Mathematical Tools for Physics

8—Multivariable Calculus 209

The problem is that thenotationis ambiguous. When I write∂f/∂xit doesn’t tell be what I’m to hold
constant. Is it to beyory′or yet something else? In some contexts the answer is clear and you won’t have any
difficulty deciding, but you’ve already encountered cases for which the distinction is crucial. In thermodynamics,
when you add heat to a gas to raise its temperature does this happen at constant pressure or at constant volume
or with some other constraint? The specific heat at constant pressure is not the same as the specific heat at
constant volume; it’s necessarily bigger because during an expansion some of the energy has to go into the work
of changing the volume. This sort of derivative depends on type of process that you’re using, and for a classical
ideal gas the difference between the two molar specific heats obeys the equation

cp−cv=R

If the gas isn’t ideal, this equation is replaced by a more complicated and general one, but the same observation
applies, that the two derivativesdQ/dT aren’t the same.
In thermodynamics there are so many variables in use that there is a standard notation for a partial derivative,
indicating exactly which other variables are to be held constant.

(

∂U

∂V

)

T

and

(

∂U

∂V

)

P

represent the change in the internal energy of an object per change in volume during processes in which respectively
the temperature and the pressure are held constant. In the previous example with the functionf=y, this says

(

∂f

∂x

)

y

= 0 and

(

∂f

∂x

)

y′

=− 1

This notation is a way to specify thedirectionin thex-yplane along which you’re taking the derivative.

8.2 Differentials
For a function of a single variable you can write

df=

df

dx

dx (3)