aarjaneiro/planetmath huggingface dataset:
proof of d⁢α⁢(X,Y)=X⁢(α⁢(Y)) - Y⁢(α⁢(X)) - α⁢([X,Y]) (local coordinates)

proof of

d

⁢

α

⁢

(

X

,

Y

)

=

X

⁢

(

α

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

(local coordinates)

Since this result is local (in other words, the

identity

holds on the whole manifold if and only if its

restriction

to every coordinate patch of the manifold holds), it suffices to demonstrate it in a local coordinate system. To do this, we shall compute coordinate expressions for the various terms and verify that the sum of terms on the right-hand side equals the left-hand side.

d

⁢

α

⁢

(

X

,

Y

)

=

(

α

j

,

i

-

α

i

,

j

)

⁢

X

i

⁢

Y

j

=

α

j

,

i

⁢

X

i

⁢

Y

j

-

α

i

,

j

⁢

X

i

⁢

Y

j

X

⁢

(

α

⁢

(

Y

)

)

=

X

i

⁢

∂

i

⁡

(

α

j

⁢

Y

j

)

=

X

i

⁢

α

j

,

i

⁢

Y

j

+

X

i

⁢

α

j

⁢

Y

j

,

i

Y

⁢

(

α

⁢

(

X

)

)

=

Y

j

⁢

∂

j

⁡

(

α

i

⁢

X

i

)

=

Y

j

⁢

α

i

,

j

⁢

X

i

+

Y

j

⁢

α

i

⁢

X

i

,

j

α

⁢

(

[

X

,

Y

]

)

=

α

i

⁢

(

X

j

⁢

Y

i

,

j

-

Y

j

⁢

X

i

,

j

)

=

α

i

⁢

X

j

⁢

Y

i

,

j

-

α

i

⁢

Y

j

⁢

X

i

,

j

Upon combining the right-hand sides of the last three equations and cancelling common terms, we obtain

X

i

⁢

α

j

,

i

⁢

Y

j

+

X

i

⁢

α

j

⁢

Y

j

,

i

-

Y

j

⁢

α

i

,

j

⁢

X

i

-

α

i

⁢

X

j

⁢

Y

i

,

j

Upon renaming dummy indices (switching

i

with

j

), the second and fourth terms cancel. What remains is exactly the right-hand side of the first equation. Hence, we have

d

⁢

α

⁢

(

X

,

Y

)

=

X

⁢

(

α

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

Title

proof of

d

⁢

α

⁢

(

X

,

Y

)

=

X

⁢

(

α

⁢

(

Y

)

)

-

Y

⁢

(

α

⁢

(

X

)

)

-

α

⁢

(

[

X

,

Y

]

)

(local coordinates)

Canonical name

ProofOfDalphaXYXalphaYYalphaXalphaXYlocalCoordinates

Date of creation

2013-03-22 15:34:01

Last modified on

2013-03-22 15:34:01

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 53-00

Generated on Fri Feb 9 22:18:07 2018 by

LaTeXML
proof of exhaustion by compact sets for ℝ^n

proof of exhaustion by compact sets for

ℝ

n

First consider

A

⊂

ℝ

n

to be a bounded open set and designate the open ball centered at

x

with radius

r

by

B

r

⁢

(

x

)

Construct

C

n

=

⋃

x

∈

∂

⁡

A

B

1

n

⁢

(

x

)

, where

∂

⁡

A

is the boundary of

A

and define

K

n

=

A

\

C

n

.

•

K

n

is compact.

It is bounded since

K

n

⊂

A

and

A

is by

assumption

bounded.

K

n

is also closed. To see this consider

x

∈

∂

⁡

K

n

but

x

∉

K

n

. Then there exists

y

∈

∂

⁡

A

and

0

<

r

<

1

n

such that

x

∈

B

r

⁢

(

y

)

.
But

B

r

⁢

(

y

)

∩

K

n

=

{

}

because

B

1

n

⁢

(

y

)

∩

K

n

=

{

}

and

0

<

r

<

1

n

⟹

B

r

⁢

(

y

)

⊂

B

1

n

⁢

(

y

)

.
This implies that

x

∉

∂

⁡

K

n

and we have a

contradiction

.

K

n

is therefore closed.

•

K

n

⊂

int

⁡

K

n

+

1

Suppose

x

∈

K

n

and

x

∉

int

⁡

K

n

+

1

. This means that for all

y

∈

∂

⁡

A

,

x

∈

B

1

n

+

1

⁢

(

y

)

¯

∨

x

∈

ℝ

n

\

A

.
Since

x

∈

K

n

⟹

x

∈

A

we must have

x

∈

B

1

n

+

1

⁢

(

y

)

¯

.
But

x

∈

B

1

n

+

1

⁢

(

y

)

¯

⊂

B

1

n

⁢

(

y

)

⟹

x

∉

K

n

and we have a contradiction.

•

⋃

n

=

1

∞

K

n

=

A

Suppose

x

∈

A

, since

A

is open there must exist

r

>

0

such that

B

r

⁢

(

x

)

⊂

A

.
Considering

n

such that

1

n

<

r

we have that

x

∉

B

1

n

⁢

(

y

)

for all

y

∈

∂

⁡

A

and thus

x

∈

K

n

.

Finally if

A

is not bounded consider

A

k

=

A

∩

B

k

⁢

(

0

)

and define

K

n

=

⋃

k

=

1

n

K

k

,

n

where

K

k

,

n

is the set resulting from the previous construction on the bounded set

A

k

.

•

K

n

will be compact because it is the finite union of compact sets.

•

K

n

⊂

int

⁡

K

n

+

1

because

K

k

,

n

⊂

int

⁡

K

k

,

n

+

1

and

int

⁡

(

A

∪

B

)

⊂

int

⁡

A

∪

int

⁡

B

•

⋃

n

=

1

∞

K

n

=

A

First find

k

such that

x

∈

A

k

.
This will always be possible since all it requires is that

k

>

|

x

|

.
Finally since

n

>

k

⟹

K

k

,

n

⊂

K

n

,

n

by construction the

argument

for the bounded case is directly applicable.

Title

proof of

exhaustion by compact sets

for

ℝ

n

Canonical name

ProofOfExhaustionByCompactSetsFormathbbRn

Date of creation

2013-03-22 15:51:38

Last modified on

2013-03-22 15:51:38

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

5

Author

cvalente (11260)

Entry type

Proof

Classification

msc 53-00

Generated on Fri Feb 9 22:18:10 2018 by

LaTeXML
proof of Poincaré lemma

proof of Poincaré lemma

Let

X

be a

smooth manifold

, and let

ω

be a

closed differential form

of degree

k

>

0

on

X

. For any

x

∈

X

, there exists a

contractible

neighbourhood

U

⊂

X

of

x

(i.e.

U

is

homotopy equivalent

to a single point), with inclusion map

ι

:

U

↪

X

.

To construct such a neighbourhood, take for example an open ball in a

coordinate chart

around

x

. Because of the

homotopy invariance

of

de Rham cohomology

, the

k

th de Rham cohomology group

H

k

⁢

(

U

)

is isomorphic to that of a point; in particular,

H

k

⁢

(

U

)

=

0

for all

k

>

0

.

Since

d

⁢

(

ι

*

⁢

ω

)

=

ι

*

⁢

(

d

⁢

ω

)

=

0

, this implies that there exists a

(

k

-

1

)

-form

η

on

U

such that

d

⁢

η

=

ι

*

⁢

ω

. In the case where

X

is a contractible

manifold

, such an

η

exists globally since we can choose

U

=

X

above.

Title

proof of Poincaré lemma

Canonical name

ProofOfPoincareLemma

Date of creation

2013-03-22 14:24:36

Last modified on

2013-03-22 14:24:36

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

4

Author

pbruin (1001)

Entry type

Proof

Classification

msc 53-00

Classification

msc 55N05

Generated on Fri Feb 9 22:18:12 2018 by

LaTeXML
pullback of a k-form

pullback of a

k

-form

If

X

is a

manifold

, let

Ω

k

⁢

(

X

)

be the vector space of

k

-forms on

X

.

Definition

Suppose

X

and

Y

are

smooth manifolds

, and suppose

f

is a

smooth mapping

f

:

X

→

Y

. Then the

pullback

induced

by

f

is the mapping

f

∗

:

Ω

k

⁢

(

Y

)

→

Ω

k

⁢

(

X

)

defined as
follows: If

ω

∈

Ω

k

⁢

(

Y

)

, then

f

∗

⁢

(

ω

)

is the

k

-form on

X

defined by the formula

(

f

*

⁢

ω

)

x

⁢

(

X

1

,

…

,

X

k

)

=

ω

f

⁢

(

x

)

⁢

(

(

D

⁢

f

)

x

⁢

(

X

1

)

,

…

,

(

D

⁢

f

)

x

⁢

(

X

k

)

)

where

x

∈

X

,

X

1

,

…

,

X

k

∈

T

x

⁢

(

X

)

, and

D

⁢

f

is the

tangent map

D

⁢

f

:

T

⁢

X

→

T

⁢

Y

.

0.0.1

Properties

Suppose

X

and

Y

are manifolds.

•

If

id

X

is the identity map on

X

, then

(

id

X

)

∗

is the identity map on

Ω

k

⁢

(

X

)

.

•

If

X

,

Y

,

Z

are manifolds, and

f

,

g

are mappings

f

:

X

→

Y

and

g

:

Y

→

Z

, then

(

g

∘

f

)

∗

=

f

∗

∘

g

∗

.

•

If

f

is a

diffeomorphism

f

:

X

→

Y

, then

f

∗

is a diffeomorphism
with inverse

(

f

-

1

)

∗

=

(

f

∗

)

∗

.

•

If

f

is a mapping

f

:

X

→

Y

, and

ω

∈

Ω

k

⁢

(

Y

)

, then

d

⁢

f

∗

⁢

ω

=

f

∗

⁢

d

⁢

ω

,

where

d

is the

exterior derivative

.

•

Suppose

f

is a mapping

f

:

X

→

Y

,

ω

∈

Ω

k

⁢

(

Y

)

, and

η

∈

Ω

l

⁢

(

Y

)

. Then

f

∗

⁢

(

ω

∧

η

)

=

f

∗

⁢

(

ω

)

∧

f

∗

⁢

(

η

)

.

•

If

g

is a

0

-form on

Y

, that is,

g

is a real valued function

g

:

Y

→

ℝ

, and

f

is a mapping

f

:

X

→

Y

,
then

f

∗

⁢

(

g

)

=

f

∘

g

.

•

Suppose

U

is a

submanifold

(or an open set) in an manifold

X

, and

ι

:

U

↪

X

is the inclusion mapping. Then

ι

∗

restricts

k

-forms on

X

to

k

-forms on

U

.

Title

pullback of a

k

-form

Canonical name

PullbackOfAKform

Date of creation

2013-03-22 14:00:34

Last modified on

2013-03-22 14:00:34

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 53-00

Related topic

Pullback2

Related topic

TangentMap

Generated on Fri Feb 9 22:18:15 2018 by

LaTeXML
submersion

submersion

A

differentiable map

f

:

X

→

Y

differential manifolds

X

and

Y

is called a

submersion

at a point

x

∈

X

if the

tangent map

T

⁢

f

⁢

(

x

)

:

T

⁢

X

⁢

(

x

)

→

T

⁢

Y

⁢

(

f

⁢

(

x

)

)

between the

tangent spaces

of

X

and

Y

at

x

and

f

⁢

(

x

)

is surjective.

If

f

is a submersion at every point of

X

, then

f

is called a

submersion

. A submersion

f

:

X

→

Y

is an open mapping, and its image is an

open submanifold

of

Y

.

A fibre bundle

p

:

X

→

B

over a

manifold

B

is an example of a submersion.

Title

submersion

Canonical name

Submersion

Date of creation

2013-03-22 15:28:49

Last modified on

2013-03-22 15:28:49

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

4

Author

pbruin (1001)

Entry type

Definition

Classification

msc 53-00

Classification

msc 57R50

Related topic

Immersion

Generated on Fri Feb 9 22:18:17 2018 by

LaTeXML
tangent map

tangent map

Definition 1

.

Suppose

X

and

Y

are

smooth manifolds

with

tangent bundles

T

⁢

X

and

T

⁢

Y

, and suppose

f

:

X

→

Y

is a

smooth mapping

. Then the

tangent map

of

f

is the map

D

⁢

f

:

T

⁢

X

→

T

⁢

Y

defined as follows: If

v

∈

T

x

⁢

(

X

)

for some

x

∈

X

, then
we can represent

v

by some curve

c

:

I

→

X

with

c

⁢

(

0

)

=

x

and

I

=

(

-

1

,

1

)

.
Now

(

D

⁢

f

)

⁢

(

v

)

is defined as the

tangent vector

in

T

⁢

(

Y

)

represented by the curve

f

∘

c

:

I

→

Y

. Thus,
since

(

f

∘

c

)

⁢

(

0

)

=

f

⁢

(

x

)

, it follows that

(

D

⁢

f

)

⁢

(

v

)

∈

T

f

⁢

(

x

)

⁢

(

Y

)

.

Properties

Suppose

X

and

Y

are a smooth manifolds.

•

If

id

X

is the identity mapping on

X

, then

D

⁢

id

X

is the identity mapping on

T

⁢

X

.

•

Suppose

X

,

Y

,

Z

are smooth manifolds, and

f

,

g

are mappings

f

:

X

→

Y

,

g

:

Y

→

Z

. Then

D

⁢

(

f

∘

g

)

=

(

D

⁢

f

)

∘

(

D

⁢

g

)

.

•

If

f

:

X

→

Y

is a

diffeomorphism

, then the

inverse

of

D

⁢

f

is a diffeomorphism,
and

(

D

⁢

f

)

-

1

=

D

⁢

(

f

-

1

)

.

Notes

Note that if

f

:

X

→

Y

is a mapping as in
the

definition

, then the tangent map is
a mapping

D

⁢

f

:

T

⁢

X

→

T

⁢

Y

,

whereas the pullback (

http://planetmath.org/PullbackOfAKForm

) of

f

is a mapping

f

∗

:

Ω

k

⁢

(

Y

)

→

Ω

k

⁢

(

X

)

.

For this reason, the tangent map is also sometimes called the pushforward map.
That is, a pullback takes objects from

Y

to

X

, and
a pushforward takes objects from

X

to

Y

.

Sometimes, the tangent map of

f

is also denoted by

f

∗

. However,
the motivation for denoting the tangent map by

D

⁢

f

is that if

X

and

Y

are open subsets in

ℝ

n

and

ℝ

m

, then

D

⁢

f

is simply
the Jacobian of

f

.

Title

tangent map

Canonical

name

TangentMap

Date of creation

2013-03-22 14:06:19

Last modified on

2013-03-22 14:06:19

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Definition

Classification

msc 53-00

Synonym

push forward map

Synonym

pushforward

Synonym

pushforward map

Related topic

PullbackOfAKForm

Related topic

FlowBoxTheorem

Generated on Fri Feb 9 22:18:20 2018 by

LaTeXML
topics in manifold theory

topics in manifold theory

A

manifold

is a space that is
locally like

ℝ

n

, however lacking a preferred system of
coordinates. Furthermore, a manifold can have global topological
properties, such as non-contractible loops (

http://planetmath.org/Curve

), that distinguish it from
the topologically trivial

ℝ

n

.

By imposing different restrictions on the

transition functions

of a manifold, one
obtain different types of manifolds:

•

topological manifolds

•

C

k

manifolds,

smooth manifolds

•

real analytic manifold

•

complex analytic manifold

•

symplectic manifolds

, where transition functions
are symplectomorphisms. On such manifolds, one can formulate the

Hamilton equations

.

Special types of manifolds

•

orientable manifolds

•

manifolds with boundary

•

compact manifolds

On manifolds, one can introduce more . Some examples are:

•

Riemannian manifolds

•

contact manifolds

•

CR manifolds

•

fiber bundles

and sheaves

Examples

•

space-time manifold in general relativity

•

phase space in mechanics

•

de Rham cohomology

in

algebraic topology

See also

For the formal definition click here (

http://planetmath.org/Manifold

)

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

Manifold entry at Wikipedia

Title

topics in manifold theory

Canonical name

TopicsInManifoldTheory

Date of creation

2013-03-22 14:11:04

Last modified on

2013-03-22 14:11:04

Owner

evin290 (5830)

Last modified by

evin290 (5830)

Numerical id

15

Author

evin290 (5830)

Entry type

Topic

Classification

msc 53-00

Generated on Fri Feb 9 22:18:25 2018 by

LaTeXML
topological manifold

topological manifold

The

definition

given in the

manifold

article is in fact the definition of

topological manifold

. If we change it to allow only smooth (

differentiable

) transition (maximal rank)

functions

, then we speak of a

smooth manifold

.
Or if we take

real analytic function

as the transition ones we get

ℝ

-analytic manifolds.

One must emphasize that the invariant number

n

appearing in the definition there is the

dimension

of the manifold which is also the dimension of the

tangent space

at a point, in a smooth manifold.

Title

topological manifold

Canonical

name

TopologicalManifold

Date of creation

2013-03-22 16:02:05

Last modified on

2013-03-22 16:02:05

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

7

Author

juanman (12619)

Entry type

Definition

Classification

msc 53-00

Classification

msc 57R50

Related topic

Manifold

Defines

topological manifold

Generated on Fri Feb 9 22:18:27 2018 by

LaTeXML
arbelos and parbelos

arbelos and parbelos

The

arbelos

is the plane region bounded by three pairwise

tangent

semicircles

with

diameters

on the same line.

The arbelos was known already in classical

Greek geometry

.  It has many interesting properties; see e.g.

http://mathworld.wolfram.com/Arbelos.html

Mathworld.  One is that the

distance

between the two outermost points along the inner semicircles of the arbelos is the same as the distance along the outer semicircle, namely, its radius times

π

.

The

parbelos

, a parabolic analog of the arbelos, is the
plane region bounded by the

latus rectum

(

http://planetmath.org/

Hyperbola

) arcs of three

parabolas

with latera recta AB, BC, AC, where the points A, B, C lie on a line.  Unlike in the arbelos, the arcs of the parbelos are not pairwise tangent: the inner two are tangent to the outer one, but not to each other.  The parbelos has several interesting properties which can be seen in Sondow’s article [1]; see also Tsukerman’s paper [2].

Some of them are analogous to the properties of the arbelos.
For example, the distance between the two outermost two points
of the parbelos along the inner arcs is the same as along the
outer arc, namely, its

semilatus rectum

times the

universal parabolic constant

(

http://planetmath.org/ArcLengthOfParabola

)

P

.

References

1

Jonathan Sondow

: The parbelos, a parabolic
analog of the arbelos. –

Amer. Math. Monthly

120

(2013) 929–935.  Also in

http://arxiv.org/abs/1210.2279

arXiv (2012).

2

Emmanuel Tsukerman

: Solution of Sondow’s
problem: a synthetic proof of the tangency property of the
parbelos.  –

Amer. Math. Monthly

121

(2014) 438–443.
Also in

http://arxiv.org/abs/1210.5580

arXiv (2012).

Title

arbelos and parbelos

Canonical name

ArbelosAndParbelos

Date of creation

2014-06-29 8:56:48

Last modified on

2014-06-29 8:56:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Topic

Classification

msc 53A04

Classification

msc 53-03

Classification

msc 51M99

Classification

msc 01A20

Classification

msc 00A08

Defines

arbelos

Defines

parbelos

Generated on Fri Feb 9 22:18:47 2018 by

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arc length of logarithmic curve

arc length of logarithmic curve

The

arc length

of the graph of

logarithm function

(

http://planetmath.org/NaturalLogarithm2

) is

expressible in closed form

(other cases are listed in the entry

arc length of parabola

).  The usual arc length

s

=

∫

a

b

1

+

(

f

′

⁢

(

x

)

)

2

⁢

𝑑

x

gives, if

0

<

a

<

b

,  for

f

⁢

(

x

)

:=

ln

⁡

x

,

f

′

⁢

(

x

)

=

1

x

,  the

expression

s

=

∫

a

b

1

+

x

2

x

⁢

𝑑

x

.

(1)

Here, finding a suitable

substitution for integration

may be a bit difficult.  E.g.

x

:=

tan

⁡

t

leads to

∫

1

+

x

2

x

⁢

𝑑

x

=

∫

d

⁢

t

sin

⁡

t

⁢

cos

2

⁡

t

,

the substitution

x

:=

sinh

⁡

t

to

∫

1

+

x

2

x

⁢

𝑑

x

=

∫

cosh

2

⁡

t

sinh

⁡

t

⁢

𝑑

t

,

which both seem to require a new substitution.  As well the Euler’s substitutions (1st and 2nd ones) lead to awkward

rational functions

as

integrands

.

But there is the straightforward substitution

1

+

x

2

:=

t

,

x

=

t

2

-

1

,

d

⁢

x

=

t

⁢

d

⁢

t

t

2

-

1

yielding

∫

1

+

x

2

x

⁢

𝑑

x

=

∫

t

2

⁢

d

⁢

t

t

2

-

1

=

t

+

1

2

⁢

ln

⁡

t

-

1

t

+

1

+

C

=

t

-

arcoth

⁡

t

+

C

(see

area functions

) and then

∫

1

+

x

2

x

⁢

𝑑

x

=

1

+

x

2

+

1

2

⁢

ln

⁡

1

+

x

2

-

1

1

+

x

2

+

1

+

C

=

1

+

x

2

+

ln

⁡

x

1

+

1

+

x

2

+

C

.

Using this

antiderivative

, one can obtain the arc length (1).  For example, if

a

=

3

and

b

=

15

,  the result is

s

=

2

+

ln

⁡

3

5

.

As for finding the arc length of the graph of the

http://planetmath.org/node/2541

exponential function

x

↦

e

x

,  which actually is the same curve as the graph of the

inverse function

x

↦

ln

⁡

x

,  one may write the expression

s

=

∫

α

β

1

+

e

2

⁢

x

⁢

𝑑

x

.

(2)

Since here the substitution

e

x

:=

t

,

x

=

ln

⁡

t

,

d

⁢

x

=

d

⁢

t

t

shows that

∫

1

+

e

2

⁢

x

⁢

𝑑

x

=

∫

1

+

t

2

t

⁢

𝑑

t

,

we see that it’s really a question of the same task as above.  The antiderivative is

∫

1

+

e

2

⁢

x

⁢

𝑑

x

=

1

+

e

2

⁢

x

-

arsinh

⁡

e

-

x

+

C

=

1

+

e

2

⁢

x

+

ln

⁡

e

x

1

+

1

+

e

2

⁢

x

+

C

.

Title

arc length of logarithmic curve

Canonical

name

ArcLengthOfLogarithmicCurve

Date of creation

2013-03-22 19:01:45

Last modified on

2013-03-22 19:01:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Example

Classification

msc 53A04

Classification

msc 26A42

Classification

msc 26A09

Classification

msc 26A06

Synonym

arc length of exponential curve

Generated on Fri Feb 9 22:18:50 2018 by

LaTeXML
arc length of parabola

arc length of parabola

The

parabola

is one of the quite few

plane curves

, the

arc length

of which is

expressible in closed form

; other ones are line, circle (

http://planetmath.org/Circle

),

semicubical parabola

, logarithmic curve (

http://planetmath.org/NaturalLogarithm2

),

catenary

,

tractrix

,

cycloid

,

clothoid

,

astroid

, Nielsen’s spiral,

logarithmic spiral

.  Determining the arc length of

ellipse

(

http://planetmath.org/PerimeterOfEllipse

) and

hyperbola

leads to

elliptic integrals

.

We evaluate the of the parabola

y

=

a

x

2

(

a

>

0

)

(1)

from the apex (the origin) to the point

(

x

,

a

⁢

x

2

)

.

The usual arc length

s

=

∫

0

x

1

+

y

′

⁣

2

⁢

𝑑

x

=

∫

0

x

1

+

4

⁢

a

2

⁢

x

2

⁢

𝑑

x

=

1

2

⁢

a

⁢

∫

0

2

⁢

a

⁢

x

t

2

+

1

⁢

𝑑

t

.

where one has made the substitution (

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

)

2

a

x

=

:

t

.  Then one can utilise the result in the entry integration of

x

2

+

1

(

http://planetmath.org/IntegrationOfSqrtx21

), whence

s

=

1

4

⁢

a

⁢

(

2

⁢

a

⁢

x

⁢

4

⁢

a

2

⁢

x

2

+

1

+

arsinh

⁡

2

⁢

a

⁢

x

)

.

(2)

This expression for the parabola arc length becomes especially when the arc is extended from the apex to the

end point

(

1

2

⁢

a

,

1

4

⁢

a

)

of the parametre, i.e. the

latus rectum

; this arc length is

1

4

⁢

a

⁢

(

2

+

arsinh

⁡

1

)

=

1

4

⁢

a

⁢

(

2

+

ln

⁡

(

1

+

2

)

)

.

Here,

2

+

ln

(

1

+

2

)

=

:

P

is called the

universal parabolic constant

, since it is common to all parabolas; it is the ratio of the arc to the semiparametre.  This constant appears also for example in the areas of some

surfaces of revolution

(see

http://mathworld.wolfram.com/UniversalParabolicConstant.html

Reese and Sondow).

Title

arc length of parabola

Canonical name

ArcLengthOfParabola

Date of creation

2013-03-22 18:57:19

Last modified on

2013-03-22 18:57:19

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Example

Classification

msc 53A04

Classification

msc 26A42

Classification

msc 26A09

Classification

msc 26A06

Synonym

closed-form arc lengths

Related topic

FamousCurvesInThePlane

Related topic

AreaFunctions

Defines

universal parabolic constant

Generated on Fri Feb 9 22:18:53 2018 by

LaTeXML
catacaustic

catacaustic

Given a

plane curve

γ

, its

catacaustic

(Greek

ϰ

⁢

α

⁢

τ

⁢

α

´

⁢

ϰ

⁢

α

⁢

υ

⁢

σ

⁢

τ

⁢

ι

⁢

ϰ

⁢

o

´

⁢

ς

‘burning along’) is the

envelope

of a family of rays reflected from

γ

after having emanated from a point (which may be infinitely far, in which case the rays are initially

parallel

).

For example, the catacaustic of a

logarithmic spiral

reflecting the rays emanating from the origin is a

congruent

spiral.  The catacaustic of the exponential curve (

http://planetmath.org/ExponentialFunction

)

y

=

e

x

reflecting the vertical rays

x

=

t

is the

catenary

y

=

cosh

⁡

(

x

+

1

)

.

Title

catacaustic

Canonical

name

Catacaustic

Date of creation

2013-03-22 18:52:56

Last modified on

2013-03-22 18:52:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A04

Classification

msc 51N20

Classification

msc 26B05

Classification

msc 26A24

Synonym

caustic

Related topic

HeronsPrinciple

Related topic

ExampleOfFindingCatacaustic

Generated on Fri Feb 9 22:18:54 2018 by

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