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

Stephen R. Addison (UCA) Partial Derivatives January 22, 2013 2 / 16
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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.


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.


Th Ch i R l
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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.


Relationships between Partial Derivatives
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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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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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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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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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p


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


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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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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leading to

dx =

x

f

y

+

x

y

f

y

f

z

df+

x

y

f

y

z

f

dz.




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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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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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04. partial derivatives

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