9-27. In thermodynamics the internal energy U of a gas is a function of pressure
P and volume V denoted by U =U(P, V ).If a gas is involved in a process where
the pressure and volume change with time, then this process can be described by
a curve called a P-V diagram of the process. Let Q=Q(t) denote the amount of
heat obtained by the gas during the process. From the first law of thermodynamics
which states that dQ =dU +P dV , show that dQ =∂U
∂PdP +[
∂U
∂V+P]dV and determine
whether the line integral
∫t 1t 0dQ , which represents the heat received during a time
interval ∆t, is independent of the path of integration or dependent upon the path of
integration.
9-28.
(a) If xand yare independent variables and you are given an equation of the form
F(x) = G(y)for all values of xand ywhat can you conclude if (i) xvaries and y
is constant and (ii) yvaries but xis constant.
(b) Assume a solution to Laplace’s equation ∇^2 φ =
∂^2 φ
∂x^2 +∂^2 φ∂y^2 = 0 in Cartesian
coordinates of the form φ=X(x)Y(y),where the variables are separated. If the
variables xand yare independent show that there results two linear differential
equations
1
Xd^2 X
dx^2=−λ and^1
Yd^2 Y
dy^2=λ,where λis termed a separation constant.
9-29. Evaluate the line integral I=
∫CF·dr, where F =yz ˆe 1 +xz ˆe 2 +xy eˆ 3 and
Cis the curve r =r (t) = cos tˆe 1 + (
t
π+ sin t)ˆe^2 +3 tπˆe^3 between the points (1 ,^0 ,0) and
(− 1 , 1 ,3).9-30. A particle moves along the x-axis subject to a restoring force −Kx. Find
the potential energy and law of conservation of energy for this type of motion.
9-31. Evaluate the line integral
I=∫K