04. partial derivatives
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The Chain Rule, Differentials and Partial Derivatives
Stephen R. Addison
University of Central Arkansas
January 22, 2013
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Outline
1 Introduction
2 The Chain Rule
3 Relationships between Partial Derivatives
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Introduction
Introduction
In this class we will review some results of basic calculus. Unfortunately, as
introductory calculus classes focus more and more on the use of thecalculator, many of you will be seeing these results for the first time. Youwill find it useful to become proficient at these sorts of manipulations.
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The Chain Rule
The Chain Rule
Consider y = f(x) and x = g(t) so y = f(g(t)). Using the chain rule we
can find dy/dt,dy
dt=
df
dx
dx
dt.
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Th Ch i R l
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The Chain Rule
Chain Rule for Partial Derivatives
Now consider a function w = f(x, y, x). Suppose we want to explore the
behavior of f along some curve C, if the curve is parameterized byx = x(t), y = y(t), and z = z(t). If we want to calculate dw
dtwe need the
chain rule for partial derivatives.
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The Chain Rule
Chain Rule for Partial Derivatives
The chain rule for partial derivatives:
dw
dt=
w
x
dx
dt+
w
y
dy
dt+
w
z
dz
dt.
Using this, we define the total differential ofw as
dw =w
xdx+
w
ydy+
w
zdz.
In this expression the wxi
dxi are called partial differentials.
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Relationships between Partial Derivatives
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Relationships between Partial Derivatives
Relationships between Partial Derivatives
We can use the definition of the total differential to develop relationshipsbetween partial derivatives. In thermal physics, we will usually want toexplicitly denote which variables are being held constant. We do this by
placing subscripts on our partial derivatives. Thus we can rewrite ourexpression for the differential ofw as
dw = w
xy,zdx+ w
yx,zdy+ w
zx,y dz.
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Relationships between Partial Derivatives
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Relationships between Partial Derivatives
Relationships between Partial Derivatives
Now suppose there is a relationship between x, y, and z such thatf(x, y, z) = 0. We can then write x = x(y, z), y = y(x, z), and
z = z(x, y). We can write the total differential of x as
dx =
x
y
z
dy+
x
z
y
dz. (1)
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Relationships between Partial Derivatives
Relationships between Partial Derivatives
Similarly, from y = y(x, z), we write the total differential of y as
dy = y
xz
dx+ y
zx
dz. (2)
Then, inserting (1) in (2) we have,
dx = x
y
z
y
x
z
dx+ x
y
z
y
z
x
+ x
z
y dz.
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Relationships between Partial Derivatives
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Relationships between Partial Derivatives
Relationships between Partial Derivatives
Only two of the variables x, y, z are independent. The above equation istrue for all values ofdx and dz. If we choose x and z as the independentvariables and dz = 0 and dx= 0. Then
dx = xy
zy
x
z
dx
or, x
y
z
y
x
z
= 1
which we rewrite as xy
z
=1
yx
z
.
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Relationships between Partial Derivatives
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p
Relationships between Partial Derivatives
Next we choose dx = 0 and dz= 0, sox
y
z
y
z
x
+
x
z
y
dz = 0
x
y
z
y
z
x
=
x
z
y
and finally we write
xy
z
yz
x
zx
y
= 1.
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p
Relationships between Partial Derivatives
Now lets consider f as a function ofx, y, and z where x, y and z are
related. This means that we can write f= f(x, y) or f= f(x, z), etc. Or,we could equally well write one of the xi as a function of f and the otherxi.
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Relationships between Partial Derivatives
If we write x = x(f, y) and y = y(f, z) then we can immediately writetheir total differentials as
dx =x
f
ydf+
xy
fdy
and
dy = y
fz
df+ y
zf
dz
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Relationships between Partial Derivatives
leading to
dx =
x
f
y
+
x
y
f
y
f
z
df+
x
y
f
y
z
f
dz.
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Relationships between Partial Derivatives
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Relationships between Partial Derivatives
We can also write x = x(f, z) and write its total differential as
dx =
x
f
z
df+
x
z
f
dz.
Then equating the coefficients ofdf and dz in the expressions for dx, the
coefficients ofdz result inx
y
f
y
z
f
=
x
z
f
which we rewrite as x
y
f
y
z
f
z
x
f
= 1.
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Relationships between Partial Derivatives
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Relationships between Partial Derivatives
We will find many of these results useful in our study of thermal physics.
You will find that you will learn some of them. However, the importantthing is to be able to derive them when they are needed.
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