aarjaneiro/planetmath huggingface dataset:
invariant scalar product

invariant scalar product

Let

𝕂

be a field and

V

a

vector space

over

𝕂

. Let

G

be a

group with

a specified

representation

on

V

denoted by

g

.

v

for

v

∈

V

and

g

∈

G

.

An

invariant scalar product

(with respect to the action of

G

) on

V

is a

scalar product

(

⋅

|

⋅

)

on

V

(i.e. a

non-degenerate

,

symmetric

𝕂

-

bilinear form

) such that for any

g

∈

G

,

u

,

v

∈

V

we have

(

g

.

u

|

g

.

v

)

=

(

u

|

v

)

Now let

𝔤

be a

Lie algebra

over

𝕂

with a representation on

V

denoted by

X

.

v

for

X

∈

𝔤

,

v

∈

V

. Then an

invariant scalar product

(with respect to the action of

𝔤

) is a scalar product on

V

such that for any

X

∈

𝔤

,

u

,

v

∈

V

we have

(

X

.

u

|

v

)

=

-

(

u

|

X

.

v

)

An invariant scalar product on a Lie algebra

𝔤

is by definition an
invariant scalar product as above where the representation is the

adjoint representation

of

𝔤

on itself. In this case invariance is usualy written

(

[

X

,

Y

]

∣

Z

)

=

(

X

∣

[

Y

,

Z

]

)

1

Examples

For example if

G

=

O

⁢

(

n

)

the

orthogonal

subgroup

of

n

×

n

real matricies and

ℝ

n

is the natural representation for

O

⁢

(

n

)

, then the standard Euclidean scalar product on

ℝ

n

is an invariant scalar product. Invariance in this example follows from the definition of

O

⁢

(

n

)

.

As another example if

𝔤

is a complex

semi-simple Lie algebra

then the

Killing form

κ

⁢

(

X

,

Y

)

:=

T

⁢

r

⁢

(

a

⁢

d

X

⋅

a

⁢

d

Y

)

is an invariant scalar product on

𝔤

itself via the adjoint representation. Invariance in this example follows from the fact that the trace operator is

associative

, i.e.

T

⁢

r

⁢

(

[

Y

,

X

]

⋅

Z

)

=

-

T

⁢

r

⁢

(

[

X

,

Y

]

⋅

Z

)

=

-

T

⁢

r

⁢

(

X

⋅

[

Y

,

Z

]

)

. Thus an invariant scalar product (with respect to a

Lie algebra representation

) is sometimes called an

associative scalar product

.

Title

invariant scalar product

Canonical name

InvariantScalarProduct

Date of creation

2013-03-22 15:30:16

Last modified on

2013-03-22 15:30:16

Owner

benjaminfjones (879)

Last modified by

benjaminfjones (879)

Numerical id

7

Author

benjaminfjones (879)

Entry type

Definition

Classification

msc 22E15

Classification

msc 22E10

Classification

msc 22E60

Classification

msc 15A63

Classification

msc 22E20

Synonym

invariant bilinear form

Synonym

associative bilinear form

Related topic

DotProduct

Defines

invariant scalar product

Defines

associative bilinear form

Defines

Killing form

Generated on Fri Feb 9 19:40:08 2018 by

LaTeXML
one-parameter subgroup

one-parameter subgroup

Let

G

be a

Lie Group

. A

one-parameter subgroup

of

G

is a

group homomorphism

ϕ

:

ℝ

→

G

that is also a differentiable
map at the same time. We view

ℝ

additively and

G

multiplicatively, so that

ϕ

⁢

(

r

+

s

)

=

ϕ

⁢

(

r

)

⁢

ϕ

⁢

(

s

)

.

Examples

.

1.

If

G

=

GL

⁡

(

n

,

k

)

, where

k

=

ℝ

or

ℂ

, then any one-parameter subgroup has the form

ϕ

⁢

(

t

)

=

e

t

⁢

A

,

where

A

=

d

⁢

ϕ

d

⁢

t

⁢

(

0

)

is an

n

×

n

matrix over

k

. The matrix

A

is just a
tangent vector to the Lie group

GL

⁡

(

n

,

k

)

. This property establishes the fact that there is a

one-to-one correspondence

between one-parameter subgroups and tangent vectors of

GL

⁡

(

n

,

k

)

. The same relationship holds for a general Lie group.
The one-to-one correspondence between tangent vectors at the

identity

(the
Lie algebra) and one-parameter subgroups is established via the

exponential
map

instead of the

matrix exponential

.

2.

If

G

=

O

⁡

(

n

,

ℝ

)

⊆

GL

⁡

(

n

,

ℝ

)

, the

orthogonal group

over

R

, then
any one-parameter subgroup has the same form as in the example above, except that

A

is

skew-symmetric

:

A

T

=

-

A

.

3.

If

G

=

SL

⁡

(

n

,

ℝ

)

⊆

GL

⁡

(

n

,

ℝ

)

, the

special linear group

over

R

,
then any one-parameter subgroup has the same form as in the example above, except that

tr

⁡

(

A

)

=

0

, where

tr

is the trace operator.

4.

If

G

=

U

⁡

(

n

)

=

O

⁡

(

n

,

ℂ

)

⊆

GL

⁡

(

n

,

ℂ

)

, the

unitary
group

over

C

, then any one-parameter subgroup has the same form as in the example above, except that

A

is skew-Hermitian (

http://planetmath.org/SkewHermitianMatrix

):

A

=

-

A

*

=

-

A

¯

T

and

tr

⁡

(

A

)

=

0

.

Title

one-parameter subgroup

Canonical name

OneparameterSubgroup

Date of creation

2013-03-22 14:54:01

Last modified on

2013-03-22 14:54:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 22E15

Classification

msc 22E10

Synonym

1-parameter subgroup

Generated on Fri Feb 9 19:40:11 2018 by

LaTeXML
Hilbert-Weyl theorem

Hilbert-Weyl theorem

Theorem:

Let

Γ

be a

compact Lie group

acting on

V

. Then there exists a finite Hilbert basis for the ring

𝒫

⁢

(

Γ

)

(the set of

invariant polynomials

)

.

[

GSS

]

proof:

In

[

GSS

]

on page 54.

Theorem:

(as stated by Hermann Weyl)

The

(absolute)

invariants corresponding to a given set of

representations

of a finite or a compact Lie group have a finite integrity basis.

[

HW

]

proof:

In

[

HW

]

on page 274.

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

HW

Hermann, Weyl: The

Classical Groups

: Their Invariants and Representations. Princeton University Press, New Jersey, 1946.

Title

Hilbert-Weyl theorem

Canonical name

HilbertWeylTheorem

Date of creation

2013-03-22 13:39:54

Last modified on

2013-03-22 13:39:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 22E20

Generated on Fri Feb 9 19:40:12 2018 by

LaTeXML
loop algebra

loop algebra

Let

𝔤

be a

Lie algebra

over a field

𝕂

. The

loop algebra

based on

𝔤

is defined to be

ℒ

⁢

(

𝔤

)

:=

𝔤

⊗

𝕂

𝕂

⁢

[

t

,

t

-

1

]

as a

vector space

over

𝕂

. The Lie bracket is determined by

[

X

⊗

t

k

,

Y

⊗

t

l

]

=

[

X

,

Y

]

𝔤

⊗

t

k

+

l

where

[

,

]

𝔤

denotes the Lie bracket from

𝔤

.

This clearly determines a Lie bracket. For

instance

the three term sum in the

Jacobi identity

(for

elements

which are

homogeneous

in

t

) simplifies to the three term sum for the Jacobi identity in

𝔤

tensored with a power of

t

and thus is zero in

ℒ

⁢

(

𝔤

)

.

The name “loop algebra” comes from the fact that this Lie algebra arises in the study of Lie algebras of loop groups. For the time being, assume that

𝕂

is the real or

complex numbers

so that the familiar

structures

of

analysis

and topology are available. Consider the set of all

mappings

from the circle

S

1

(we may think of this circle more concretely as the

unit circle

of the

complex plane

) to a

finite-dimensional

Lie group

G

with Lie algebra is

𝔤

. We may make this set into a group by defining

multiplication

pointwise

: given

a

,

b

:

S

1

→

G

, we define

(

a

⋅

b

)

⁢

(

x

)

=

a

⁢

(

x

)

⋅

b

⁢

(

x

)

.

References

1

Victor Kac,

Infinite Dimensional Lie Algebras

, Third edition. Cambridge University Press, Cambridge, 1990.

Title

loop algebra

Canonical name

LoopAlgebra

Date of creation

2013-03-22 15:30:07

Last modified on

2013-03-22 15:30:07

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 22E60

Classification

msc 22E65

Classification

msc 22E67

Defines

loop algebra

Generated on Fri Feb 9 19:40:14 2018 by

LaTeXML
representation theory of 𝔰⁢𝔩_2⁢ℂ

representation theory of

𝔰

⁢

𝔩

2

⁢

ℂ

The

special linear Lie algebra

of

2

×

2

matricies, denoted by

𝔰

⁢

𝔩

2

⁢

ℂ

, is defined to be the span (over

ℂ

) of the matricies

E

=

(

0

1

0

0

)

,

H

=

(

1

0

0

-

1

)

,

F

=

(

0

0

-

1

0

)

with Lie bracket given by the

commutator

of matricies:

[

X

,

Y

]

:=

X

⋅

Y

-

Y

⋅

X

. The matricies

E

,

F

,

H

satisfy the commutation

relations

:

[

E

,

F

]

=

H

,

[

H

,

E

]

=

2

⁢

E

,

[

H

,

F

]

=

-

2

⁢

F

.

The

representation theory

of

𝔰

⁢

𝔩

2

⁢

ℂ

is a very important tool for understanding the structure theory and representation theory of other

Lie algebras

(

semi-simple

finite dimensional

Lie algebras, as well as infinite dimensional Kac-Moody Lie algebras).

The finite dimensional,

irreducible

, representations of

𝔰

⁢

𝔩

2

⁢

ℂ

are in bijection with the non-negative integers

ℤ

≥

0

as follows. Let

k

∈

ℤ

≥

0

,

V

be a

ℂ

-

vector space

spanned by vectors

v

0

,

…

,

v

k

. The following action of

E

,

H

,

F

on

V

define the unique (up to

isomorphism

)

irreducible representation

of

𝔰

⁢

𝔩

2

⁢

ℂ

of

dimension

k

+

1

(or of

highest weight

k

):

E

.

v

0

=

0

E

.

v

i

=

(

i

-

1

)

⁢

(

k

-

i

+

1

)

⁢

v

i

-

1

∀

1

≤

i

≤

k

H

.

v

i

=

(

k

-

2

⁢

i

)

⁢

v

i

∀

1

≤

i

≤

k

F

.

v

i

=

v

i

+

1

∀

0

≤

i

<

k

F

.

v

k

=

0

The main points are that the one dimensional spaces

ℂ

⋅

v

i

are

eigenspaces

for

H

with

eigenvalue

k

-

2

⁢

i

, the operator corresponding to

E

kills

v

0

and otherwise sends

ℂ

⋅

v

i

→

ℂ

⋅

v

i

-

1

, while

F

kills

v

k

and otherwise sends

ℂ

⋅

v

i

→

ℂ

⋅

v

i

+

1

. The operator corresponding to

E

is often called a

raising operator

since it raises the eigenvalue for

H

, and that of

F

is called a

lowering operator

since it lowers the eigenvalue for

H

.

𝔰

⁢

𝔩

2

⁢

ℂ

is a

simple Lie algebra

, thus by Weyl’s Theorem all finite dimensional representations for

𝔰

⁢

𝔩

2

⁢

ℂ

are

completely reducible

. So any finite dimensional representation of

𝔰

⁢

𝔩

2

⁢

ℂ

splits into a

direct sum

of irreducible representations for various non-negative integers as described above.

Title

representation theory of

𝔰

⁢

𝔩

2

⁢

ℂ

Canonical name

RepresentationTheoryOfmathfraksl2mathbbC

Date of creation

2013-03-22 15:30:29

Last modified on

2013-03-22 15:30:29

Owner

benjaminfjones (879)

Last modified by

benjaminfjones (879)

Numerical id

7

Author

benjaminfjones (879)

Entry type

Definition

Classification

msc 22E60

Classification

msc 22E47

Defines

sl_2

Defines

special linear Lie algebra of 2x2 matricies

Generated on Fri Feb 9 19:40:16 2018 by

LaTeXML
the connection between Lie groups and Lie algebras

the connection between Lie groups and Lie algebras

Given a finite dimensional

Lie group

G

, it has an associated Lie algebra

𝔤

=

Lie

⁢

(

G

)

. The Lie algebra encodes a great deal of information about the Lie group. I’ve collected a few results on this topic:

Theorem 1

(

Existence

) Let

g

be a finite dimensional Lie algebra over

R

or

C

. Then there exists a finite dimensional real or complex Lie group

G

with

Lie

⁢

(

G

)

=

g

.

Theorem 2

(Uniqueness) There is a unique

connected

simply-connected Lie group

G

with any given finite-dimensional Lie algebra. Every connected Lie group with this Lie algebra is a

quotient

G

/

Γ

by a discrete central subgroup

Γ

.

Even more important, is the fact that the correspondence

G

↦

𝔤

is functorial: given a

homomorphism

φ

:

G

→

H

of Lie groups, there is

natural
homomorphism

defined on Lie algebras

φ

*

:

𝔤

→

𝔥

, which just the

derivative

of the map

φ

at the

identity

(since the Lie algebra is canonically identified with the tangent space at the identity).

There are analogous existence and

uniqueness theorems

for maps:

Theorem 3

(Existence) Let

ψ

:

g

→

h

be a homomorphism of Lie algebras. Then if

G

is the unique connected, simply-connected group with Lie algebra

g

, and

H

is any Lie group with Lie algebra

h

, there exists a homomorphism of Lie groups

φ

:

G

→

H

with

φ

*

=

ψ

.

Theorem 4

(Uniqueness)
Let

G

be connected Lie group and

H

an arbitrary Lie group. Then if two maps

φ

,

φ

′

:

G

→

H

induce the same maps on Lie algebras, then they are equal.

Essentially, what these

theorems

tell us is the correspondence

𝔤

↦

G

from Lie algebras to simply-connected Lie groups is functorial, and right adjoint (

http://planetmath.org/AdjointFunctor

) to the functor

H

↦

Lie

⁢

(

H

)

from Lie groups to Lie algebras.

Title

the

connection between Lie groups and Lie algebras

Canonical name

TheConnectionBetweenLieGroupsAndLieAlgebras

Date of creation

2013-03-22 13:20:56

Last modified on

2013-03-22 13:20:56

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

10

Author

bwebste (988)

Entry type

Definition

Classification

msc 22E60

Generated on Fri Feb 9 19:40:18 2018 by

LaTeXML
Lie groupoid

Lie groupoid

Definition 0.1

.

A

Lie groupoid

is is a

category

𝒢

L

=

(

G

0

,

G

1

)

in which every arrow or

morphism

is invertible, and also such that the following conditions are satisfied:

1.

The space of objects

G

0

and the space of arrows

G

1

are both

smooth manifolds

2.

Both

structure maps

s

,

t

:

G

1

⟶

G

0

are smooth

3.

All structure maps are

submersions

:

s

,

t

:

G

1

⟶

G

0

,

u

:

G

0

⟶

G

1

,

i

:

G

1

⟶

G

1

,

and

m

:

G

1

×

s

,

t

G

1

⟶

G

1

.

Notes:

A Lie groupoid can be considered as a generalization of a

Lie group

, but it does have the additional requirements for the

groupoid

’s structure maps that do not have corresponding conditions in the simpler case of the Lie group structure. Because the object space

G

0

of a Lie groupoid

𝒢

L

is a smooth manifold,

G

0

is denoted in this case as

M

.

Title

Lie groupoid

Canonical name

LieGroupoid

Date of creation

2013-03-22 19:19:21

Last modified on

2013-03-22 19:19:21

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

15

Author

bci1 (20947)

Entry type

Definition

Classification

msc 22E70

Classification

msc 22E60

Classification

msc 20F40

Classification

msc 22A22

Classification

msc 20L05

Related topic

Groupoid

Related topic

GroupoidRepresentation4

Related topic

RepresentationsOfLocallyCompactGroupoids

Related topic

Functor

Generated on Fri Feb 9 19:40:26 2018 by

LaTeXML
spheres that are Lie groups

spheres that are Lie groups

\PMlinkescapephrase

spheres

\PMlinkescapephrase

satisfy

Theorem -

The only spheres (

http://planetmath.org/Sphere

) that are

Lie groups

are

S

0

,

S

1

and

S

3

.

Proof:

It is known that

S

0

,

S

1

and

S

3

have a Lie group .

On the other , we have seen in the parent entry (

http://planetmath.org/CohomologyOfCompactConnectedLieGroups

) that the

cohomology groups

(

http://planetmath.org/DeRhamCohomology

) of a

compact

connected

Lie group

G

satisfy

H

1

⁢

(

G

;

ℝ

)

=

0

⟹

H

3

⁢

(

G

;

ℝ

)

≠

0

The result then follows from the fact that the of spheres satisfy

H

1

⁢

(

S

n

;

ℝ

)

=

0

and

H

3

⁢

(

S

n

;

ℝ

)

=

0

for

n

≠

1

,

3

.

□

Title

spheres that are Lie groups

Canonical name

SpheresThatAreLieGroups

Date of creation

2013-03-22 17:50:02

Last modified on

2013-03-22 17:50:02

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Corollary

Classification

msc 22E99

Classification

msc 57T10

Generated on Fri Feb 9 19:40:28 2018 by

LaTeXML
abelian function

abelian function

An abelian or

hyperelliptic function

is a generalisation of an

elliptic function

. It is a function of two variables with four periods.
In a

similar

way to an elliptic function it can also be regarded as the

inverse function

to certain

integrals

(called abelian or hyperelliptic integrals) of the form

∫

d

⁢

z

R

⁢

(

z

)

where

R

is a

polynomial

of degree

greater

than 4.

Title

abelian function

Canonical name

AbelianFunction

Date of creation

2013-03-22 14:43:34

Last modified on

2013-03-22 14:43:34

Owner

vmoraru (1243)

Last modified by

vmoraru (1243)

Numerical id

5

Author

vmoraru (1243)

Entry type

Definition

Classification

msc 26-00

Synonym

hyperelliptic function

Generated on Fri Feb 9 19:40:42 2018 by

LaTeXML
alternative proof of the fundamental theorem of calculus

alternative proof of the fundamental theorem of calculus

An alternative proof for the first part involves the use of a formula derived by the

method of exhaustion

:

∫

a

b

f

⁢

(

t

)

⁢

𝑑

t

=

(

b

-

a

)

⁢

∑

n

=

1

∞

∑

m

=

1

2

n

-

1

(

-

1

)

m

+

1

⁢

2

-

n

⁢

f

⁢

(

a

+

m

⁢

(

b

-

a

)

/

2

n

)

.

Given that

F

⁢

(

x

)

=

∫

a

x

f

⁢

(

t

)

⁢

𝑑

t

,

and

F

′

⁢

(

x

)

=

lim

Δ

⁢

x

→

0

⁡

F

⁢

(

x

+

Δ

⁢

x

)

-

F

⁢

(

x

)

Δ

⁢

x

=

lim

Δ

⁢

x

→

0

⁡

1

Δ

⁢

x

⁢

∫

x

x

+

Δ

⁢

x

f

⁢

(

t

)

⁢

𝑑

t

,

the above formula leads to:

F

′

⁢

(

x

)

=

lim

Δ

⁢

x

→

0

⁡

(

x

+

Δ

⁢

x

-

x

)

Δ

⁢

x

⁢

∑

n

=

1

∞

∑

m

=

1

2

n

-

1

(

-

1

)

m

+

1

⁢

2

-

n

⁢

f

⁢

(

x

+

m

⁢

Δ

⁢

x

/

2

n

)

,

or

F

′

⁢

(

x

)

=

∑

n

=

1

∞

∑

m

=

1

2

n

-

1

(

-

1

)

m

+

1

⁢

2

-

n

⁢

f

⁢

(

x

)

.

Since it can be shown that

∑

n

=

1

∞

∑

m

=

1

2

n

-

1

(

-

1

)

m

+

1

⁢

2

-

n

=

∑

n

=

1

∞

2

-

n

=

1

,

It follows that

F

′

⁢

(

x

)

=

f

⁢

(

x

)

.

The second part of the proof is identical to the parent.

Title

alternative proof of the

fundamental theorem of calculus

Canonical name

AlternativeProofOfTheFundamentalTheoremOfCalculus

Date of creation

2013-03-22 15:55:24

Last modified on

2013-03-22 15:55:24

Owner

ruffa (7723)

Last modified by

ruffa (7723)

Numerical id

4

Author

ruffa (7723)

Entry type

Proof

Classification

msc 26-00

Generated on Fri Feb 9 19:40:45 2018 by

LaTeXML
bibliography for real analysis

bibliography for real analysis

1

Introductory

Ralph P. Boas,

A Primer of

Real Functions

, MAA, 1972.

Edward Gaughan,

Introduction to

Analysis

, Brooks/Cole, 1968.

Serge Lang,

Undergraduate Analysis

, Springer-Verlag, 1968.

Walter Rudin,

Principles of Mathematical Analysis

, McGraw-Hill, 1964.

Georgi A. Shilov,

Elementary Real and Complex Analysis

, MIT, 1973.

2

Advanced

Sterling K. Berberian,

Measure and Integration

, Chelsea, 1970.

Edwin Hewitt and Karl Stromberg,

Real and Abstract Analysis

, Springer-Verlag, 1969.

H. L. Royden,

Real Analysis

, MacMillan, 1968.

Walter Rudin,

Real & Complex Analysis

, McGraw-Hill, 1987.

Ernst Lindelöf,

Differentiali- ja integralilasku ja sen sovellutukset

I – IV (

Differential

and integral calculus and its applications), WSOY and Mercatorin Kirjapaino, Helsinki, 1928–1946.

G. E. Shilov and B. L. Gurevich,

Integral

, Measure and

Derivative

: A Unified Approach

, Prentice-Hall, 1966.

3

Additional Topic

non-Newtonian

calculus

Title

bibliography for real analysis

Canonical name

BibliographyForRealAnalysis

Date of creation

2013-03-22 15:39:55

Last modified on

2013-03-22 15:39:55

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

14

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 26-00

Generated on Fri Feb 9 19:40:47 2018 by

LaTeXML
derivative notation

derivative notation

This is the list of known

standard representations

and their nuances.

d

⁢

u

d

⁢

v

,

d

⁢

f

d

⁢

x

,

d

⁢

y

d

⁢

x

-

The most common notation, this is read as the

derivative

of

u

with respect to

v

.

Exponents

relate which derivative, for example,

d

2

⁢

y

d

⁢

x

2

is the

second derivative

of

y

with respect to

x

.

f

′

⁢

(

x

)

,

f

→

′

⁢

(

𝐱

)

,

y

′′

-

This is read as

f

prime of

x

. The number of primes tells the derivative, ie.

f

′′′

⁢

(

x

)

is the third derivative of

f

⁢

(

x

)

with respect to

x

. Note that in higher

dimensions

, this may be a tensor of a rank equal to the derivative.

D

x

⁢

f

⁢

(

𝐱

)

,

F

y

⁢

(

𝐱

)

,

f

x

⁢

y

⁢

(

𝐱

)

-

These notations are rather arcane, and should not be used generally, as they have other meanings. For example

F

y

can easily by the

y

component

of a

vector-valued function

. The subscript in this case means “with respect to”, so

F

y

⁢

y

would be the second derivative of

F

with respect to

y

.

D

1

⁢

f

⁢

(

𝐱

)

,

F

2

⁢

(

𝐱

)

,

f

12

⁢

(

𝐱

)

-

The subscripts in these cases refer to the derivative with respect to the nth variable. For example,

F

2

⁢

(

x

,

y

,

z

)

would be the derivative of

F

with respect to

y

. They can easily represent higher derivatives, ie.

D

21

⁢

f

⁢

(

𝐱

)

is the derivative with respect to the first variable of the derivative with respect to the second variable.

∂

⁡

u

∂

⁡

v

,

∂

⁡

f

∂

⁡

x

-

The

partial derivative

of

u

with respect to

v

. This symbol can be manipulated as in

d

⁢

u

d

⁢

v

for higher partials.

d

d

⁢

v

,

∂

∂

⁡

v

-

This is the operator version of the derivative. Usually you will see it acting on something such as

d

d

⁢

v

⁢

(

v

2

+

3

⁢

u

)

=

2

⁢

v

.

[

𝐉𝐟

⁢

(

𝐱

)

]

,

[

𝐃𝐟

⁢

(

𝐱

)

]

-

The first of these represents the

http://planetmath.org/node/842

Jacobian

of

𝐟

, which is a matrix of partial derivatives such that

[

𝐉𝐟

⁢

(

𝐱

)

]

=

[

D

1

⁢

f

1

⁢

(

𝐱

)

…

D

n

⁢

f

1

⁢

(

𝐱

)

⋮

⋱

⋮

D

1

⁢

f

m

⁢

(

𝐱

)

…

D

n

⁢

f

m

⁢

(

𝐱

)

]

where

f

n

represents the nth function of a vector valued function. The second of these notations represents the derivative matrix, which in most cases is the Jacobian, but in some cases, does not exist, even though the Jacobian exists. Note that the

directional derivative

in the

direction

v

→

is simply

[

𝐉𝐟

⁢

(

𝐱

)

]

⁢

v

→

.

Title

derivative notation

Canonical name

DerivativeNotation

Date of creation

2013-03-22 11:58:27

Last modified on

2013-03-22 11:58:27

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Topic

Classification

msc 26-00

Related topic

Derivative

Related topic

Gradient

Related topic

PartialDerivative

Related topic

DirectionalDerivative

Related topic

JacobianMatrix

Related topic

LeibnizNotationForVectorFields

Generated on Fri Feb 9 19:40:49 2018 by

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derivative of logarithm with respect to base

derivative of logarithm with respect to base

The

∂

∂

⁡

a

⁢

log

a

⁡

x

=

-

ln

⁡

x

(

ln

⁡

a

)

2

⁢

a

(1)

for the

partial derivative

of

logarithm

expression

with respect to the base

a

may be derived by denoting first

log

a

⁡

x

=

y

.

By the

definition

of logarithm, this equation means the same as

a

y

=

x

,

where we can take the

natural logarithms

y

⁢

ln

⁡

a

=

ln

⁡

x

solving then

y

=

ln

⁡

x

ln

⁡

a

.

Then, the

differentiation

is easy:

∂

⁡

y

∂

⁡

a

=

0

⁢

ln

⁡

a

-

1

a

⁢

ln

⁡

x

(

ln

⁡

a

)

2

=

-

ln

⁡

x

(

ln

⁡

a

)

2

⁢

a

.

Title

derivative of logarithm with respect to base

Canonical

name

DerivativeOfLogarithmWithRespectToBase

Date of creation

2013-03-22 19:11:28

Last modified on

2013-03-22 19:11:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Definition

Classification

msc 26-00

Classification

msc 26A09

Classification

msc 26A06

Related topic

Derivative

Generated on Fri Feb 9 19:40:51 2018 by

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