# Inexact/exact differentials

1. Feb 9, 2005

### krusty the clown

What is the difference between an exact and and inexact differential?
These were introduced in my physics 2 book with the first law of thermodynamics represented by differentials,

dEint= dQ + dW

Then, it has a note that says
"Note that dQ and dW are not true differential quantities because Q and W are not state variables; however dEint is. Because dQ and dW are inexact differentials, they are often represented by the symbols dQ and dW ,both with lines through the vertical part of the d's. Sorry, I it wouldn't let me open the latex instructions.

Thanks- Erik

2. Feb 9, 2005

### dextercioby

Well,the short version is that
If dU is an exact differential,and U(T,V,N) is the function,then

$$dU=\frac{\partial U}{\partial T} dT+\frac{\partial U}{\partial V} dV+\frac{\partial U}{\partial N} dN$$ (1)

$\delta Q$ and $\delta L$ are not total differentials,maening the functional dependence of the functions Q and L cannot put us in the position to write an equality similar to (1).I'm sure that every serious thermodynamics text (Callen,Greiner) discusses this mathematical feature.

Daniel.

3. Feb 9, 2005

### mathwonk

When I hear that language it makes me think of the following case: an exact differential is an expression of form df where f is a single valued function. another characterization is that a differential is exact if its integral around every closed loop is zero.

sometimes however one encounters expressions like dtheta, where theta of course is the angle function. now since the angle function is multivalued, not single valued, and since the integral of dtheta around the unit circle is 2pi and not zero, dtheta is not exact although it looks like one.

As an ignorant bystander I am going to guess this is also what is going on in your situation. but i mgiht be wrong since i do not understand any of the other words in your post. like "state variable"....

4. Feb 10, 2005

### dextercioby

That's merely a convention.Still the mathematics behind 1 forms is essential.Names can change from book to book.

Daniel.

5. Feb 10, 2005

### krusty the clown

I still don't understand completely, but at this point we are just treating them like exact differentials so it isn't that important right now. I was just a little currious.

Thaks for you help.
Erik

6. Feb 10, 2005

### mathwonk

if the integral over every closed loop is zero then they are exact, otherwise not. thats it.

these are sometimes called conservative force fields.

i.e. if that quantity you are calling E(int) cannot change when you traverse a closed path, then it is conservative, and dE(int) is exact.