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Differentials. Multi-Dimensional Spaces. depending on a single parameter . Differentials are a powerful mathematical tool. They require, however, precise introduction. exact. differentials. In particular we have to distinguish between. inexact.

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Differentials

Multi-Dimensional Spaces

depending on a single

parameter

Differentials are a powerful mathematical tool

They require, however, precise introduction

exact

differentials

In particular we have to distinguish between

inexact

Remember some important mathematical background

Point (in D-dimensional space)

Line: parametric representation

D functions


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for

Example in D=2 from classical mechanics

where t0=0 and tf=2vy/g

x2

0

x1


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Scalar field: a single function of D coordinates

For example: the electrostatic potential of a charge or the gravitational potential

of the mass M (earth for instance)

r


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z

y

x

Vector field: specified by the D components of a vector.

Each component is a function of D coordinates

Well-known vector fields in D=3

Graphical example in D=3

Force F(r) in a gravitational field

Electric field: E(r)

Magnetic field: B(r)

3 component entity

Each point in space


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Line integral:

scalar product

If the line has the parameter representation:

i=1,2,…,D

for

The line integral can be evaluated like an ordinary 1-dimensional definite Integral


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y

x

z

t

Let’s explore an example:

Consider the electric field created by a changing magnetic field

where

y

y

y

Line of integration

R

0

x

x

f

x


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Parameter representation of the line:

Counter clockwise walk along the semicircle of radius R

y

x

1

Note: Result is independent

of the parameterization


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y

Line of integration

Let’s also calculate the integral around the full circle:

Parameter representation of the line:

R

x

Faraday’s law of electrodynamics

Have a closer look to

or

Differential form


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Meaning of an equation that relates one differential form to another

Equation valid for all lines

Must be true for all sets of coordinate differentials

Example:

Particular set of differentials

Relationships valid for vector fields are

also valid for differentials


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Exact and Inexact Differentials another

y

x

z

t

is an exact differential

A differential form

if for all i and j it is true that

.

An equivalent condition reads:

also written as

Let’s do these Exactness tests in the case of our example

Is the differential form

exact


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- another

Check of the cross-derivatives

but

Not exact

Alternatively we can also show:

+

=

= 0

=


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Example from thermodynamics another

Exactness of

Transfer of notation:

T , V are the coordinates of the space

1

Functions corresponding to the vector components:

Check of the cross-derivatives

2

=

exact


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Differential of a function another

and

Independent of the path

between

Scalar field: a single function of D coordinates

or in compact notation

where

Differentials of functions are exact

Proof:

x2

Or alternatively:

x1

Line integral of a differential of a function


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We are familiar with this property from varies branches of physics:

Conservative forces:

Remember: A force which is given by the negative gradient

of a scalar potential is known to be conservative

Gravitational force derived from

Example:

Pot. energy

depends

on h, not how

to get there.

h


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Exact differential theorem physics:

for all closed contours

and

Independent of the line connecting

x2

x1

The following 4 statements imply each other

dA is the differential of a function

1

dA is exact

2

3

4


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How to find the physics:function underlying an exact differential

Consider:

Since dA exact

Aim:

Find A(x,y) by integration

Comparison

constant

Unknown function depending on y only

Apart from

one const.

A(x,y)

Unknown function depending on x only

constant


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Example: physics:

where a,b and c are constants

First we check exactness

Comparison

Check:


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Inexact Differentials of Thermodynamics physics:

are

Values of W,Q and

We know:

Equilibrium processes can be represented by lines in state space

Consider infinitesimal short sub-process

Quantities of infinitesimal short sub-processes

With first law

for all lines L

Since U is a state function we can express

U=U(T,V)

dU differential form of a function

dU exact

inexact

However:


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inexact physics:

How can we see that

Compare with the general differential form

for coordinates P and V

and

inexact

=

Line dependence of W and line independence of U

Example:

P0

Work:

isothermal

Pf

Vf

V0


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Coordinates on common isotherm physics:

=

P

V

P0

Pf

Vf

V0

Internal energy:

Isothermal process from

1

U=U(T)

Ideal gas

1

2

2

Across constant volume and constant pressure path


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T physics:0= Tf

is inexact

How can we see that

-R

+R

Consider U=U(P,V)

where P and V are the coordinates

with

Since

inexact

inexact

Alternatively inspection of

exact

inexact


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Coordinate transformations physics:

Example:

Changing coordinates of state space from (P,V)

(T,P)

V=V(T,P)

If U=U(T,P)

With

+


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Let’s collect terms of common differentials physics:

Remember: Enthalpy

H=U+PV

with

Similar for changing coordinates of state space from (P,V)

(T,V)


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Heat capacities expressed in terms of differentials physics:

From

P=const.

and

V=const.

are alternate notation for the components

Note:

and

(of the above vector fields which correspond to the differential forms)

and

Do not confuse with partial derivatives, since there is no functionQ(T,P)

.

is inexact

whose differential is


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