When I first
learned this concept in
Vector Calculus and
Ordinary Differential Equations, it seemed really pointless.
Hardcore math
geeks don't solve diff eq's with no pansy-ass Integrating Factors!
Then I took Thermodynamics.
Basically, in Physics they are important because they illustrate the idea of
path independancy. The definition of an exact differential is as follows:
If there exists a function T(x,y) such that the partial of T with respect to x equals P(x,y) and the partial of T with respect to y equals Q(x,y), the
differential equation formed by P(x,y) + Q(x,y) * y' = 0 is exact. This is because the function y(x) that satisfied the equation obtained by differentiating T(x,y) with
respect to x by the
chain rule. (Looks a hell of a lot like
Green's Theorem).
Now, this just sounds more and more complicated and pointless. What's the point?
For equations that are exact differentials, there is a general solution. And general solutions, as everybody knows, are very good things.
To make a non-exact differential into an exact differential for easier solving involves an
Integrating Factor. You multiply the whole equation by an integrating factor, which is either previously
determined or just guess one that seems to 'fit' with the equation.
In
thermo, these exact differentials are known as
state functions because they depend only on the inital and final
state of the system and not the
path traversed.
Entropy is an exact differential. Heat is not.
I thought that was kind of cool. It turns out to be extremely
logical because entropy is defined as a state of a system, similar to
mass or
volume, while heat is not. Therefore, the integrating factor to express
heat is 1/T.
So the whole point of
thermodynamics is to simply invent a set of functions which describe the system but have
exact differentials with respect to variables P,T,V and S.