aarjaneiro/planetmath huggingface dataset:
hypoelliptic

hypoelliptic

Definition

.

Let

P

be a partial differential operator defined in an open subset

U

⊂

ℝ

n

.
If
for every

distribution

(

http://planetmath.org/Distribution4

)

u

defined in an open subset

V

⊂

U

such that

P

⁢

u

is

C

∞

(smooth),

u

must also be

C

∞

, then

P

is
called

hypoelliptic

.

Similarly, if the same assertion holds with

C

∞

replaced by
real analytic,
then

P

is said to be

analytically hypoelliptic

.

Note that some authors use “hypoelliptic” to mean “analytically hypoelliptic.” Hence, if it is not clear from context, it is best to specify the regularity when using the term. For example,

C

∞

-hypoelliptic instead of just hypoelliptic.

References

1

J. Barros-Neto, Ralph A. Artino.
,
Lecture Notes in Pure and Applied Mathematics, 53. Marcel Dekker, Inc., New York, 1980.

http://www.ams.org/mathscinet-getitem?mr=81k:35031

MR 81k:35031

2

Bernard Helffer, Francis Nier.
,
Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005.

http://www.ams.org/mathscinet-getitem?mr=2006a:58039

MR 2006a:58039

3

Norio Shimakura.
,
Kinokuniya Company Ltd., Tokyo, Japan, 1978.

Title

hypoelliptic

Canonical

name

Hypoelliptic

Date of creation

2013-03-22 16:01:13

Last modified on

2013-03-22 16:01:13

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 35H10

Defines

analytically hypoelliptic

Defines

analytic hypoelliptic

Generated on Fri Feb 9 20:43:38 2018 by

LaTeXML
differential equations of Jacobi ϑ functions

differential equations of Jacobi

ϑ

functions

The

theta functions

the following

partial differential equation

:

π

⁢

i

4

⁢

∂

2

⁡

ϑ

i

∂

⁡

z

2

+

∂

⁡

ϑ

i

∂

⁡

τ

=

0

It is easy to check that each in the series which define the theta functions this

differential equation

. Furthermore, by the

Weierstrass M-test

, the series obtained by differentiating the series which define the theta functions term-by-term

converge absolutely

, and hence one may compute

derivatives

of the theta functions by taking derivatives of the series term-by-term.

Students of mathematical physics will recognize this equation as a one-dimensional diffusion equation. Furthermore, as may be seen by examining the series defining the theta functions, the theta functions approach periodic delta distributions in the limit

τ

→

0

. Hence, the theta functions are the Green’s

functions

of the one-dimensional diffusion equation with periodic boundary conditions.

Title

differential equations of Jacobi

ϑ

functions

Canonical name

DifferentialEquationsOfJacobivarthetaFunctions

Date of creation

2013-03-22 14:41:19

Last modified on

2013-03-22 14:41:19

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 35H30

Generated on Fri Feb 9 20:43:40 2018 by

LaTeXML
Poisson’s equation

Poisson’s equation

Poisson’s equation is a second-order

partial differential equation

which arises in physical problems such as finding the electric potential of a given charge

distribution

. Its general form in

n

is

∇

2

⁡

ϕ

⁢

(

𝐫

)

=

ρ

⁢

(

𝐫

)

where

∇

2

is the

Laplacian

and

ρ

:

D

→

ℝ

, often called a

source

, is a given

function

on some subset

D

of

ℝ

n

. If

ρ

is identically zero, the

Poisson equation

reduces to the

Laplace equation

.

The Poisson equation is linear, and therefore obeys the

superposition principle

: if

∇

2

⁡

ϕ

1

=

ρ

1

and

∇

2

⁡

ϕ

2

=

ρ

2

, then

∇

2

⁡

(

ϕ

1

+

ϕ

2

)

=

ρ

1

+

ρ

2

. This fact can be used to construct solutions to Poisson’s equation from

fundamental solutions

, or

Green’s functions

, where the source distribution is a

delta function

.

A very important case is the one in which

n

=

3

,

D

is all of

ℝ

3

, and

ϕ

⁢

(

𝐫

)

→

0

as

|

𝐫

|

→

∞

. The

general solution

is then given by

ϕ

⁢

(

𝐫

)

=

-

1

4

⁢

π

⁢

∫

ℝ

3

ρ

⁢

(

𝐫

′

)

|

𝐫

-

𝐫

′

|

⁢

d

3

⁢

𝐫

′

.

Title

Poisson’s equation

Canonical name

PoissonsEquation

Date of creation

2013-03-22 13:38:28

Last modified on

2013-03-22 13:38:28

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

6

Author

pbruin (1001)

Entry type

Definition

Classification

msc 35J05

Related topic

HelmholtzDifferentialEquation

Related topic

LaplacesEquation

Related topic

GreensFunction

Generated on Fri Feb 9 20:43:52 2018 by

LaTeXML
derivation of heat equation

derivation of heat equation

Let us consider the heat conduction in a

ϱ

and specific heat capacity

c

.  Denote by

u

⁢

(

x

,

y

,

z

,

t

)

the temperature in the point

(

x

,

y

,

z

)

at the time

t

.  Let

a

be a surface in the matter and

v

the spatial region by it.

When the growth of the temperature of a

volume element

d

⁢

v

in the time

d

⁢

t

is

d

⁢

u

, the element releases the amount

-

d

⁢

u

⁢

c

⁢

ϱ

⁢

d

⁢

v

=

-

u

t

′

⁢

d

⁢

t

⁢

c

⁢

ϱ

⁢

d

⁢

v

of heat, which is the heat flux through the surface of

d

⁢

v

.  Thus if there are no sources and sinks of heat in

v

, the heat flux through the surface

a

in

d

⁢

t

is

-

d

⁢

t

⁢

∫

v

c

⁢

ϱ

⁢

u

t

′

⁢

𝑑

v

.

(1)

On the other hand, the flux through

d

⁢

a

in the time

d

⁢

t

must be proportional to

a

, to

d

⁢

t

and to the

derivative

of the temperature in the direction of the normal line of the surface element

d

⁢

a

, i.e. the flux is

-

k

⁢

∇

⁡

u

⋅

d

⁢

a

→

⁢

d

⁢

t

,

where

k

is a

positive

(because the heat always from higher temperature to lower one).  Consequently, the heat flux through the whole surface

a

is

-

d

⁢

t

⁢

∮

a

k

⁢

∇

⁡

u

⋅

d

⁢

a

→

,

which is, by the Gauss’s theorem, same as

-

d

⁢

t

⁢

∫

v

k

⁢

∇

⋅

∇

⁡

u

⁢

d

⁢

v

=

-

d

⁢

t

⁢

∫

v

k

⁢

∇

2

⁡

u

⁢

d

⁢

v

.

(2)

Equating the expressions (1) and (2) and dividing by

d

⁢

t

, one obtains

∫

v

k

⁢

∇

2

⁡

u

⁢

d

⁢

v

=

∫

v

c

⁢

ϱ

⁢

u

t

′

⁢

𝑑

v

.

Since this equation is valid for any region

v

in the matter, we infer that

k

⁢

∇

2

⁡

u

=

c

⁢

ϱ

⁢

u

t

′

.

Denoting

k

c

⁢

ϱ

=

α

2

,  we can write this equation as

α

2

⁢

∇

2

⁡

u

=

∂

⁡

u

∂

⁡

t

.

(3)

This is the

differential equation

of heat conduction, first derived by Fourier.

References

1

K. Väisälä:

Matematiikka IV

.  Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).

Title

derivation of heat equation

Canonical name

DerivationOfHeatEquation

Date of creation

2013-03-22 18:45:04

Last modified on

2013-03-22 18:45:04

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Derivation

Classification

msc 35K05

Classification

msc 35Q99

Related topic

DerivationOfWaveEquation

Generated on Fri Feb 9 20:43:54 2018 by

LaTeXML
Feynman-Kac formula

Feynman-Kac formula

Let

X

t

be the

n

-dimensional Itō process satisfying the

stochastic differential
equation

d

⁢

X

t

=

μ

⁢

(

X

t

)

⁢

d

⁢

t

+

σ

⁢

(

X

t

)

⁢

d

⁢

W

t

and let

A

be its infinitesimal generator.

Further suppose that

q

is a lower-bounded

continuous function

on

ℝ

n

, and

f

is a twice-differentiable
function on

ℝ

n

with compact support.
Then

u

(

t

,

x

)

=

𝔼

[

e

-

∫

0

t

q

⁢

(

X

s

)

⁢

𝑑

s

f

(

X

t

)

∣

X

0

=

x

]

,

t

≥

0

,

x

∈

ℝ

n

is a solution to the

partial differential equation

∂

⁡

u

∂

⁡

t

=

A

⁢

u

⁢

(

x

)

-

u

⁢

q

⁢

(

x

)

with

initial condition

u

⁢

(

0

,

x

)

=

f

⁢

(

x

)

.

(The

expectation

for

u

is to be taken with respect
to the

probability measure

under which

W

t

is a

Brownian motion

.)

References

1

Bernt Øksendal.

,
An Introduction with Applications

. 5th ed., Springer 1998.

2

Hui-Hsiung Kuo.

Introduction to

Stochastic Integration

.
Springer 2006.

Title

Feynman-Kac formula

Canonical name

FeynmanKacFormula

Date of creation

2013-03-22 17:16:11

Last modified on

2013-03-22 17:16:11

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

6

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 35K15

Classification

msc 60H30

Classification

msc 60H10

Related topic

RichardFeynman

Generated on Fri Feb 9 20:43:56 2018 by

LaTeXML
spherical mean

spherical mean

Let

h

be a function (usually real or complex valued) on

ℝ

n

(

n

≥

1

).
Its

spherical mean

at point

x

over a sphere of radius

r

is defined as

M

h

⁢

(

x

,

r

)

=

1

A

⁢

(

n

-

1

)

⁢

∫

∥

ξ

∥

=

1

h

⁢

(

x

+

r

⁢

ξ

)

⁢

𝑑

S

=

1

A

⁢

(

n

-

1

,

r

)

⁢

∫

∥

ξ

∥

=

|

r

|

h

⁢

(

x

+

ξ

)

⁢

𝑑

S

,

where the

integral

is over the surface of the unit

n

-

1

-sphere. Here

A

⁢

(

n

-

1

)

is is the area of the unit sphere, while

A

⁢

(

n

-

1

,

r

)

=

r

n

-

1

⁢

A

⁢

(

n

-

1

)

is the area of a sphere of radius

r

(

http://planetmath.org/AreaOfTheNSphere

). In essense, the spherical mean

M

h

⁢

(

x

,

r

)

is just the

average

of

h

over the surface of a sphere of radius

r

centered at

x

, as the name suggests.

The spherical mean is defined for both

positive

and negative

r

and is
independent of its sign. As

r

→

0

, if

h

is

continuous

,

M

h

⁢

(

x

,

r

)

→

h

⁢

(

x

)

. If

h

has two continuous

derivatives

(is in

C

2

⁢

(

ℝ

n

)

) then the
following identity holds:

∇

x

2

⁡

M

h

⁢

(

x

,

r

)

=

(

∂

2

∂

⁡

r

2

+

n

-

1

r

⁢

∂

∂

⁡

r

)

⁢

M

h

⁢

(

x

,

r

)

,

where

∇

2

is the

Laplacian

.

Spherical means are used to obtain an explicit

general solution

for the

wave
equation

in

n

space and one time

dimensions

.

Title

spherical mean

Canonical name

SphericalMean

Date of creation

2013-03-22 14:09:04

Last modified on

2013-03-22 14:09:04

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 35L05

Classification

msc 26E60

Related topic

WaveEquation

Generated on Fri Feb 9 20:44:08 2018 by

LaTeXML
wave equation

wave equation

The

wave equation

is a

partial differential equation

which
describes certain kinds of waves. It arises in various physical
situations, such as vibrating , waves, and
electromagnetic waves.

The wave equation in one is

∂

2

⁡

u

∂

⁡

t

2

=

c

2

⁢

∂

2

⁡

u

∂

⁡

x

2

.

The

general solution

of the one-dimensional wave equation can be
obtained by a

change of coordinates

:

(

x

,

t

)

⟶

(

ξ

,

η

)

,
where

ξ

=

x

-

c

⁢

t

and

η

=

x

+

c

⁢

t

. This gives

∂

2

⁡

u

∂

⁡

ξ

⁢

∂

⁡

η

=

0

, which we can integrate to get

d’Alembert’s solution

:

u

⁢

(

x

,

t

)

=

F

⁢

(

x

-

c

⁢

t

)

+

G

⁢

(

x

+

c

⁢

t

)

where

F

and

G

are twice

differentiable functions

.

F

and

G

represent waves traveling in the positive and negative

x

directions, respectively, with velocity

c

. These functions can be
obtained if appropriate

initial conditions

and

boundary conditions

are given. For example, if

u

⁢

(

x

,

0

)

=

f

⁢

(

x

)

and

∂

⁡

u

∂

⁡

t

⁢

(

x

,

0

)

=

g

⁢

(

x

)

are given, the solution is

u

⁢

(

x

,

t

)

=

1

2

⁢

[

f

⁢

(

x

-

c

⁢

t

)

+

f

⁢

(

x

+

c

⁢

t

)

]

+

1

2

⁢

c

⁢

∫

x

-

c

⁢

t

x

+

c

⁢

t

g

⁢

(

s

)

⁢

d

s

.

In general, the wave equation in

n

is

∂

2

⁡

u

∂

⁡

t

2

=

c

2

⁢

∇

2

⁡

u

.

where

u

is a function of the location variables

x

1

,

x

2

,

…

,

x

n

, and time

t

. Here,

∇

2

is the

Laplacian

with respect to the location variables, which in

Cartesian coordinates

is given by

∇

2

=

∂

2

∂

⁡

x

1

2

+

∂

2

∂

⁡

x

2

2

+

⋯

+

∂

2

∂

⁡

x

n

2

.

Title

wave equation

Canonical name

WaveEquation

Date of creation

2013-03-22 13:10:12

Last modified on

2013-03-22 13:10:12

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 35L05

Related topic

HelmholtzDifferentialEquation

Related topic

SphericalMean

Defines

d’Alembert’s solution to the wave equation

Generated on Fri Feb 9 20:44:15 2018 by

LaTeXML
d’Alembert and D. Bernoulli solutions of wave equation

d’Alembert and D. Bernoulli solutions of wave equation

Let’s consider the d’Alembert’s solution (

http://planetmath.org/WaveEquation

)

u

⁢

(

x

,

t

)

:=

1

2

⁢

[

f

⁢

(

x

-

c

⁢

t

)

+

f

⁢

(

x

+

c

⁢

t

)

]

+

1

2

⁢

c

⁢

∫

x

-

c

⁢

t

x

+

c

⁢

t

g

⁢

(

s

)

⁢

𝑑

s

(1)

of the

wave equation

in one

dimension

in the special case when the other

initial condition

is

u

t

′

⁢

(

x

,

0

)

:=

g

⁢

(

x

)

≡

0

.

(2)

We shall see that the solution is equivalent with the solution of D. Bernoulli.

We the given function

f

to the

Fourier sine series

on the interval

[

0

,

p

]

:

f

(

y

)

=

∑

n

=

1

∞

A

n

sin

n

⁢

π

⁢

y

p

with

A

n

=

2

p

∫

0

p

f

(

x

)

sin

n

⁢

π

⁢

x

p

d

x

(

n

=

1

,

2

,

…

)

Thus we may write

{

f

⁢

(

x

-

c

⁢

t

)

=

∑

n

=

1

∞

A

n

⁢

sin

⁡

(

n

⁢

π

⁢

x

p

-

n

⁢

π

⁢

c

⁢

t

p

)

=

∑

n

=

1

∞

A

n

⁢

(

sin

⁡

n

⁢

π

⁢

x

p

⁢

cos

⁡

n

⁢

π

⁢

c

⁢

t

p

-

cos

⁡

n

⁢

π

⁢

x

p

⁢

sin

⁡

n

⁢

π

⁢

c

⁢

t

p

)

,

f

⁢

(

x

+

c

⁢

t

)

=

∑

n

=

1

∞

A

n

⁢

sin

⁡

(

n

⁢

π

⁢

x

p

+

n

⁢

π

⁢

c

⁢

t

p

)

=

∑

n

=

1

∞

A

n

⁢

(

sin

⁡

n

⁢

π

⁢

x

p

⁢

cos

⁡

n

⁢

π

⁢

c

⁢

t

p

+

cos

⁡

n

⁢

π

⁢

x

p

⁢

sin

⁡

n

⁢

π

⁢

c

⁢

t

p

)

.

Adding these equations and dividing by 2 yield

u

⁢

(

x

,

t

)

=

1

2

⁢

[

f

⁢

(

x

-

c

⁢

t

)

+

f

⁢

(

x

+

c

⁢

t

)

]

=

∑

n

=

1

∞

A

n

⁢

cos

⁡

n

⁢

π

⁢

c

⁢

t

p

⁢

sin

⁡

n

⁢

π

⁢

x

p

,

(3)

which indeed is the solution of D. Bernoulli (

http://planetmath.org/SolvingTheWaveEquationByDBernoulli

) in the case

g

⁢

(

x

)

≡

0

.

Note.

The solution (3) of the wave equation is especially in the special case where one has besides (2) the sine-formed initial condition

u

⁢

(

x

,

0

)

:=

f

⁢

(

x

)

≡

sin

⁡

π

⁢

x

p

.

(4)

Then

A

n

=

0

for every

n

except 1, and one obtains

u

⁢

(

x

,

t

)

=

cos

⁡

π

⁢

c

⁢

t

p

⁢

sin

⁡

π

⁢

x

p

.

(5)

Remark.

In the case of quantum systems one has Schrödinger’s wave equation (

http://planetmath.org/SchrodingersWaveEquation

)
whose solutions are different from the above.

Title

d’Alembert and D. Bernoulli solutions of wave equation

Canonical

name

DAlembertAndDBernoulliSolutionsOfWaveEquation

Date of creation

2013-03-22 18:23:15

Last modified on

2013-03-22 18:23:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Derivation

Classification

msc 35L15

Classification

msc 35L05

Related topic

AdditionFormulasForSineAndCosine

Related topic

SchrodingersWaveEquation

Generated on Fri Feb 9 20:44:18 2018 by

LaTeXML
telegraph equation

telegraph equation

Both the electric voltage and the satisfy the

telegraph equation

f

x

⁢

x

′′

-

a

⁢

f

t

⁢

t

′′

-

b

⁢

f

t

′

-

c

⁢

f

=

0

,

(1)

where

x

is distance,

t

is time and

a

,

b

,

c

are non-negative constants.  The equation is a generalised form of the

wave equation

.

If the

initial conditions

are

f

⁢

(

x

,

0

)

=

f

t

′

⁢

(

x

,

0

)

=

0

and the

boundary conditions

f

⁢

(

0

,

t

)

=

g

⁢

(

t

)

,

f

⁢

(

∞

,

t

)

=

0

,  then the

Laplace transform

of the solution

function

f

⁢

(

x

,

t

)

is

F

⁢

(

x

,

s

)

=

G

⁢

(

s

)

⁢

e

-

x

⁢

a

⁢

s

2

+

b

⁢

s

+

c

.

(2)

In the special case

b

2

-

4

⁢

a

⁢

c

=

0

,  the solution is

f

⁢

(

x

,

t

)

=

e

-

b

⁢

x

2

⁢

a

⁢

g

⁢

(

t

-

x

⁢

a

)

⁢

H

⁢

(

t

-

x

⁢

a

)

.

(3)

Justification of

(2).  Transforming the

partial differential equation

(1) (

x

may be regarded as a parametre) gives

F

x

⁢

x

′′

⁢

(

x

,

s

)

-

a

⁢

[

s

2

⁢

F

⁢

(

x

,

s

)

-

s

⁢

f

⁢

(

x

,

0

)

-

f

t

′

⁢

(

x

,

0

)

]

-

b

⁢

[

s

⁢

F

⁢

(

x

,

s

)

-

f

⁢

(

x

,

0

)

]

-

c

⁢

F

⁢

(

x

,

s

)

=

0

,

which due to the initial conditions simplifies to

F

x

⁢

x

′′

⁢

(

x

,

s

)

=

(

a

⁢

s

2

+

b

⁢

s

+

c

⏟

K

2

)

⁢

F

⁢

(

x

,

s

)

.

The solution of this

ordinary differential equation

is

F

⁢

(

x

,

s

)

=

C

1

⁢

e

K

⁢

x

+

C

2

⁢

e

-

K

⁢

x

.

Using the latter boundary condition, we see that

F

⁢

(

∞

,

s

)

=

∫

0

∞

e

-

s

⁢

t

⁢

f

⁢

(

∞

,

t

)

⁢

𝑑

t

≡

0

,

whence

C

1

=

0

.  Thus the former boundary condition implies

C

2

=

F

⁢

(

0

,

s

)

=

ℒ

⁢

{

g

⁢

(

t

)

}

=

G

⁢

(

s

)

.

So we obtain the equation (2).

Justification of

(3).  When the discriminant (

http://planetmath.org/QuadraticFormula

) of the quadratic equation

a

⁢

s

2

+

b

⁢

s

+

c

=

0

vanishes, the roots (

http://planetmath.org/Equation

) coincide to

s

=

-

b

2

⁢

a

,  and

a

⁢

s

2

+

b

⁢

s

+

c

=

a

⁢

(

s

+

b

2

⁢

a

)

2

.  Therefore (2) reads

F

⁢

(

x

,

s

)

=

G

⁢

(

s

)

⁢

a

-

x

⁢

a

⁢

(

s

+

b

2

⁢

a

)

=

e

-

b

⁢

x

2

⁢

a

⁢

e

-

x

⁢

a

⁢

s

⁢

G

⁢

(

s

)

.

According to the

delay theorem

, we have

ℒ

-

1

⁢

{

e

-

k

⁢

s

⁢

G

⁢

(

s

)

}

=

g

⁢

(

t

-

k

)

⁢

H

⁢

(

t

-

k

)

.

Thus we obtain for

ℒ

-

1

⁢

{

F

⁢

(

x

,

s

)

}

the expression of (3).

Title

telegraph equation

Canonical name

TelegraphEquation

Date of creation

2013-03-22 18:03:15

Last modified on

2013-03-22 18:03:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 35L15

Classification

msc 35L20

Synonym

telegrapher’s equation

Related topic

HeavisideStepFunction

Related topic

SecondOrderLinearDifferentialEquation

Related topic

DelayTheorem

Related topic

MellinsInverseFormula

Related topic

TableOfLaplaceTransforms

Generated on Fri Feb 9 20:44:21 2018 by

LaTeXML
soliton

soliton

A soliton is a non-linear object which moves through space without dispersion at

constant

speed. They occur naturally as solutions to the Korteweg - de Vries equation. They were first observed by John Scott Russell in the 19th century and then by Martin Kruskal and Norman Zabusky (who coined the term soliton) in a famous

computer simulation

in 1965. Insight into solitons can be obtained by noting that the Korteweg - de Vries equation

satisfies

D’Alembert’s solution:

u

⁢

(

x

,

t

)

=

f

⁢

(

x

-

c

⁢

t

)

+

g

⁢

(

x

+

c

⁢

t

)

We see at once that this satisfies two important criteria: it has a constant velocity

c

, and it can also be shown that the two

functions

f

and

g

can collide without altering shape. Solitons also

occur in

non-linear optics and as solutions to field equations in

quantum field theory

.

Title

soliton

Canonical name

Soliton

Date of creation

2013-03-22 17:47:50

Last modified on

2013-03-22 17:47:50

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

10

Author

invisiblerhino (19637)

Entry type

Definition

Classification

msc 35Q51

Classification

msc 37K40

Synonym

solitary wave

Generated on Fri Feb 9 20:44:26 2018 by

LaTeXML
derivation of Coulomb’s Law from Gauss’ Law

derivation of Coulomb’s Law from Gauss’ Law

As an example of the statement that Maxwell’s equations completely define electromagnetic phenomena, it will be shown that Coulomb’s Law may be derived from Gauss’ law for electrostatics. Consider a point charge. We can obtain an expression for the electric field surrounding the charge. We surround the charge with a ”virtual” sphere of radius

R

, then use Gauss’ law in integral form:

∮

S

𝐄

⋅

d

𝐀

=

q

ϵ

0

We rewrite this as a volume integral in

spherical polar coordinates

over the ”virtual” sphere mentioned above, which has the point charge at its centre. Since the electric field is spherically

symmetric

(by

assumption

) the electric field is

constant

over this volume.

∮

S

𝐄

⋅

d

𝐀

=

∫

0

R

∫

0

2

⁢

π

∫

0

π

E

⁢

r

⁢

sin

⁡

θ

⁢

d

⁢

r

⁢

d

⁢

θ

⁢

d

⁢

ϕ

Hence

4

⁢

π

⁢

R

2

⁢

E

=

q

ϵ

0

Or

E

=

q

4

⁢

π

⁢

ϵ

0

⁢

R

2

The usual form can then be recovered from the Lorentz force law,

𝐅

=

𝐄

⁢

q

+

𝐯

×

𝐁

noting the absence of magnetic field.

Title

derivation of Coulomb’s Law from Gauss’ Law

Canonical name

DerivationOfCoulombsLawFromGaussLaw

Date of creation

2013-03-22 17:53:23

Last modified on

2013-03-22 17:53:23

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

8

Author

invisiblerhino (19637)

Entry type

Derivation

Classification

msc 35Q60

Classification

msc 78A25

Defines

Coulomb’s Law

Generated on Fri Feb 9 20:44:33 2018 by

LaTeXML
derivation of wave equation from Maxwell’s equations

derivation of wave equation from Maxwell’s equations

Maxwell was the first to note that Ampère’s Law does not

satisfy

conservation of charge (his corrected form is given in Maxwell’s equation). This can be shown using the equation of conservation of electric charge:

∇

⋅

𝐉

+

∂

⁡

ρ

∂

⁡

t

=

0

Now consider Faraday’s Law in

differential form

:

∇

×

𝐄

=

-

∂

⁡

𝐁

∂

⁡

t

Taking the curl of both sides:

∇

×

(

∇

×

𝐄

)

=

∇

×

(

-

∂

⁡

𝐁

∂

⁡

t

)

The right-hand side may be simplified by noting that

∇

×

(

∂

⁡

𝐁

∂

⁡

t

)

=

-

∂

∂

⁡

t

⁢

(

∇

×

𝐁

)

Recalling Ampère’s Law,

-

∂

∂

⁡

t

⁢

(

∇

×

𝐁

)

=

-

μ

0

⁢

ϵ

0

⁢

∂

2

⁡

𝐄

∂

⁡

t

2

Therefore

∇

×

(

∇

×

𝐄

)

=

-

μ

0

⁢

ϵ

0

⁢

∂

2

⁡

𝐄

∂

⁡

t

2

The left hand side may be simplified by the following

vector identity

:

∇

×

(

∇

×

𝐄

)

=

-

∇

2

⁡

𝐄

Hence

∇

2

⁡

𝐄

=

μ

0

⁢

ϵ

0

⁢

∂

2

⁡

𝐄

∂

⁡

t

2

Applying the same

analysis

to Ampére’s Law then substituting in Faraday’s Law leads to the result

∇

2

⁡

𝐁

=

μ

0

⁢

ϵ

0

⁢

∂

2

⁡

𝐄

∂

⁡

t

2

Making the substitution

μ

0

⁢

ϵ

0

=

1

/

c

2

we note that these equations take the form of a transverse wave travelling at

constant

speed

c

. Maxwell evaluated the constants

μ

0

and

ϵ

0

according to their known values at the time and concluded that

c

was approximately equal to 310,740,000

ms

-

1

, a value within 3% of today’s results!

Title

derivation of wave equation

from Maxwell’s equations

Canonical name

DerivationOfWaveEquationFromMaxwellsEquations

Date of creation

2013-03-22 17:52:09

Last modified on

2013-03-22 17:52:09

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

10

Author

invisiblerhino (19637)

Entry type

Derivation

Classification

msc 35Q60

Classification

msc 78A25

Generated on Fri Feb 9 20:44:35 2018 by

LaTeXML