1. both A and B give the area under the integral curve for y(x) where y is zero at x = 0 and dy/dx is always positive.
dA = y dx
dB = We can call that area also U but
dU = where stands for inexact differential. Inexact differentials are used in thermodynamics where only the sum of the two differentials have meaning, not their individual values. The individual values are path dependant. B stands for mechanical and A for thermal energy.  2.

3. Originally Posted by Rudiger von Massow Inexact differentials are used in thermodynamics where only the sum of the two differentials have meaning, not their individual values.
This appears to me to be a misunderstanding of the nature of inexact differentials. An exact differential is a differential of a function of the variables of the relevant space (in thermodynamics, the state). A path-integral evaluates to the difference of the function at the two endpoints and is independent of the path itself. The key issue is the existence of the function itself. One may have a function over paths in a space but the function over the space itself does not exist. This would be the case if the path-integral of the corresponding differential evaluates to different values for different paths between the same two endpoints. A simple example is distance. The distance between two points depends on the path between those two points. Therefore, there is no distance function over space. But one can still have a differential of distance for any given path. Such a differential will be inexact.

If such a function exists, then the chain rule says: and the differential is exact. But for the differential , if the partial derivatives , , etc do not exist, then is inexact.  4. The two differentials dA and dB are both limited to one dimension, so the other partial derivatives are missing. That, according to you, makes them inexact.  5. Originally Posted by Rudiger von Massow The two differentials dA and dB are both limited to one dimension, so the other partial derivatives are missing. That, according to you, makes them inexact.
Well, no. I was assuming the space in which the differentials exist was at least two-dimensional. This is the usual context for discussing inexact differentials. One normally doesn't refer to the definite integral of a function of the real numbers as a "path integral". Furthermore, the space of states in thermodynamics is typically two-dimensional (pressure and temperature).

What would make dA inexact in any case is the non-existence of A (and similarly for dB).

A point I was attempting to correct with my original reply is the suggestion in the opening post that the sum of two inexact differentials is exact. In other words, the first law of thermodynamics is not the result of a property of inexact differentials.  6. Sorry, I have to go to thermodynamics to prevent confusing myself and others. In thermodynamics
dU = is a well accepted formula supported by steam tables. Putting the steam into a tube with movable plug I can plot U versus entropy and column length using steam table data in the region of steam stability.
Sorry, I don't know how to include a figure.
I can read of that figure that:  This is based the equation above. It means that mechanical energy only alters the temperatures, heat requirements of entropy differentials, of the steam, not the entropy. Heat addition increases entropy. Thus mechanical energy and heat are inexact quantities since their differentials are inexact.
I tried to get confirmation from a mathematician.  7. I admit the double integral is problematic, since has the form of a curved splinter with its point at the origin. Laying these next to one another until the wide ends sum up to T makes: That interpretation may not be quite kosher. What do you think?
By the way: Can I send you an email with the figures to demonstrate ?
The in post 5 should also be   8. It is so simple: The energy of a steam is That integral extended to more S is
dQ = TdS if no work is exchanged
That integral extended to higher T is
dW = FdX = if no heat is exchanged.
Both work W and heat Q are inexact if any time during the process the other energy is also varied:
dU = U = can be reached by any combination of one or the other energy increases.  9. Originally Posted by Rudiger von Massow In thermodynamics
dU = is a well accepted formula supported by steam tables.
Yes. That is the First Law of Thermodynamics. And you are correct in saying that dQ and dW are inexact and dU is exact. But you seem to be suggesting that dU is exact because dQ and dW are inexact, implying that the sum of two inexact differentials is exact. This is not mathematically true. The First Law of Thermodynamics is a law of physics and not the mathematical consequence of properties of inexact differentials. Originally Posted by Rudiger von Massow Thus mechanical energy and heat are inexact quantities since their differentials are inexact.
This is a misrepresentation of the terms "exact" and "inexact" as they are applied to differentials.

It's worth noting that in thermodynamics, one defines dQreversible and dWreversible as the corresponding differentials along paths in state space that are reversible (the direction can be reversed by an infinitesimal change of state). The restriction to reversible paths allows functions of state to be defined from the differentials and therefore these differentials are exact. They are used to define functions of state such as enthalpy, entropy, Gibbs free energy, etc.  10. In my above arguments I have always implied that dQ and dW are reversible. The first law dU = is applicable to reversible and irreversible processes. A steam locomotive can operate even if the process may be almost reversible. Even if reversible the change in mechanical energy and heat are path dependant and therefore convertible.
Enthalpy, the Helmholtz function and the Gibbs function are functions which result from the partial or total application of Legendre transformation on energy. They do not enter the above discussion.  11. The definition of 'inexact', though that is poor nomenclature, is that it applies to a pair such that dU = The two individually are variable and in their variation define a different path.  12. The first Law of thermodynamics is the result of numerous experimental data. If mathematics can not handle it, that is too bad. I would like to prove that if are expressed as: and then they are inexact differentials and follow all paths possible in dU = . I can do that graphically but I do not know how to transfer a drawing to a reply.  13. I am not at all sure that discussion of Carnot cycles has a real place in this sub-forum, especially as you are committing one the major sins in marhetmatics - failure to declare your symbols. Naughty!!

However I do agree with this Originally Posted by KJW This is a misrepresentation of the terms "exact" and "inexact" as they are applied to differentials.
Let's see. Suppose a differential p-form. Then if one says this is a closed differential (form)

Recall that the operator maps p-forms onto (p+1)-forms.

So if, for some and a p-form there exists a (p - 1)-form such that one calls this an exact differential (form). Obviously for an inexact differential form this is not true

Clearly all exact differential forms are closed - . This is the Poincare Lemma

The converse is true - for every closed p-form there exists a (p+1)-form - only if the space over which these differential forms are defined is topologically trivial (or only locally otherwise).

The implications are profound, especially in applications  14. We do not have to be scared of 'inexact differentials' defining an exact quantity. My joined bank account has two inexact differentials. . As the money is paid in it looses its identity and when it is taken out it gets a new unrelated identity in my or my wife's wallet. This is only a very simplified comparison. The BA depends only on the sum, not on the individual values.  15. It is interesting that the most important and best proven law of thermodynamics gets poohooed by two mathematicians. Interesting.  16. Originally Posted by Rudiger von Massow It is interesting that the most important and best proven law of thermodynamics gets poohooed by two mathematicians. Interesting.
I don't think they are saying any such thing. And, from the way he writes, I would guess KJW is a very competent physical chemist or physicist.  inexact differentials, path dependency 