# Question about total derivative/chain rule

1. Mar 23, 2012

### mSSM

The following actually comes from Landau's 3rd Edit. Statistical Physics Part 1, Paragraph on Adiabatic Processes, Page 39.

I have the following two equations, where $\lambda=\lambda(t)$. I am not so sure about $S$ (which is somewhat my problem):
$$\frac{\mathrm{d}S}{\mathrm{d}t} = \left( \frac{\mathrm{d}\lambda}{\mathrm{d}t} \right)^2$$

Which is supposed to mean:
$$\frac{\mathrm{d}S}{\mathrm{d}\lambda} = \frac{\mathrm{d}\lambda}{\mathrm{d}t}$$

Now, I thought that what is essentially being done there is multiplying the first equation such that we get:
$$\frac{\mathrm{d}S}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\lambda} = \frac{\mathrm{d}\lambda}{\mathrm{d}t}$$

But if I now assume that:
$$\frac{\mathrm{d}S}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\lambda} = \frac{\mathrm{d}S}{\mathrm{d}\lambda}$$
doesn't that in turn mean that $t$ is a function of $\lambda$? Mathematically this seams sounds (to me), but physically this does not make so much sense, if $t$ is the time.

2. Mar 23, 2012

### tiny-tim

welcome to pf!

hi mSSM! welcome to pf!
suppose λ is distance

why shouldn't time be a function of distance?

3. Mar 23, 2012

### mSSM

Re: welcome to pf!

sin
Thanks! :) Okay, I guess you could turn it that way. So essentially you say that we simply associate every instant of time with a certain "distance"... Since in my case the distance (=external condition) is a mechanical quantity I guess this would be sound.

Would something like that still be acceptable for a macroscopic quantity?

4. Mar 23, 2012

### tiny-tim

i'm not sure i understand that sentence

a quantity is a quantity

why would length (or any other quantity) be unacceptable?

5. Mar 23, 2012

### mSSM

Yeah, you are right. I can't think of a reason why that shouldn't work. :)