aarjaneiro/planetmath huggingface dataset:
sources and sinks of vector field

sources and sinks of vector field

Let the

vector field

U

→

of

ℝ

3

be interpreted, as in the remark of the parent entry (

http://planetmath.org/Flux

), as the velocity of a liquid.  Then the flux

∮

a

U

→

⋅

𝑑

a

→

of

U

→

through a closed surface

a

expresses how much more liquid per time-unit it comes from inside of

a

to outside than contrarily.  Since for a usual

incompressible

liquid, the outwards flow and the inwards flow are equal, we must think in the case that the flux differs from 0 either that the flowing liquid is suitably compressible or that there are inside the surface some

sources

creating liquid and

sinks

annihilating liquid.  Ordinarily, one uses the latter idea.  Both the sources and the sinks may be called sources, when the sinks are

negative

sources

.  The flux of the vector

U

→

through

a

is called the

productivity

or the

strength

of the sources inside

a

.

For example, the sources and sinks of an electric field (

E

→

) are the locations containing positive and negative charges, respectively.  The gravitational field has only sinks, which are the locations containing .

The

expression

1

Δ

⁢

v

⁢

∮

∂

⁡

Δ

⁢

v

U

→

⋅

𝑑

a

→

,

where

Δ

⁢

v

means a region in the vector field and also its volume, is the productivity of the sources in

Δ

⁢

v

per a volume-unit.  When we let

Δ

⁢

v

to shrink towards a point

P

in it, to an

infinitesimal

volume-element

d

⁢

v

, we get the limiting value

ϱ

:=

1

d

⁢

v

⁢

∮

∂

⁡

d

⁢

v

U

→

⋅

𝑑

a

→

,

(1)

called the

source density

in

P

.  Thus the productivity of the source in

P

is

ϱ

⁢

d

⁢

v

.  If

ϱ

=

0

, there is in

P

neither a source, nor a sink.

The Gauss’s

theorem

∫

v

∇

⋅

U

→

⁢

𝑑

v

=

∮

a

U

→

⋅

𝑑

a

→

applied to

d

⁢

v

says that

∇

⋅

U

→

=

1

d

⁢

v

⁢

∮

∂

⁡

d

⁢

v

U

→

⋅

𝑑

a

→

.

(2)

Accordingly,

ϱ

=

∇

⋅

U

→

(3)

and

∮

a

U

→

⋅

𝑑

a

→

=

∫

v

ϱ

⁢

𝑑

v

.

(4)

This last can be read that

the flux of the vector through a closed surface equals to the total productivity of the sources inside the surface.

For example, if

U

→

is the electric flux

density

D

→

, (4) means that the electric flux through a closed surface equals to the total charge inside.

Title

sources and sinks of vector field

Canonical

name

SourcesAndSinksOfVectorField

Date of creation

2013-03-22 18:47:03

Last modified on

2013-03-22 18:47:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B15

Classification

msc 26B12

Related topic

Divergence

Related topic

SolenoidalField

Related topic

CirculationAndVorticity

Defines

source

Defines

sink

Defines

source of vector field

Defines

productivity

Defines

strength

Defines

source density

Generated on Fri Feb 9 19:59:19 2018 by

LaTeXML
volume of spherical cap and spherical sector

volume of spherical cap and spherical sector

Theorem

1.

The volume of a

spherical cap

is

π

⁢

h

2

⁢

(

r

-

h

3

)

,  when

h

is its height and

r

is the radius of the sphere.

Proof.

The sphere may be formed by letting the circle

(

x

-

r

)

2

+

y

2

=

r

2

,  i.e.

y

=

(

±

)

⁢

r

⁢

x

-

x

2

,  rotate about the

x

-axis.  Let the spherical cap be the portion from the sphere on the left of the plane at

x

=

h

perpendicular

to the

x

-axis.

Then the for the

volume of solid of revolution

yields the volume in question:

V

=

π

⁢

∫

0

h

(

r

⁢

x

-

x

2

)

2

⁢

𝑑

x

=

π

⁢

∫

0

h

(

2

⁢

r

⁢

x

-

x

2

)

⁢

𝑑

x

=

π

⁢

/

x

=

0

h

⁡

(

r

⁢

x

2

-

x

3

3

)

=

π

⁢

h

2

⁢

(

r

-

h

3

)

.

Theorem 2.

The volume of a

spherical sector

is

2

3

⁢

π

⁢

r

2

⁢

h

,  where

h

is the height of the spherical cap of the spherical sector and

r

is the radius of the sphere.

Proof.

The volume

V

of the spherical sector equals to the sum or

difference

of the spherical cap and the

circular cone

depending on whether

h

<

r

or

h

>

r

.  If the radius of the base circle of the cone is

ϱ

, then

V

=

{

π

⁢

h

2

⁢

(

r

-

h

3

)

+

1

3

⁢

π

⁢

ϱ

2

⁢

(

r

-

h

)

when

h

<

r

,

π

⁢

h

2

⁢

(

r

-

h

3

)

-

1

3

⁢

π

⁢

ϱ

2

⁢

(

h

-

r

)

when

h

>

r

.

But one can see that both

expressions

of

V

are identical.  Moreover, if

c

is the

great circle

of the sphere having as a

diameter

the line of the axis of the cone and if

P

is the

midpoint

of the base of the cone, then in both cases, the power of the point

P

with respect to the circle

c

is

ϱ

2

=

(

2

⁢

r

-

h

)

⁢

h

.

Substituting this to the expression of

V

and simplifying give

V

=

2

3

⁢

π

⁢

r

2

⁢

h

,  Q.E.D.

Title

volume of spherical cap and spherical sector

Canonical

name

VolumeOfSphericalCapAndSphericalSector

Date of creation

2013-03-22 18:19:14

Last modified on

2013-03-22 18:19:14

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26B15

Classification

msc 53A05

Classification

msc 51M04

Synonym

volume of spherical cap

Synonym

volume of spherical sector

Related topic

SubstitutionNotation

Related topic

GreatCircle

Related topic

Diameter2

Related topic

PowerOfPoint

Generated on Fri Feb 9 19:59:28 2018 by

LaTeXML
area of plane region

area of plane region

Let the contour of the region in the

x

⁢

y

-plane be a

closed curve

P

.  Then the area of the region equals to the

path integral

A

=

1

2

⁢

∮

P

(

x

⁢

d

⁢

y

-

y

⁢

d

⁢

x

)

(1)

taken in the

positive

(i.e. anticlockwise) circling direction.

Remarks

1.

The (1) can be gotten as a special case of Green’s theorem by setting

F

→

:=

1

2

⁢

(

-

y

,

x

)

.

2.

Because

x

⁢

d

⁢

y

+

y

⁢

d

⁢

x

=

d

⁢

(

x

⁢

y

)

,  we have

0

=

1

2

⁢

∮

P

(

x

⁢

d

⁢

y

+

y

⁢

d

⁢

x

)

.

This equation may be added to or subtracted from (1), giving the alternative forms

A

=

∮

P

x

⁢

𝑑

y

=

-

∮

P

y

⁢

𝑑

x

.

(2)

3.

The formulae (1) and (2) all other formulae concerning the planar area computing, e.g.

A

=

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

,

A

=

1

2

⁢

∫

φ

1

φ

2

[

r

⁢

(

φ

)

]

2

⁢

𝑑

φ

,

the former of which is factually same as the latter form of (2).

Example.

The

ellipse

x

2

a

2

+

y

2

b

2

=

1

has the parametric

x

=

a

⁢

cos

⁡

t

,

y

=

b

⁢

sin

⁡

t

(

0

≦

t

<

2

⁢

π

).  We have

x

⁢

d

⁢

y

-

y

⁢

d

⁢

x

=

[

a

⁢

cos

⁡

t

⋅

b

⁢

cos

⁡

t

+

b

⁢

sin

⁡

t

⋅

a

⁢

sin

⁡

t

]

⁢

d

⁢

t

=

a

⁢

b

⁢

d

⁢

t

,

and hence (1) gives for the area of the ellipse

A

=

1

2

⁢

a

⁢

b

⁢

∫

0

2

⁢

π

𝑑

t

=

π

⁢

a

⁢

b

.

Title

area of plane region

Canonical name

AreaOfPlaneRegion

Date of creation

2013-03-22 15:17:46

Last modified on

2013-03-22 15:17:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Topic

Classification

msc 26B20

Classification

msc 26A42

Synonym

planar area

Related topic

Area2

Related topic

DefiniteIntegral

Related topic

PolarCurve

Related topic

RiemannMultipleIntegral

Related topic

PropertiesOfEllipse

Related topic

AreaBoundedByArcAndTwoLines

Generated on Fri Feb 9 19:59:30 2018 by

LaTeXML
Gauss Green theorem

Gauss Green theorem

Theorem 1

(Gauss-Green)

Let

Ω

⊂

R

n

be a

bounded

open set with

C

1

boundary, let

ν

Ω

:

∂

⁡

Ω

→

R

n

be the exterior

unit normal

vector to

Ω

in the point

x

and let

f

:

Ω

¯

→

R

n

be a

vector function

in

C

0

⁢

(

Ω

¯

,

R

n

)

∩

C

1

⁢

(

Ω

,

R

n

)

. Then

∫

Ω

div

⁢

f

⁢

(

x

)

⁢

𝑑

x

=

∫

∂

⁡

Ω

⟨

f

⁢

(

x

)

,

ν

Ω

⁢

(

x

)

⟩

⁢

𝑑

σ

⁢

(

x

)

.

Some remarks on notation.
The operator

div

⁢

f

is the

divergence

of the

vector field

f

, which is sometimes written as

∇

⋅

f

.
In the right-hand side we have a

surface integral

,

d

⁢

σ

is the corresponding area measure on

∂

⁡

Ω

.
The

scalar product

in the second

integral

is sometimes written as

f

n

⁢

(

x

)

and represents the

normal

component

of

f

with respect to

∂

⁡

Ω

; hence the whole integral represents the

flux

of the vector field

f

through

∂

⁡

Ω

;

This theorem can be easily extended to

piecewise

regular

domains.
However the more general statement of this Theorem involves the theory of

perimeters

and

B

⁢

V

functions.

Theorem 2

(generalized Gauss-Green)

Let

E

⊂

R

n

be any

measurable set

.
Then

∫

E

div

⁢

f

⁢

(

x

)

⁢

𝑑

x

=

∫

∂

*

⁡

E

⟨

ν

E

⁢

(

x

)

,

f

⁢

(

x

)

⟩

⁢

𝑑

ℋ

n

-

1

⁢

(

x

)

holds for every

continuously differentiable function

f

:

R

n

→

R

n

with compact support (i.e.

f

∈

C

c

1

⁢

(

R

n

,

R

n

)

) where

•

∂

*

⁡

E

is the

essential boundary

of

E

which is a subset of the topological boundary

∂

⁡

E

;

•

ν

E

⁢

(

x

)

is the exterior normal vector to

E

, which is defined when

x

∈

ℱ

⁢

E

;

•

ℋ

n

-

1

is the

(

n

-

1

)

-dimensional

Hausdorff measure

.

Title

Gauss Green theorem

Canonical name

GaussGreenTheorem

Date of creation

2013-03-22 15:01:51

Last modified on

2013-03-22 15:01:51

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

13

Author

paolini (1187)

Entry type

Theorem

Classification

msc 26B20

Synonym

divergence theorem

Related topic

GreensTheorem

Related topic

GeneralStokesTheorem

Related topic

IntegrationWithRespectToSurfaceArea

Related topic

ClassicalStokesTheorem

Related topic

FluxOfVectorField

Generated on Fri Feb 9 19:59:35 2018 by

LaTeXML
Green’s theorem

Green’s theorem

Green’s theorem provides a connection between

path integrals

over a well-connected region in the plane and the area of the region

bounded

in the plane. Given a closed path

P

bounding a region

R

with area

A

, and a

vector-valued function

F

→

=

(

f

⁢

(

x

,

y

)

,

g

⁢

(

x

,

y

)

)

over the plane,

∮

P

F

→

⋅

𝑑

x

→

=

∫

∫

R

[

g

1

⁢

(

x

,

y

)

-

f

2

⁢

(

x

,

y

)

]

⁢

𝑑

A

where

a

n

is the

derivative

of

a

with respect to the

n

th variable.

Corollary:

The closed path integral over a

gradient

of a function with

continuous

partial derivatives

is always zero. Thus, gradients are conservative vector fields. The

smooth function

is called the

potential

of the vector field.

Proof:

The corollary states that

∮

P

∇

→

h

⋅

𝑑

x

→

=

0

We can easily prove this using Green’s theorem.

∮

P

∇

→

h

⋅

𝑑

x

→

=

∫

∫

R

[

g

1

⁢

(

x

,

y

)

-

f

2

⁢

(

x

,

y

)

]

⁢

𝑑

A

But since this is a gradient…

∫

∫

R

[

g

1

⁢

(

x

,

y

)

-

f

2

⁢

(

x

,

y

)

]

⁢

𝑑

A

=

∫

∫

R

[

h

21

⁢

(

x

,

y

)

-

h

12

⁢

(

x

,

y

)

]

⁢

𝑑

A

Since

h

12

=

h

21

for any function with continuous partials, the corollary is proven.

Title

Green’s theorem

Canonical name

GreensTheorem

Date of creation

2013-03-22 12:15:55

Last modified on

2013-03-22 12:15:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 26B20

Related topic

GaussGreenTheorem

Related topic

ClassicalStokesTheorem

Generated on Fri Feb 9 19:59:37 2018 by

LaTeXML
real and imaginary parts of contour integral

real and imaginary parts of contour integral

If

f

⁢

(

z

)

is

continuous

on the

contour

γ

of the

complex plane

and

z

=

x

+

i

y

(

x

,

y

∈

ℝ

)

,

ℜ

f

(

z

)

=

u

(

x

,

y

)

,

ℑ

f

(

z

)

=

v

(

x

,

y

)

,

then the contour integral of

f

⁢

(

z

)

along

γ

may be expressed via two real

path integrals

as

∫

γ

f

⁢

(

z

)

⁢

𝑑

z

=

∫

γ

(

u

⁢

d

⁢

x

-

v

⁢

d

⁢

y

)

+

i

⁢

∫

γ

(

v

⁢

d

⁢

x

+

u

⁢

d

⁢

y

)

.

Title

real and imaginary parts of contour integral

Canonical name

RealAndImaginaryPartsOfContourIntegral

Date of creation

2013-03-22 19:14:32

Last modified on

2013-03-22 19:14:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Result

Classification

msc 26B20

Classification

msc 30E20

Classification

msc 30A99

Related topic

ContourIntegral

Related topic

PathIntegral

Related topic

RealPartSeriesAndImaginaryPartSeries

Generated on Fri Feb 9 19:59:39 2018 by

LaTeXML
continuity of convex functions, alternate proof

continuity of convex functions, alternate proof

Let

f

be convex and

y

∈

(

a

,

b

)

be arbitrary but fixed. Then

f

⁢

(

λ

⁢

x

+

(

1

-

λ

)

⁢

y

)

≤

λ

⁢

f

⁢

(

x

)

+

(

1

-

λ

)

⁢

f

⁢

(

y

)

(1)

f

⁢

(

λ

⁢

x

+

(

1

-

λ

)

⁢

y

)

-

f

⁢

(

y

)

≤

λ

⁢

(

f

⁢

(

x

)

-

f

⁢

(

y

)

)

≤

λ

⁢

|

f

⁢

(

x

)

-

f

⁢

(

y

)

|

.

(2)

Fix a number

c

>

sup

⁡

{

|

f

⁢

(

u

)

-

f

⁢

(

v

)

|

:

u

,

v

∈

(

a

,

b

)

}

. Then

|

f

⁢

(

λ

⁢

x

+

(

1

-

λ

)

⁢

y

)

-

f

⁢

(

y

)

|

≤

λ

⁢

|

f

⁢

(

x

)

-

f

⁢

(

y

)

|

<

λ

⁢

c

.

(3)

Given

ϵ

>

0

, let

λ

range over

(

0

,

ϵ

/

c

)

if

ϵ

/

c

<

1

, or

λ

=

1

otherwise. Then it is easy to see
that

f

⁢

(

λ

⁢

x

+

(

1

-

λ

)

⁢

y

)

and

f

⁢

(

y

)

lie within

ϵ

distance

of each other when

λ

varies as specified.

Continuity

of

f

now follows–for

x

<

y

, the left-hand limit equals

f

⁢

(

y

)

and for

y

<

x

, the right-hand limit also equals

f

⁢

(

y

)

, hence
the limit is

f

⁢

(

y

)

.

Title

continuity of convex functions

, alternate proof

Canonical name

ContinuityOfConvexFunctionsAlternateProof

Date of creation

2013-03-22 18:25:28

Last modified on

2013-03-22 18:25:28

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

4

Author

yesitis (13730)

Entry type

Proof

Classification

msc 26B25

Classification

msc 26A51

Generated on Fri Feb 9 19:59:41 2018 by

LaTeXML
local minimum of convex function is necessarily global

local minimum of convex function is necessarily global

Theorem 1

.

A

local minimum

(resp.

local maximum

) of a

convex function

(resp.

concave function

) on a

convex subset

of a

topological vector space

, is always a

global extremum

.

Proof.

Let

f

:

S

→

ℝ

be a convex function
on a

convex set

S

in a topological vector space.

Suppose

x

is a local minimum for

f

;
that is, there is an open neighborhood

U

of

x

where

f

⁢

(

x

)

≤

f

⁢

(

ξ

)

for all

ξ

∈

U

.
We prove

f

⁢

(

x

)

≤

f

⁢

(

y

)

for arbitrary

y

∈

S

.

Consider the

convex combination

(

1

-

t

)

⁢

x

+

t

⁢

y

for

0

≤

t

≤

1

:

Since

scalar multiplication

and

vector addition

are, by definition,

continuous

in a topological vector space, the convex combination approaches

x

as

t

→

0

. Therefore for small enough

t

,

(

1

-

t

)

⁢

x

+

t

⁢

y

is in the neighborhood

U

.
Then

f

⁢

(

x

)

≤

f

⁢

(

t

⁢

y

+

(

1

-

t

)

⁢

x

)

for small

t

>

0

≤

t

⁢

f

⁢

(

y

)

+

(

1

-

t

)

⁢

f

⁢

(

x

)

since

f

is convex.

Rearranging , we have

f

⁢

(

x

)

≤

f

⁢

(

y

)

.

To show the analogous situation for a concave function

f

,
the above reasoning can be applied after replacing

f

with

-

f

.
∎

Title

local minimum of convex function is necessarily global

Canonical name

LocalMinimumOfConvexFunctionIsNecessarilyGlobal

Date of creation

2013-03-22 13:33:37

Last modified on

2013-03-22 13:33:37

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

13

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 26B25

Synonym

extremal value of convex/concave functions

Synonym

local maximum of concave function is necessarily global

Generated on Fri Feb 9 19:59:43 2018 by

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absolutely continuous function

absolutely continuous function

is the precise condition one needs to
impose in order for the

fundamental theorem of calculus

to hold for the

Lebesgue integral

.

Definition

Suppose

[

a

,

b

]

be a closed

bounded interval

of

ℝ

.
Then a

function

f

:

[

a

,

b

]

→

ℂ

is

absolutely continuous

on

[

a

,

b

]

,
if for any

ε

>

0

, there is a

δ

>

0

such that the following
condition holds:

(

∗

)

If

(

a

1

,

b

1

)

,

…

,

(

a

n

,

b

n

)

is a finite

collection

of

disjoint

open intervals

in

[

a

,

b

]

such that

∑

i

=

1

n

(

b

i

-

a

i

)

<

δ

,

then

∑

i

=

1

n

|

f

⁢

(

b

i

)

-

f

⁢

(

a

i

)

|

<

ε

.

Theorem 1

().

Let

f

:

[

a

,

b

]

→

C

be a
function. Then

f

is absolutely continuous if and only if
there is a function

g

∈

L

1

⁢

(

a

,

b

)

(i.e. a

g

:

(

a

,

b

)

→

C

with

∫

a

b

|

g

|

<

∞

), such that

f

⁢

(

x

)

=

f

⁢

(

a

)

+

∫

a

x

g

⁢

(

t

)

⁢

𝑑

t

for all

x

∈

[

a

,

b

]

.
What is more, if

f

and

g

are as above, then

f

is

differentiable

almost everywhere and

f

′

=

g

almost everywhere. (Above, both integrals are Lebesgue integrals.)

See

[

2

,

3

]

for proof.

See also

[

1

]

, and

[

4

]

for a discussion
about different proofs.

References

1

Wikipedia, entry on

http://en.wikipedia.org/wiki/Absolute_continuity

Absolute continuity.

2

F. Jones,

Lebesgue Integration on

Euclidean Spaces

,
Jones and Barlett Publishers, 1993.

3

C.D. Aliprantis, O. Burkinshaw,

Principles of

Real Analysis

,
2nd ed., Academic Press, 1990.

4

D. B’arcenas,

The

Fundamental Theorem of
Calculus for Lebesgue Integral

,
Divulgaciones Matemáticas, Vol. 8, No. 1, 2000, pp. 75-85.

Title

absolutely continuous function

Canonical name

AbsolutelyContinuousFunction

Date of creation

2013-03-22 15:18:47

Last modified on

2013-03-22 15:18:47

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

13

Author

matte (1858)

Entry type

Definition

Classification

msc 26B30

Classification

msc 26A46

Related topic

SingularFunction

Related topic

AbsolutelyContinuous

Defines

fundamental theorem of calculus for the Lebesgue integral

Generated on Fri Feb 9 19:59:46 2018 by

LaTeXML
BV function

BV function

Functions of bounded variation

,

B

⁢

V

functions

, are functions whose distributional

derivative

is a finite

Radon measure

. More precisely:

Definition 1

(functions of bounded variation).

Let

Ω

⊂

R

n

be an open set. We say that a function

u

∈

L

1

⁢

(

Ω

)

has

bounded variation

, and write

u

∈

B

⁢

V

⁢

(

Ω

)

, if there exists a finite Radon

vector measure

D

⁢

u

∈

M

⁢

(

Ω

,

R

n

)

such that

∫

Ω

u

⁢

(

x

)

⁢

div

⁢

ϕ

⁢

(

x

)

⁢

𝑑

x

=

-

∫

Ω

⟨

ϕ

⁢

(

x

)

,

D

⁢

u

⁢

(

x

)

⟩

for every function

ϕ

∈

C

c

1

⁢

(

Ω

,

R

n

)

. The

measure

D

⁢

u

,
represents the distributional derivative of

u

since the above equality holds true for every

ϕ

∈

C

c

∞

⁢

(

Ω

,

R

n

)

.

Notice that

W

1

,

1

⁢

(

Ω

)

⊂

B

⁢

V

⁢

(

Ω

)

. In fact if

u

∈

W

1

,

1

⁢

(

Ω

)

one can choose

μ

:=

∇

⁡

u

⁢

ℒ

(where

ℒ

is the

Lebesgue measure

on

Ω

).
The equality

∫

u

⁢

div

⁢

ϕ

=

-

∫

ϕ

⁢

𝑑

μ

=

-

∫

ϕ

⁢

∇

⁡

u

is nothing else than the definition of

weak derivative

, and hence holds true.
One can easily find an example of a

B

⁢

V

functions which is not

W

1

,

1

.

An

equivalent

definition can be given as follows.

Definition 2

(variation).

Given

u

∈

L

1

⁢

(

Ω

)

we define the

variation

of

u

in

Ω

as

V

⁢

(

u

,

Ω

)

:=

sup

⁡

{

∫

Ω

u

⁢

div

⁢

ϕ

:

ϕ

∈

𝒞

c

1

⁢

(

Ω

,

ℝ

n

)

,

∥

ϕ

∥

L

∞

⁢

(

Ω

)

≤

1

}

.

We define

B

⁢

V

⁢

(

Ω

)

=

{

u

∈

L

1

⁢

(

Ω

)

:

V

⁢

(

u

,

Ω

)

<

+

∞

}

.

.

Title

BV function

Canonical name

BVFunction

Date of creation

2013-03-22 15:12:32

Last modified on

2013-03-22 15:12:32

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

11

Author

paolini (1187)

Entry type

Definition

Classification

msc 26B30

Synonym

function of bounded variation

Related topic

TotalVariation

Defines

total variation

Generated on Fri Feb 9 19:59:48 2018 by

LaTeXML
example of a B⁢V function which is not W^{1,1}

example of a

B

⁢

V

function which is not

W

1

,

1

The following example presents a

function

u

∈

B

⁢

V

⁢

(

Ω

)

∖

W

1

,

1

⁢

(

Ω

)

.

Example 1

.

Let

Ω

:=

(

-

1

,

1

)

×

(

-

1

,

1

)

⊂

ℝ

2

. We will show that the function

u

⁢

(

x

,

y

)

=

{

1

,

if

x

≥

0

0

,

if

x

<

0

belongs to

B

⁢

V

⁢

(

Ω

)

.
Given

ϕ

∈

C

c

1

⁢

(

Ω

,

ℝ

2

)

,

ϕ

=

(

ϕ

1

,

ϕ

2

)

, one has

∬

Ω

u

⁢

(

x

,

y

)

⁢

div

⁢

ϕ

⁢

(

x

,

y

)

⁢

𝑑

x

⁢

𝑑

y

=

∫

-

1

1

[

∫

0

1

ϕ

x

1

⁢

(

x

,

y

)

⁢

𝑑

x

]

⁢

𝑑

y

+

∫

0

1

[

∫

-

1

1

ϕ

y

2

⁢

(

x

,

y

)

⁢

𝑑

y

]

⁢

𝑑

x

=

∫

-

1

1

ϕ

1

⁢

(

1

,

y

)

-

ϕ

1

⁢

(

0

,

y

)

⁢

d

⁢

y

+

∫

0

1

ϕ

2

⁢

(

x

,

1

)

-

ϕ

2

⁢

(

x

,

-

1

)

⁢

d

⁢

x

=

-

∫

-

1

1

ϕ

1

⁢

(

0

,

y

)

+

0

=

-

∫

ϕ

⁢

(

x

,

y

)

⁢

𝑑

μ

⁢

(

x

,

y

)

if we choose

μ

:=

(

μ

1

,

μ

2

)

:=

(

ℋ

1

⁢

⌞

⁢

(

{

0

}

×

(

-

1

,

1

)

)

,

0

)

. So we notice that

u

∈

B

⁢

V

⁢

(

Ω

)

and

D

⁢

u

=

μ

is

singular

with respect to the

Lebesgue measure

ℒ

.

Title

example of a

B

⁢

V

function which is not

W

1

,

1

Canonical name

ExampleOfABVFunctionWhichIsNotW11

Date of creation

2013-03-22 15:12:59

Last modified on

2013-03-22 15:12:59

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

5

Author

paolini (1187)

Entry type

Example

Classification

msc 26B30

Generated on Fri Feb 9 19:59:51 2018 by

LaTeXML
boundedly homogeneous function

boundedly homogeneous function

A

function

f

:

ℝ

n

→

ℝ

,  where

n

is a positive integer, is called

boundedly

homogeneous

with respect to a set

Λ

of

positive

reals and a real number

r

, if the equation

f

⁢

(

λ

⁢

x

→

)

=

λ

r

⁢

f

⁢

(

x

→

)

is true for all and

x

→

∈

ℝ

n

and

λ

∈

Λ

.  Then

Λ

is the

set of homogeneity

and

r

the

degree of homogeneity

of

f

.

Example.

The function

x

↦

x

r

⁢

sin

⁡

(

ln

⁡

x

)

is boundedly homogeneous with respect to the set

Λ

=

{

e

2

⁢

π

⁢

ν

⋮

ν

∈

ℤ

}

and  with degree of homogeneity

r

.

Theorem

.

Let

f

:

ℝ

+

→

ℝ

be a boundedly

homogeneous function

with the degree of homogeneity

r

and the set of homogeneity

Λ

⊃

{

1

}

.  Then

f

is of the form

f

⁢

(

x

)

=

x

r

⁢

f

1

⁢

(

ln

⁡

x

)

(1)

where

f

1

:

ℝ

→

ℝ

is a periodic

real function

depending on

f

.

Proof.

Defining

g

⁢

(

x

)

:=

f

⁢

(

x

)

x

r

,  we obtain

g

⁢

(

λ

⁢

x

)

=

f

⁢

(

λ

⁢

x

)

(

λ

⁢

x

)

r

=

λ

r

⁢

f

⁢

(

x

)

λ

r

⁢

x

r

=

λ

0

⁢

g

⁢

(

x

)

∀

λ

∈

Λ

.

Thus

g

is a boundedly homogeneous function with the set of homogeneity

Λ

and the degree of homogeneity 0.  Moreover, define

f

1

⁢

(

x

)

:=

g

⁢

(

e

x

)

.  If

λ

∈

Λ

∖

{

1

}

and

p

:=

ln

⁡

λ

,
we see that

f

1

⁢

(

x

+

p

)

=

g

⁢

(

e

x

⁢

e

p

)

=

g

⁢

(

e

x

⁢

λ

)

=

g

⁢

(

e

x

)

=

f

1

⁢

(

x

)

∀

x

∈

ℝ

+

.

Therefore,

f

1

is periodic and (1) is in .

References

1

Konrad Schlude

: “Bemerkung zu beschränkt homogenen Funktionen”.  –

Elemente der Mathematik

54

(1999).

Title

boundedly homogeneous function

Canonical

name

BoundedlyHomogeneousFunction

Date of creation

2013-03-22 19:13:17

Last modified on

2013-03-22 19:13:17

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B35

Classification

msc 15-00

Synonym

boundedly homogeneous

Defines

set of homogeneity

Defines

degree of homogeneity

Generated on Fri Feb 9 19:59:54 2018 by

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