Yep, it's that time again. I have an introductory thermodynamics (Physic 103) exam in two weeks, and I have some general questions about differentail equations.
While there's a parallel course in basic multivariable calculus that is somewhat illuminating, and I'm reasonably comfortable with partial differential equations, gradients, Jacobians, multiple integrals and so on (I mean, it's mostly just extentions of introductory caluclus and linear algebra), the thermodynamics course material is, as I understand is usual, very handwavy in its treatment of (exact and inexact) differentials, differentiation and multivariable calculus (indeed banishing anything resembling a detailed treatment to a three-page appendix) - especially since the calculus material I'm used to uses standard epsilon-delta methodology.
There are more differential equations than I can count, but quite a few of them are on this form:
dU = (δU/δT)VdT + (δU/δV)TdV
This is, of course, not an unfamiliar form of equation. Indeed, since we have U = f(T, V), then the above equation would be equivalent to
dU = grad(U) dot (dT, dV)
Hum, so this seems analogous to Cauchy's definition dy = f'(x) dx. So, can I understand this equation in a sense more mathematical than "Some small change in the internal energy dU is given by the partial derivative with respect to temperature times some small change in temperature etc etc", as it is usually given? The equation U = f(T,V) apparently describes a three-dimensional surface (and f(T,V) = constant some two-dimensional level curve). How should I then understand the vectors (dT, dV, dU) and (dT,dV)? Can I think of (dT, 0, 0), (0, dV, 0), (0, 0, dU) as unit vectors in the dimension of their respective physical quantity? Or will I have to make peace with dT, dV, dU representing abstract, but real quantities of change?
And that's not even getting into the inexact differentials... but as I understand it, you get that when you don't have an actual equation Q = f(p,T,V) that you differentiate with respect to each quantity, but rather different equations for the various quantities.
I would be thankful for anyone who could help shed some light on this, or link to a good resource.
While there's a parallel course in basic multivariable calculus that is somewhat illuminating, and I'm reasonably comfortable with partial differential equations, gradients, Jacobians, multiple integrals and so on (I mean, it's mostly just extentions of introductory caluclus and linear algebra), the thermodynamics course material is, as I understand is usual, very handwavy in its treatment of (exact and inexact) differentials, differentiation and multivariable calculus (indeed banishing anything resembling a detailed treatment to a three-page appendix) - especially since the calculus material I'm used to uses standard epsilon-delta methodology.
There are more differential equations than I can count, but quite a few of them are on this form:
dU = (δU/δT)VdT + (δU/δV)TdV
This is, of course, not an unfamiliar form of equation. Indeed, since we have U = f(T, V), then the above equation would be equivalent to
dU = grad(U) dot (dT, dV)
Hum, so this seems analogous to Cauchy's definition dy = f'(x) dx. So, can I understand this equation in a sense more mathematical than "Some small change in the internal energy dU is given by the partial derivative with respect to temperature times some small change in temperature etc etc", as it is usually given? The equation U = f(T,V) apparently describes a three-dimensional surface (and f(T,V) = constant some two-dimensional level curve). How should I then understand the vectors (dT, dV, dU) and (dT,dV)? Can I think of (dT, 0, 0), (0, dV, 0), (0, 0, dU) as unit vectors in the dimension of their respective physical quantity? Or will I have to make peace with dT, dV, dU representing abstract, but real quantities of change?
And that's not even getting into the inexact differentials... but as I understand it, you get that when you don't have an actual equation Q = f(p,T,V) that you differentiate with respect to each quantity, but rather different equations for the various quantities.
I would be thankful for anyone who could help shed some light on this, or link to a good resource.
via International Skeptics Forum http://ift.tt/1FKCSIK
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