Exact differential
In multivariate calculus, a differential or differential form is said to be exact or perfect (so called an "exact differential"), as contrasted with an inexact differential, if it is equal to the general differential "dQ" for some differentiable function "Q". An exact differential is sometimes also called a "total differential", or a "full differential", or, in the study of differential geometry, it is termed an exact form. The integral of an exact differential over any integral path is path-independent, and this fact is used to identify state functions in thermodynamics. Overview. Definition. Even if we work in  three dimensions here, the definitions of exact differentials for other dimensions are similar to the three dimensional definition. In three dimensions, a form of the type is called a differential form. This form is called "exact" on a open domain formula_2 in space if there exists some differentiable scalar function formula_3 defined on formula_4 such that throughout formula_4, where formula_8 are orthogonal coordinates (e.g., Cartesian, cylindrical, or spherical coordinates). In other words, in some domain of three dimensional space, a differential form is an "exact differential" if it is equal to the general differential of a differentiable function. Integral path  independence. The exact differential for a differentiable function formula_9 is equal to formula_10, that is the scalar product between the conservative vector field formula_11 (where the right hand side is the gradient of formula_9) for the corresponding potential formula_9 and the general differential displacement vector formula_14. The gradient theorem states that does not depend on which integral path between the given path endpoints formula_16 and formula_17 is chosen. So it is concluded that the integral of an exact differential is independent of the choice of an integral path between given path endpoints (path independence). For three dimensional spaces, this integral  path independence can also be proved by using the vector calculus identity formula_18 and the Stokes' theorem. for a closed loop formula_20 with the smooth oriented surface formula_21 in it. Thermodynamic state function. In thermodynamics, when formula_22 is exact, the function formula_9 is a state function (a mathematical function depending only on the current equilibrium state, not depending on the system path taken to reach the equilibrium state) of the system. Internal energy formula_24, Entropy formula_25, Enthalpy formula_26, Helmholtz free energy formula_27, and Gibbs free energy formula_28 are state functions. Generally, neither work formula_29 nor heat formula_9 (A common character  representing heat, don't confuse this as an exact differential even if the same character formula_9 is used to represent exact differentials.) is a state function. One dimension. In one dimension, a differential form is exact if and only if formula_27 has an antiderivative (but not necessarily one in terms of elementary functions). If formula_27 has an antiderivative and let formula_9 be an antiderivative of formula_27 so formula_37, then formula_32 obviously satisfies the condition for exactness. If formula_27 does "not" have an antiderivative, then we cannot write formula_40 with formula_41 for a differentiable function formula_9 so formula_32 is inexact . Two and three dimensions. By symmetry of second derivatives, for any "well-behaved" (non-pathological) function formula_9, we have Hence, in a simply-connected region "R" of the "xy"-plane, a differential form is an exact differential if and only if the equation holds. If it is an exact differential so formula_48 and formula_49, then formula_9 is a differentiable (smoothly continuous) function along formula_51 and formula_52, so formula_53. If formula_47 holds, then formula_27 and formula_56 are differentiable (again, smoothly continuous) functions along formula_52 and formula_51 respectively, and formula_53 is only the case. For three dimensions, in a simply-connected region "R  of the xyz"-coordinate system, by a similar reason, a differential is an exact differential if and only if between the functions "A", "B" and "C" there exist the relations These conditions are equivalent to the following sentence: If "G" is the graph of this vector valued function then for all tangent vectors "X","Y" of the "surface" "G" then "s"("X", "Y") = 0 with "s" the symplectic form. These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives. So, in order for a differential "dQ", that is a  function of four variables, to be an exact differential, there are six conditions (the combination formula_64) to satisfy. Partial differential relations. If a differentiable function formula_65 is one-to-one (injective) for each independent variable, e.g., formula_65 is one-to-one for formula_51 at a fixed formula_52 while it is not necessarily one-to-one for formula_69, then the following total differentials exist because each independent variable is a differentiable function for the other variables, e.g., formula_70. Substituting the first equation into the second and rearranging, we obtain Since formula_52 and formula_77 are independent variables, formula_78 and formula_79 may be  chosen without restriction. For this last equation to generally hold, the bracketed terms must be equal to zero. The left bracket equal to zero leads to the reciprocity relation while the right bracket equal to zero goes to the cyclic relation as shown below. Reciprocity relation. Setting the first term in brackets equal to zero yields A slight rearrangement gives a reciprocity relation,