aarjaneiro/planetmath huggingface dataset:
proof of properties of derivatives by pure algebra

proof of properties of derivatives by pure algebra

Theorem 1

.

The

derivative

satisfies the following rules:

Linearity

d

d

⁢

x

⁢

(

f

⁢

(

x

)

+

g

⁢

(

x

)

)

=

d

⁢

f

d

⁢

x

+

d

⁢

g

d

⁢

x

,

d

d

⁢

x

⁢

(

a

⁢

f

⁢

(

x

)

)

=

a

⁢

d

⁢

f

d

⁢

x

,

for

f

⁢

(

x

)

,

g

⁢

(

x

)

∈

R

⁢

[

x

]

and

a

∈

R

.

Power Rule

d

d

⁢

x

⁢

(

x

n

)

=

n

⁢

x

n

-

1

.

Product Rule

d

d

⁢

x

⁢

(

f

⁢

(

x

)

⁢

g

⁢

(

x

)

)

=

d

⁢

f

d

⁢

x

⁢

g

⁢

(

x

)

+

f

⁢

(

x

)

⁢

d

⁢

g

d

⁢

x

.

Remark 2

.

The following proofs apply to

derivatives by pure algebra

(

http://

planetmath

.org/DerivativesByPureAlgebra

). While the nature of the proofs are
similar to the usual proofs, the notion of a limit is replaced by

modular
arithmetic

in

R

⁢

[

x

,

h

]

/

(

h

)

.

Proof.

Power rule.

d

d

⁢

x

⁢

(

x

n

)

≡

(

x

+

h

)

n

-

x

n

h

=

∑

j

=

1

n

(

i

j

)

⁢

x

n

-

j

⁢

h

j

-

1

≡

(

n

1

)

⁢

x

n

-

1

=

n

⁢

x

n

-

1

.

Linearity rule.

For all

f

⁢

(

x

)

,

g

⁢

(

x

)

∈

R

⁢

[

x

]

≅

R

⁢

[

x

,

h

]

/

(

h

)

, it follows

(

f

+

g

)

⁢

(

x

+

h

)

-

(

f

+

g

)

⁢

(

x

)

h

≡

f

⁢

(

x

+

h

)

+

g

⁢

(

x

+

h

)

-

f

⁢

(

x

)

-

g

⁢

(

x

)

h

≡

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

+

g

⁢

(

x

+

h

)

-

g

⁢

(

x

)

h

.

Furthermore, for all

a

∈

R

(

a

⁢

f

)

⁢

(

x

+

h

)

-

(

a

⁢

f

)

⁢

(

x

)

h

≡

a

⁢

f

⁢

(

x

+

h

)

-

a

⁢

f

⁢

(

x

)

h

=

a

⁢

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

.

Product rule.

In

R

⁢

[

x

,

h

]

modulo

(

h

)

we have:

d

d

⁢

x

⁢

(

f

⁢

g

)

≡

f

(

x

+

h

)

g

(

x

+

h

)

)

-

f

(

x

)

g

(

x

)

h

≡

f

⁢

(

x

+

h

)

⁢

g

⁢

(

x

+

h

)

-

f

⁢

(

x

)

⁢

g

⁢

(

x

+

h

)

+

f

⁢

(

x

)

⁢

g

⁢

(

x

+

h

)

-

f

⁢

(

x

)

⁢

g

⁢

(

x

)

h

≡

(

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

)

⁢

g

⁢

(

x

+

h

)

+

f

⁢

(

x

)

⁢

(

g

⁢

(

x

+

h

)

-

g

⁢

(

x

)

)

h

≡

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

⁢

g

⁢

(

x

+

h

)

+

f

⁢

(

x

)

⁢

g

⁢

(

x

+

h

)

-

g

⁢

(

x

)

h

≡

d

⁢

f

d

⁢

x

⁢

g

⁢

(

x

)

+

f

⁢

(

x

)

⁢

d

⁢

g

d

⁢

x

.

∎

Title

proof of properties of derivatives by pure algebra

Canonical name

ProofOfPropertiesOfDerivativesByPureAlgebra

Date of creation

2013-03-22 16:00:03

Last modified on

2013-03-22 16:00:03

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

5

Author

Algeboy (12884)

Entry type

Proof

Classification

msc 26B05

Classification

msc 46G05

Classification

msc 26A24

Related topic

RulesOfCalculusForDerivativeOfPolynomial

Generated on Fri Feb 9 19:56:17 2018 by

LaTeXML
Rademacher’s theorem

Rademacher’s theorem

Let

f

:

ℝ

n

→

ℝ

be any

Lipschitz continuous

function. Then

f

is

differentiable

in almost every

x

∈

ℝ

n

.

Title

Rademacher’s theorem

Canonical name

RademachersTheorem

Date of creation

2013-03-22 13:32:01

Last modified on

2013-03-22 13:32:01

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

6

Author

paolini (1187)

Entry type

Theorem

Classification

msc 26B05

Synonym

differentiability of Lipschitz functions

Related topic

LipschitzCondition

Generated on Fri Feb 9 19:56:19 2018 by

LaTeXML
smooth functions with compact support

smooth functions with compact support

Definition

Let

U

be an

open set

in

ℝ

n

. Then the set of

smooth functions with compact support

(in

U

) is the set
of

functions

f

:

ℝ

n

→

ℂ

which are smooth
(i.e.,

∂

α

⁡

f

:

ℝ

n

→

ℂ

is a

continuous function

for all multi-indices

α

)
and

supp

⁡

f

is

compact

and contained in

U

.
This

function space

is denoted by

C

0

∞

⁢

(

U

)

.

0.0.1

Remarks

1.

A proof that

C

0

∞

⁢

(

U

)

is non-trivial (that is, it contains other functions
than the zero function) can be found
here (

http://planetmath.org/Cinfty_0UIsNotEmpty

).

2.

With the usual point-wise addition and point-wise multiplication
by a scalar,

C

0

∞

⁢

(

U

)

is a vector space over the field

ℂ

.

3.

Suppose

U

and

V

are open subsets in

ℝ

n

and

U

⊂

V

.
Then

C

0

∞

⁢

(

U

)

is a vector subspace of

C

0

∞

⁢

(

V

)

.
In particular,

C

0

∞

⁢

(

U

)

⊂

C

0

∞

⁢

(

V

)

.

It is possible to equip

C

0

∞

⁢

(

U

)

with a

topology

, which makes

C

0

∞

⁢

(

U

)

into a

locally convex topological vector space

. The idea is
to exhaust

U

with compact sets. Then, for each compact set

K

⊂

U

,
one defines a topology of

smooth functions

on

U

with

support

on

K

. The topology for

C

0

∞

⁢

(

U

)

is the inductive
limit topology of these topologies. See e.g.

[

1

]

.

References

1

W. Rudin,

Functional Analysis

,
McGraw-Hill Book Company, 1973.

Title

smooth functions with compact support

Canonical name

SmoothFunctionsWithCompactSupport

Date of creation

2013-03-22 13:44:00

Last modified on

2013-03-22 13:44:00

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

10

Author

matte (1858)

Entry type

Definition

Classification

msc 26B05

Related topic

Cn

Generated on Fri Feb 9 19:56:21 2018 by

LaTeXML
tangent line

tangent line

If the curve

y

=

f

⁢

(

x

)

of

x

⁢

y

-plane is sufficiently smooth in its point

(

x

0

,

y

0

)

and in a

neighborhood

of this, the curve may have a

tangent line

(or simply ) in

(

x

0

,

y

0

)

.  Then the

tangent line

of the curve

y

=

f

⁢

(

x

)

in the point

(

x

0

,

y

0

)

is the limit position of the

secant line

through the two points

(

x

0

,

y

0

)

and

(

x

,

f

⁢

(

x

)

)

of the curve, when

x

limitlessly tends to the value

x

0

(i.e.

x

→

x

0

)

.  Due to the smoothness,

f

⁢

(

x

)

→

f

⁢

(

x

0

)

=

y

0

,

(

x

,

f

⁢

(

x

)

)

→

(

x

0

,

y

0

)

,

and the slope

m

of the secant (

http://planetmath.org/SecantLine

) tends to

lim

x

→

x

0

⁡

f

⁢

(

x

)

-

f

⁢

(

x

0

)

x

-

x

0

=

f

′

⁢

(

x

0

)

which will be the slope of the tangent line.

Note.

Because the tangency is a local property on the curve, the

tangent

with the

tangency point

(

x

0

,

y

0

)

may intersect the curve in another point, and then the tangent is a secant (

http://planetmath.org/SecantLine

), too.  For example, the curve

y

=

x

3

-

3

⁢

x

2

has the line

y

=

0

as its tangent in the point

(

0

,

0

)

but this line the curve also in the point

(

3

,

0

)

.

Title

tangent line

Canonical

name

TangentLine

Date of creation

2013-03-22 14:50:31

Last modified on

2013-03-22 14:50:31

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 26B05

Classification

msc 26A24

Synonym

tangent

Synonym

tangent of the curve

Synonym

tangent to the curve

Related topic

Curve

Related topic

TangentOfConicSection

Related topic

Hyperbola2

Defines

tangency point

Generated on Fri Feb 9 19:56:25 2018 by

LaTeXML
circular segment

circular segment

A chord of a circle the corresponding disk into two

circular segments

.  The perimetre of a circular segment consists thus of the chord (

c

) and a circular arc (

a

).

The magnitude

r

of the radius of circle and the magnitude

α

of a

central angle

naturally determine uniquely the magnitudes of the corresponding arc and chord, and these may be directly calculated from

{

a

=

r

⁢

α

,

c

=

2

⁢

r

⁢

sin

⁡

α

2

.

(1)

Conversely, the magnitudes of

a

and

c

(

<

a

) uniquely determine

r

and

α

from the pair of equations (1), but

r

and

α

are generally not in a

closed form

; this becomes clear from the relationship

c

a

⋅

α

2

=

sin

⁡

α

2

implied by (1).

The area of a circular segment is obtained by subtracting from [resp. adding to] the area of the corresponding sector the area of the

isosceles triangle

having the chord as base (

http://planetmath.org/BaseAndHeightOfTriangle

) [the adding concerns the case where the central angle is

greater

than the

straight angle

]:

A

=

α

2

⁢

π

⋅

π

⁢

r

2

∓

1

2

⁢

r

2

⁢

sin

⁡

α

=

r

2

2

⁢

(

α

∓

sin

⁡

α

)

The of the circular segment, i.e. the

distance

of the

midpoints

(

http://planetmath.org/ArcLength

) of the arc and the chord, may be expressed in the following forms:

h

=

(

1

-

cos

⁡

α

2

)

⁢

r

=

r

-

r

2

-

c

2

4

=

c

2

⁢

tan

⁡

α

4

Title

circular segment

Canonical name

CircularSegment

Date of creation

2013-03-22 19:05:02

Last modified on

2013-03-22 19:05:02

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B10

Classification

msc 51M04

Related topic

LineSegment

Related topic

SphericalSegment

Related topic

ExampleOfCalculusOfVariations

Defines

height of circular segment

Generated on Fri Feb 9 19:56:41 2018 by

LaTeXML
implicit differentiation

implicit differentiation

Implicit differentiation

is a tool used to analyze the solution sets of equations of the form

f

⁢

(

𝐱

,

𝐲

)

=

0

that cannot be conveniently put into a form

𝐲

=

g

⁢

(

𝐱

)

that explicitly shows the dependence of

𝐲

on the

variable

𝐱

. To use implicit differentiation meaningfully, you must be certain that

𝐲

is actually a

function

of

𝐱

in a

neighborhood

of the point

(

𝐚

,

𝐛

)

where you plan to apply the

derivative

. This means you want to ensure that

f

⁢

(

𝐱

,

𝐲

)

satisfies the

implicit function theorem

in that neighborhood (

f

must be

continuously differentiable

at

(

𝐚

,

𝐛

)

, and its derivative must be non-singular at

(

𝐚

,

𝐛

)

). If this is not the case, the quantity you get from naively differentiating both sides of the equation may not represent the derivative. To actually

differentiate

implicitly, we use the

chain rule

to differentiate the entire equation.

Example:

The first step is to identify the

implicit function

. Suppose we have simplified an equation to the form

x

2

+

y

2

+

x

⁢

y

=

0

(Since this is a two dimensional equation, all one has to check is that the graph of

y

may be an implicit function of

x

in local neighborhoods.) Then, to differentiate implicitly, we differentiate both sides of the equation with respect to

x

, using the chain rule whenever we encounter

y

, as

y

is treated as a function of

x

. We will get

2

⁢

x

+

2

⁢

y

⋅

d

⁢

y

d

⁢

x

+

x

⋅

1

⋅

d

⁢

y

d

⁢

x

+

y

=

0

Next, we simply solve for our implicit derivative

d

⁢

y

d

⁢

x

=

-

2

⁢

x

+

y

2

⁢

y

+

x

. Note that the derivative depends on both the variable and the implicit function

y

. Most of your derivatives will be functions of one or all the variables, including the implicit function itself.

Title

implicit differentiation

Canonical

name

ImplicitDifferentiation

Date of creation

2013-03-22 12:28:22

Last modified on

2013-03-22 12:28:22

Owner

slider142 (78)

Last modified by

slider142 (78)

Numerical id

8

Author

slider142 (78)

Entry type

Definition

Classification

msc 26B10

Generated on Fri Feb 9 19:56:43 2018 by

LaTeXML
implicit function theorem

implicit function theorem

Theorem

.

Let

Ω

be an open subset of

R

n

×

R

m

and let

f

∈

C

1

⁢

(

Ω

,

R

m

)

. Let

(

x

0

,

y

0

)

∈

Ω

⊂

R

n

×

R

m

.
If the matrix

D

y

⁢

f

⁢

(

x

0

,

y

0

)

defined by

D

y

⁢

f

⁢

(

x

0

,

y

0

)

=

(

∂

⁡

f

j

∂

⁡

y

k

⁢

(

x

0

,

y

0

)

)

j

,

k

j

=

1

,

…

,

m

k

=

1

,

…

,

m

is

invertible

, then there exists a

neighborhood

U

⊂

R

n

of

x

0

and a

function

g

∈

C

1

⁢

(

U

,

R

m

)

such that

f

⁢

(

x

,

g

⁢

(

x

)

)

=

f

⁢

(

x

0

,

y

0

)

∀

x

∈

U

.

Moreover

D

⁢

g

⁢

(

x

)

=

-

(

D

y

⁢

f

⁢

(

x

,

g

⁢

(

x

)

)

)

-

1

⁢

D

x

⁢

f

⁢

(

x

,

g

⁢

(

x

)

)

.

Title

implicit function theorem

Canonical name

ImplicitFunctionTheorem

Date of creation

2013-03-22 12:58:33

Last modified on

2013-03-22 12:58:33

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

12

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 26B10

Related topic

FlowBoxTheorem

Related topic

DerivativeAsParameterForSolvingDifferentialEquations

Generated on Fri Feb 9 19:56:45 2018 by

LaTeXML
Jacobian matrix

Jacobian matrix

The

Jacobian matrix

[

𝐉

⁢

f

⁢

(

𝐚

)

]

of a function

f

:

ℝ

n

→

ℝ

m

at the point

𝐚

with respect to some choice of bases for

ℝ

n

and

ℝ

m

is the matrix of the

linear map

from

ℝ

n

into

ℝ

m

that generalizes the definition of the

derivative

of a function on

ℝ

. It can be defined as the matrix of the linear map

D

, such that

lim

𝐡

→

𝟎

⁡

∥

f

⁢

(

𝐚

+

𝐡

)

-

f

⁢

(

𝐚

)

-

D

⁢

(

𝐡

)

∥

∥

𝐡

∥

=

0

The linear map that satisfies the above limit is called the derivative of

f

at

𝐚

. It is easy to show that the Jacobian matrix of a given

differentiable function

at

𝐚

with respect to chosen bases is just the matrix of

http://planetmath.org/node/841

partial derivatives

of the

component

functions of

f

at

𝐚

:

[

𝐉

⁢

f

⁢

(

𝐱

)

]

=

[

D

1

⁢

f

1

⁢

(

𝐱

)

…

D

n

⁢

f

1

⁢

(

𝐱

)

⋮

⋱

⋮

D

1

⁢

f

m

⁢

(

𝐱

)

…

D

n

⁢

f

m

⁢

(

𝐱

)

]

A more concise way of writing it is

[

𝐉

⁢

f

⁢

(

𝐱

)

]

=

[

D

1

⁢

f

→

,

⋯

,

D

n

⁢

f

→

]

=

[

∇

⁡

f

1

⋮

∇

⁡

f

m

]

where

D

n

⁢

𝐟

→

is the partial derivative with respect to the

n

’th variable and

∇

⁡

f

m

is the

gradient

of the

m

’th component of

𝐟

.

Note that the Jacobian matrix represents the matrix of the derivative

D

of

f

at

𝐱

iff

f

is

differentiable

at

𝐱

. Also, if

f

is differentiable at

𝐱

, then the

directional derivative

in the

direction

v

→

is

D

⁢

(

v

→

)

=

[

𝐉

⁢

f

⁢

(

𝐱

)

]

⁢

v

→

.

Given

local coordinates

for some

real submanifold

M

of

ℝ

n

, it is easy to show that the effect of a

change of coordinates

on

volume forms

is a local

scaling

of the volume form by the

determinant

of the Jacobian matrix of the derivative of the backwards change of coordinates, which is called the

inverse

Jacobian

. The determinant of the inverse Jacobian is thus commonly seen in integration over a change of coordinates.

∫

Ω

x

f

⁢

(

x

)

⁢

𝑑

Ω

x

=

∫

Ω

ξ

f

⁢

(

ξ

)

⁢

|

J

-

1

|

⁢

𝑑

Ω

ξ

where

x

∈

Ω

x

⊂

M

,

ξ

∈

Ω

ξ

⊂

M

, and

𝐉

=

∇

⁡

ξ

⁢

(

x

)

is the derivative of the function

ξ

⁢

(

x

)

which maps points in

Ω

x

to points in

Ω

ξ

.

Title

Jacobian matrix

Canonical name

JacobianMatrix

Date of creation

2013-03-22 11:58:33

Last modified on

2013-03-22 11:58:33

Owner

PhysBrain (974)

Last modified by

PhysBrain (974)

Numerical id

18

Author

PhysBrain (974)

Entry type

Definition

Classification

msc 26B10

Synonym

derivative of a multivariable function

Synonym

derivative of a

vector-valued function

Synonym

Jacobi matrix

Related topic

PartialDerivative

Related topic

Derivative2

Related topic

DerivativeNotation

Related topic

Gradient

Related topic

ChainRuleSeveralVariables

Related topic

DirectionalDerivative

Related topic

JordanBanachAndJordanLieAlgebras

Defines

Jacobian

Generated on Fri Feb 9 19:56:47 2018 by

LaTeXML
proof of implicit function theorem

proof of implicit function theorem

We state the

Theorem

with a different notation:

Theorem 1

.

Let

Ω

be an open subset of

R

n

×

R

m

and let

f

∈

C

1

⁢

(

Ω

,

R

m

)

. Let

(

x

0

,

y

0

)

∈

Ω

⊂

R

n

×

R

m

.
If the matrix

D

y

⁢

f

⁢

(

x

0

,

y

0

)

defined by

D

y

⁢

f

⁢

(

x

0

,

y

0

)

=

(

∂

⁡

f

j

∂

⁡

y

k

⁢

(

x

0

,

y

0

)

)

j

,

k

j

=

1

,

…

,

m

k

=

1

,

…

,

m

is

invertible

, then there exists a neighbourhood

U

⊂

R

n

of

x

0

,
a neighbourhood

V

⊂

R

m

of

y

0

and a

function

g

∈

C

1

⁢

(

U

,

V

)

such that

f

⁢

(

x

,

y

)

=

f

⁢

(

x

0

,

y

0

)

⇔

y

=

g

⁢

(

x

)

∀

(

x

,

y

)

∈

U

×

V

.

Moreover

D

⁢

g

⁢

(

x

)

=

-

(

D

y

⁢

f

⁢

(

x

,

g

⁢

(

x

)

)

)

-

1

⁢

D

x

⁢

f

⁢

(

x

,

g

⁢

(

x

)

)

.

Proof.

Consider the function

F

∈

𝒞

1

⁢

(

Ω

,

ℝ

n

×

ℝ

m

)

defined by

F

⁢

(

x

,

y

)

=

(

x

,

f

⁢

(

x

,

y

)

)

.

One finds that

D

⁢

F

⁢

(

x

,

y

)

=

(

I

m

0

D

x

⁢

f

D

y

⁢

f

)

.

Being

D

y

⁢

f

⁢

(

x

0

,

y

0

)

invertible,

D

⁢

F

⁢

(

x

0

,

y

0

)

is invertible too.
Applying the

inverse function Theorem

to

F

we find that there exist a neighbourhood

U

of

x

0

and

V

of

y

0

and
a function

G

∈

C

1

⁢

(

U

×

V

,

ℝ

n

+

m

)

such that

F

⁢

(

G

⁢

(

x

,

y

)

)

=

(

x

,

y

)

for all

(

x

,

y

)

∈

U

×

V

. Letting

G

⁢

(

x

,

y

)

=

(

G

1

⁢

(

x

,

y

)

,

G

2

⁢

(

x

,

y

)

)

(so that

G

1

:

V

×

W

→

ℝ

n

,

G

2

:

V

×

W

→

ℝ

m

)
we hence have

(

x

,

y

)

=

F

⁢

(

G

1

⁢

(

x

,

y

)

,

G

2

⁢

(

x

,

y

)

)

=

(

G

1

⁢

(

x

,

y

)

,

f

⁢

(

G

1

⁢

(

x

,

y

)

,

G

2

⁢

(

x

,

y

)

)

)

and hence

x

=

G

1

⁢

(

x

,

y

)

and

y

=

f

⁢

(

G

1

⁢

(

x

,

y

)

,

G

2

⁢

(

x

,

y

)

)

=

f

⁢

(

x

,

G

2

⁢

(

x

,

y

)

)

.
So we only have to set

g

⁢

(

x

)

=

G

2

⁢

(

x

,

f

⁢

(

x

0

,

y

0

)

)

to obtain

f

⁢

(

x

,

g

⁢

(

x

)

)

=

f

⁢

(

x

0

,

y

0

)

,

∀

x

∈

U

.

Differentiating with respect to

x

we obtain

D

x

⁢

f

⁢

(

x

,

g

⁢

(

x

)

)

+

D

y

⁢

f

⁢

(

x

,

g

⁢

(

x

)

)

⁢

D

⁢

g

⁢

(

x

)

=

0

which gives the desired

formula

for the computation of

D

⁢

g

.
∎

Title

proof of implicit function theorem

Canonical name

ProofOfImplicitFunctionTheorem

Date of creation

2013-03-22 13:31:23

Last modified on

2013-03-22 13:31:23

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

8

Author

paolini (1187)

Entry type

Proof

Classification

msc 26B10

Generated on Fri Feb 9 19:56:53 2018 by

LaTeXML
Cauchy condition for limit of function

Cauchy condition for limit of function

A real function

f

has the limit

lim

x

→

x

0

⁡

f

⁢

(

x

)

if and only if for every positive number

ε

there exists another positive number

δ

⁢

(

ε

)

satisfying

|

f

⁢

(

u

)

-

f

⁢

(

v

)

|

<

ε

when

0

<

|

u

-

x

0

|

<

δ

⁢

(

ε

)

⁢

and

⁢

0

<

|

v

-

x

0

|

<

δ

⁢

(

ε

)

.

References

1

Л. Д. Кудрявцев:

Математический анализ. I том

.  Издательство
‘‘Высшая школа’’. Москва (1970).

Title

Cauchy condition for limit of function

Canonical name

CauchyConditionForLimitOfFunction

Date of creation

2013-03-22 17:42:18

Last modified on

2013-03-22 17:42:18

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26B12

Classification

msc 26A06

Classification

msc 54E35

Synonym

necessary and

sufficient

condition of limit

Related topic

Complete

Related topic

CauchyCriterionForTheExistenceOfALimitOfAFunction

Generated on Fri Feb 9 19:56:59 2018 by

LaTeXML
Clairaut’s theorem

Clairaut’s theorem

Clairaut’s Theorem

.

If

f

:

R

n

→

R

m

is a

function

whose second

partial derivatives

exist and are

continuous

on a set

S

⊆

R

n

, then

∂

2

⁡

f

∂

⁡

x

i

⁢

∂

⁡

x

j

=

∂

2

⁡

f

∂

⁡

x

j

⁢

∂

⁡

x

i

on

S

, where

1

≤

i

,

j

≤

n

.

This

theorem

is commonly referred to as

the

equality of mixed partials

.
It is usually first presented in a vector calculus course,
and is useful in this

context

for proving basic properties of the interrelations of

gradient

,

divergence

, and curl.
For example, if

𝐅

:

ℝ

3

→

ℝ

3

is a function satisfying the

hypothesis

, then

∇

⋅

(

∇

×

𝐅

)

=

0

.
Or, if

f

:

ℝ

3

→

ℝ

is a function satisfying the hypothesis, then

∇

×

∇

⁡

f

=

𝟎

.

Title

Clairaut’s theorem

Canonical name

ClairautsTheorem

Date of creation

2013-03-22 13:53:44

Last modified on

2013-03-22 13:53:44

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

18

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 26B12

Synonym

equality of mixed partials

Generated on Fri Feb 9 19:57:03 2018 by

LaTeXML
converse of Euler’s homogeneous function theorem

converse of Euler’s homogeneous function theorem

Theorem.

If the function

f

of the

real variables

x

1

,

…

,

x

k

satisfies the

identity

x

1

⁢

∂

⁡

f

∂

⁡

x

1

+

…

+

x

k

⁢

∂

⁡

f

∂

⁡

x

k

=

n

⁢

f

,

(1)

then

f

is a

homogeneous function of degree

n

.

Proof.

Let

f

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

:=

φ

⁢

(

t

)

.  Differentiating with respect to

t

we obtain

φ

′

⁢

(

t

)

=

x

1

⁢

f

x

1

′

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

+

…

+

x

k

⁢

f

x

k

′

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

=

1

t

⁢

[

t

⁢

x

1

⁢

f

x

1

′

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

+

…

+

t

⁢

x

k

⁢

f

x

k

′

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

]

,

which by (1) may be written

φ

′

⁢

(

t

)

=

n

t

⁢

f

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

=

n

t

⁢

φ

⁢

(

t

)

.

Accordingly,

φ

′

⁢

(

t

)

φ

⁢

(

t

)

=

n

t

,

which implies the integrated form

ln

⁡

|

φ

⁢

(

t

)

|

=

ln

⁡

t

n

+

ln

⁡

C

for any

positive

t

.  Thus we have

φ

⁢

(

t

)

=

C

⁢

t

n

,  where

C

is

independent

on

t

.  Choosing

t

=

1

we see that

C

=

φ

⁢

(

1

)

,  and therefore

φ

⁢

(

t

)

=

t

n

⁢

φ

⁢

(

1

)

.  This last equation means that

f

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

=

t

n

⁢

f

⁢

(

x

1

,

…

,

x

k

)

saying that

f

is a (positively) homogeneous function of degree

n

.

References

1

Ernst Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset II

.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).

Title

converse of Euler’s homogeneous function theorem

Canonical name

ConverseOfEulersHomogeneousFunctionTheorem

Date of creation

2013-03-22 18:07:56

Last modified on

2013-03-22 18:07:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26B12

Classification

msc 26A06

Classification

msc 15-00

Synonym

converse of Euler’s theorem on homogeneous functions

Related topic

ConverseTheorem

Related topic

ChainRuleSeveralVariables

Related topic

Logarithm

Generated on Fri Feb 9 19:57:06 2018 by

LaTeXML