What is the difference between exact and inexact differentials

Those who love analysis will, with joy, see mechanics become a new branch, and I am grateful that I have extended its domain. The analytical mechanics of Lagrange is founded on the principle of virtual work, which was developed by Johann Bernoulli, in the static version, and by d'Alembert, in the dynamic version.

The principle of virtual work stated by Johann Bernoulli generalized the previous formulations given by his predecessor [3] [3] R. The principle gives the condition for the static equilibrium of a mechanical system acted by several forces.

In a letter written in to Varignon, which was reproduced in the treatise of Varignon on statics [39] [39] P. In all equilibrium of any forces, in whatever way they are applied, and in whatever direction that they act on each other, indirectly or directly, the sum of the positive energies will be equal to the sum of the negative energies taken positively.

In this statement, energy means the product of the magnitude of a force by a small displacement of the point acted by the force. The small displacement is parallel to the force, being positive if in the direction of the force and negative in the opposite direction. It should be noted that the small displacements Johann Bernoulli called virtual velocities.

In his treatise on analytical mechanics, Lagrange adopts the principle of virtual work, as a fundamental principle of statics. He writes the principle in the analytical form as. Lagrange states that the equation 43 is the condition of equilibrium of a system of forces. It should be remarked that forces of reaction, which occurs for instance when a body rests on a surface, do not enter this equation because their works perform no work.

Lagrange shows furthermore that if the function is a minimum then the equilibrium is stable and the system will display small oscillations. If on the contrary the function is a maximum, the equilibrium is unstable and, being once disturbed, the system will depart from equilibrium.

The static principle of virtual work of Johann Bernoulli was extended to dynamic problems by d'Alembert [40] [40] J. According to d'Alembert, in dynamic problems one should take into account the inertial forces in addition to the actual forces. An analytical expression of the d'Alembert principle was given by Lagrange, and was obtained as follows. Lagrange argues that, the laws of motion of a body will be reduced to the laws of equilibrium if, following d'Alembert, one includes the inertial force, which is the mass of the body multiplied by the acceleration 4.

Adding the work of all bodies and all Cartesian coordinates to the left hand side of 43 ,. The equations of motion were obtained by Lagrange from the principle expressed by equation 44 as follows.

The relation between the differentials is. Defining the quantity. Comparing 48 with 51 and 52 one reaches the result. The crucial step in the derivation of equation 53 is found in the second equality of equation Since these two quantities are equal, the following relation is obtained.

Using this result and the relation 49 between B i j and A i j , it is straightforward to reach the second equality of equation Lagrange considers next the case where the forces Q i in equation 44 are such that. The replacement of this expression and the expression 53 in equation 44 , yields the result. Thermodynamics emerged around the middle of the nineteenth century and presented two changes in the way the heat was conceived [1] [1] M. The first was the recognition that heat should be understood as a form of work, which lead to the law of conservation of energy.

The second was the way in which heat was transformed in mechanical work, which allowed Clausius to define entropy and lead to the law of the increase in entropy. The heat absorbed minus the work performed by a system along a thermodynamic process is equal to the increase in the internal energy, and is independent of the trajectory connecting the the final and initial states.

In his first paper on thermodynamics [41] [41] R. Clausius, Annalen der Physik und Chemie 79, Clausius starts his reasoning by assuming that a small quantity d Q of heat exchanged when the volume V and temperature T of a gas changes by d V and d T is given by.

Then he considers the total heat exchanged in a clockwise cycle. Considering that the cycle is small Clausius argues that the heat exchanged is. To reach result 61 , Clausius argues as follows. Along the isotherms of the small cycle shown in figure 4. The quantities M 1 and M 2 are related to M by. An analogous result for the net work was obtained by Clausius.

However in a comment to this paper [2] [2] R. Clausius carried out an original derivation of results 61 and 71 , but they can be understood as a direct application of a theorem formulated by Cauchy in [42] [42] A. According to this theorem of Cauchy, the contour integral of a region in a plane is related to an integral over this region as follows. Next Clausius uses the law of Mayer and Joule according to which the work is always transformed in the same quantity of heat.

The reciprocal of A is the mechanical equivalent of heat. Clausius calls this differential d U and writes. As it is, equation 74 cannot be integrated unless one knows M , N and p as a function of V and T. This was accomplished by Clausius for an ideal gas. In addition to the equation of state, Clausius relies on another law which he says is valid as much a the equation of state.

The way in which heat is transformed in mechanical work was the main subject of research on heat carried out by Carnot. His investigations lead him to the following fundamental principle. When a system undergoes a cyclic process composed by two isothermal and two adiabatic processes the ratio of the work produced and the heat depends only on the two temperatures. In this principle of Carnot, heat was understood as a conserved quantity which in this case descends from a high temperature to a low temperature.

Clausius modified this principle by replacing heat by heat absorbed. Clausius, Annalen der Physik und Chemie 93, Clausius, Annalen der Physik und Chemie , We have analyzed the role of exact and inexact differentials in the early developments of mechanics and thermodynamics.

We have also examined the evolution of differential calculus in relation to the concepts related to exact and inexact differentials. Euler introduced the concept of integrating factor, which he used to solve an ordinary linear differential equation of the first-order. Euler also found the condition for a complete differential by examining the differential of a function of two variables. In an independent way, Clairaut also reached the same condition. When this condition is not fulfilled the differential is incomplete or inexact.

In the analytical treatment of mechanics, Lagrange considered forces whose differential work is an exact differential, in which case it is possible to define a work function and reach the conservation of energy.

When the differential is inexact, it is possible to transform it into and an exact differential as long as an integrating factor can found.

The law of conservation of energy was also written by Clausius in terms of exact differentials. The process of finding exact differentials was a fundamental procedure which allowed the formulation of thermodynamics in terms of state functions such as energy and entropy.

Open menu Brazil. Open menu. Text EN Text English. Abstract We give an account and a critical analysis of the use of exact and inexact differentials in the early development of mechanics and thermodynamics, and the emergence of differential calculus and how it was applied to solve some mechanical problems, such as those related to the cycloidal pendulum.

Introduction It is usual to formulate the basic equations of thermodynamics in terms of differentials. The ratio between d y and d x equals the ratio between the ordinate MP and the subtangent TP. The figure is based on figures 1 and 3 of reference [15] [15] G. The segments CM and BM are the abscissa y and ordinate of point M whereas nm and Mn are the differentials d y and d x.

The figure is based on figure 1 of reference [36] [36] Johann Bernoulli, Acta Eruditorum, Maji , p. An easy way that I got to think of this is by thinking of the differentials as the one forms of a vector field. Now, the vector field would only have a potential function if the differential is exact and this condition is equivalent to the vector field having zero curl.

Now, what I don't understand is how the above idea above idea of curl, vector fields etc relate to the original idea of approximating the surface? This is a standard notion in beginning differential equations courses.

In higher dimensions, this is not the case. If this condition fails, there are no integral manifolds at all, so you cannot "weld" in any meaningful way. But your question is far more specific.

There is a standard "physics-y" argument here. So you cannot build a well-defined surface. Sign up to join this community.

The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? For example, is just the difference between the mean energy of the system in the final macrostate and the initial macrostate , in the limit where these two states are nearly the same.

It follows that if the system is taken from an initial macrostate to any final macrostate the mean energy change is given by Consider, now, the infinitesimal work done by the system in going from some initial macrostate to some neighbouring final macrostate. In general, is not the difference between two numbers referring to the properties of two neighbouring macrostates.

Instead, it is merely an infinitesimal quantity characteristic of the process of going from state to state. In other words, the work is in general an inexact differential. The total work done by the system in going from any macrostate to some other macrostate can be written as Recall that in going from macrostate to macrostate the change does not depend on the process used whereas the work , in general, does.