of Nikolai Kochin (see the first formula on page 198 of the reprint).
^Sometimes the local and the global forms are also called respectively differential and non-differential, but this is not appropriate in all cases. For example, this is appropriate for Euler equations, while it is not for Navier-Stokes equations since in their global form there are some residual spatial first-order derivative operators in all the caractheristic transport terms that in the local form contains second-order spatial derivatives.
^Chorin, Marsden, A mathematical introduction to fluid mechanics, par. 3.2 Shocks, p. 122