PHGN311 Homework #4
Due Friday, Sep. 20, 2013 at the beginning of class
Complex numbers and Differentials
Show your work on all problems.
Finish Chapter 2 on complex numbers and read Chapter 4 on partial differentiation
1. Boas 2.15.5
2. Boas 2.16.8
3. Boas 2.17.25
4. Using complex variables, find the general solution to the homogeneous differential
d 3z
equation: 3 − 8z = 0 . Write the solution in real form since its likely in physical
dx
problems we want a real solution.
5. Boas 4.1.6
6. Boas 4.4.3 (these kinds of approximation problems using differentials are commonly
used in physics).
7. Boas 4.4.7
8. Boas 4.4.15
9. Boas 4.13.10 (use the chain rule)
10. In thermodynamics, it’s common to work with differentials. We can write
dU = TdS − PdV where U is the internal energy, T is temperature, S is entropy, P is
pressure and V is volume.
€
! ∂U $
a. Find # &
" ∂ S %V
b. If we define the Gibbs free energy G as G = U − TS + PV find simple
expressions (involving things like P, S, V or T, etc. not complicated partial
! ∂G $
! ∂G $
derivatives) for # & and #
&
" ∂ T %P
" ∂ P %T
c. Assuming G is a smooth nice function, use your results from b to derive the
" ∂ S % " ∂V %
Maxwell relation − $ ' = $ '
# ∂ P &T # ∂ T &P