differential field

differential field

Let

F

be a field (ring) together with a derivation

(

⋅

)

′

:

F

→

F

.
The derivation must satisfy two properties:

Additivity

(

a

+

b

)

′

=

a

′

+

b

′

;

Leibniz’ Rule

(

a

⁢

b

)

′

=

a

′

⁢

b

+

a

⁢

b

′

.

A derivation is the algebraic abstraction of a

derivative

from ordinary

calculus

. Thus the terms

derivation

,

derivative

, and

differential

are often used interchangeably.

Together,

(

F

,

)

′

is referred to as a

differential field

(ring)

.
The

subfield

(subring) of all elements with vanishing derivative,

K

=

{

a

∈

F

∣

a

′

=

0

}

,
is called the

field (ring) of constants

. Clearly,

(

⋅

)

′

is

K

-linear.

There are many notations for the derivation symbol, for example

a

′

may
also be denoted as

d

⁢

a

,

δ

⁢

a

,

∂

⁡

a

, etc. When there is more
than one derivation

∂

i

,

(

F

,

{

∂

i

}

)

is referred to as a

partial differential field (ring)

.

1

Examples

Differential fields and rings (together under the name of

differential algebra

)
are a natural setting for the study of algebraic properties of derivatives
and anti-derivatives (

indefinite integrals

), as well as ordinary and

partial differential
equations

and their

solutions

. There is an abundance of examples drawn
from these areas.

•

The trivial example is a field

F

with

a

′

=

0

for each

a

∈

F

. Here,
nothing new is gained by introducing the derivation.

•

The most common example is the

field of rational functions

ℝ

⁢

(

z

)

over an indeterminant satisfying

z

′

=

1

. The field of constants is

ℝ

. This is the setting for ordinary calculus.

•

Another example is

ℝ

⁢

(

x

,

y

)

with two derivations

∂

x

and

∂

y

. The field of constants is

ℝ

and the
derivations are extended to all elements from the properties

∂

x

⁡

x

=

1

,

∂

y

⁡

y

=

1

, and

∂

x

⁡

y

=

∂

y

⁡

x

=

0

.

•

Consider the set of

smooth functions

C

∞

⁢

(

M

)

on a manifold

M

. They
form a ring (or a field if we allow formal

inversion

of functions
vanishing in some places). Vector fields on

M

act naturally as
derivations on

C

∞

⁢

(

M

)

.

•

Let

A

be an

algebra

and

U

t

=

exp

⁡

(

t

⁢

u

)

be a one-parameter subgroup of

automorphisms

of

A

. Here

u

is the infinitesimal generator of these
automorphisms. From the properties of

U

t

,

u

must be a linear operator
on

A

that satisfies the

Leibniz rule

u

⁢

(

a

⁢

b

)

=

u

⁢

(

a

)

⁢

b

+

a

⁢

u

⁢

(

b

)

. So

(

A

,

u

)

can be considered a differential ring.

Title

differential field

Canonical name

DifferentialField

Date of creation

2013-03-22 14:18:47

Last modified on

2013-03-22 14:18:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 13N15

Classification

msc 12H05

Related topic

DifferentialPropositionalCalculus

Defines

differential ring

Defines

partial differential field

Defines

partial differential ring

Defines

field of constants

Defines

ring of constants

Generated on Fri Feb 9 17:34:48 2018 by

LaTeXML