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bio website physics.stackexchange.com/…
location Waterloo, Canada
age 22
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Mathematical Physics Major at the University of Waterloo


Apr
23
awarded  Popular Question
Mar
21
awarded  Popular Question
Feb
4
awarded  Notable Question
Feb
3
awarded  Nice Question
Feb
3
accepted How do we know that heat is a differential form?
Feb
1
comment How do we know that heat is a differential form?
Thank you for your answer. Just a small question of mine. I had always assumed (and been told) that the only restriction on $\delta Q = T\ dS$ was that the process needs to be quasi-static, regardless of whether its reversible or irreversible. Can you give me an example of an irreversible quasi-static process for which $\delta Q = T\ dS$ does not hold?
Jan
31
awarded  Nice Question
Jan
31
comment How do we know that heat is a differential form?
I have little experience with thermodynamics of non-quasi-static processes, but it seems from your addendum that the differential formulation of the first law $dU = \delta Q - \delta W$ is practically limited to quasi-static thermodynamics. This has been a deeper question than I had anticipated. I appreciate everyone's help with this.
Jan
31
awarded  Commentator
Jan
31
comment How do we know that heat is a differential form?
Thank you for the revision. If you could perhaps provide a proof or a reference that $\delta W$ is a form field then I would be happy to call it day. That fact seems intuitive but is not obvious to me. In my original question I also questioned whether work was a differential form. I only knew that if one was, then the other must be due to the first law.
Jan
31
comment How do we know that heat is a differential form?
I really appreciate all the time and patience you've put into this. I admit this is rather counter-intuitive to how I understood $\delta Q$. If I understood you correctly, a curve $\gamma$ through state space is not all there is to a "process". So there are different "processes" which traverse the same curve $\gamma$ but given different $Q$. For each process, we have a separate $\delta Q$, which for that process is a well defined differential.
Jan
31
awarded  Yearling
Jan
31
comment How do we know that heat is a differential form?
I apologize for the confusion. I think my question got a bit side tracked somewhere along the way. I had always assumed that there exists some universal $\delta Q$. Each process is simply some curve $\gamma$ in the state space which links some initial state $\mathbf{x}_1$ and some final state $\mathbf{x}_2$ and the net heat transfer $Q$ is the integral $\int_\gamma \delta Q$. That is to say, while the net heat $Q$ depends on the process, the $\delta Q$ differential itself does not. You seem to be saying that each process is linked with a separate differential form.
Jan
31
comment How do we know that heat is a differential form?
Thank you for the correction. I had meant to say that differentials are linear at each point in the state space, i.e. linear on the tangent space. I have always heard $\delta Q$ being called an inexact differential. That seems to imply that $$\delta Q(d\mathbf{x}_1 + d\mathbf{x}_2) = \delta Q(d\mathbf{x}_1) + \delta Q(d\mathbf{x}_2)$$ at some fixed point, just by the definition of "differential form". If I've understood you correctly, you're saying that this doesn't necessarily hold? Is it inappropriate to call $\delta Q$ a differential form then?
Jan
31
asked How do we know that heat is a differential form?
Dec
13
awarded  Popular Question
Dec
3
awarded  Constituent
Dec
3
awarded  Caucus
Sep
5
comment Dimensional analysis restricted to rational exponents?
I've chosen to accept this question instead as I feel the answer is more concrete than the previously accepted one. I appreciate the examples in particular. Still, I wish that the other aspect of my question: "Why is it beneficial to restrict dimensional analysis to rational exponents?" was discussed more.
Sep
5
accepted Dimensional analysis restricted to rational exponents?