Math 18.02 (Spring 2009): Lecture 14
Nonindependent variables. Partial differential equations
March 6
Reading Material:
From
Simmons
: 19.6. From
Course Notes
: N.
Last time:
Lagrange Multipliers
Today:
Nonindependent variables. Partial differential equations.
2
Nonindependent variables
If you go to OCW and look at Lecture 21 of 3.00: Thermodynamics of Materials, you will see that
the title of this class is ”Mathematics of Thermodynamics”.
In the middle of the lecture notes
relative to this lecture you will find an equation involving differentials that looks like this:
dU
=
TdS

PdV
+
C
i
=1
μ
i
dN
i
,
(2.1)
where all the functions are measuring the following quantities relative to a gas:
T
=
Temperature
U
=
Internal energy
S
=
Entropy
V
=
Volume
N
i
=
Numbers of molecules of type
i
.
The relationship among these quantities described by (2.1) is a combination of the First and Second
Laws of Thermodynamics that I am not going to state here. The point though is that if one thinks
about
T, U, S, V
and
N
i
, for
i
= 1
, ..., C
as variables, then (2.1) is a way to describe how they are
related.
From the definition of differential we can deduce the partial derivatives of a function. In fact recall
that for a function of two variables
z
=
f
(
x, y
)
,
where
x
and
y
are the independent variables, we
have the differential
dz
=
f
x
dx
+
f
y
dy.
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 Spring '06
 HartleyRogers
 Equations, Derivative, Multivariable Calculus, Partial Differential Equations, Partial differential equation

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