aarjaneiro/planetmath huggingface dataset:
directional derivative

directional derivative

Let

U

be an open set in

ℝ

n

and

f

:

U

→

ℂ

is a

differentiable
function

. If

u

∈

U

and

v

∈

ℝ

n

, then the

directional derivative

of

f

in the

direction

of

v

is

(

D

v

⁢

f

)

⁢

(

u

)

=

d

d

⁢

s

⁢

f

⁢

(

u

+

s

⁢

v

)

|

s

=

0

.

In other words,

(

D

v

⁢

f

)

⁢

(

u

)

measures how

f

changes in the direction of

v

from

u

.

Alternatively,

(

D

v

⁢

f

)

⁢

(

u

)

=

lim

h

→

0

⁡

f

⁢

(

u

+

h

⁢

v

)

-

f

⁢

(

u

)

h

=

D

⁢

f

⁢

(

u

)

⋅

v

,

where

D

⁢

f

is the

Jacobian matrix

of

f

.

Properties

Let

u

∈

U

.

1.

D

v

⁢

f

is linear in

v

. If

v

,

w

∈

ℝ

n

and

λ

,

μ

∈

ℝ

,
then

D

λ

⁢

v

+

μ

⁢

w

⁢

f

⁢

(

u

)

=

λ

⁢

D

v

⁢

f

⁢

(

u

)

+

μ

⁢

D

w

⁢

f

⁢

(

u

)

.

In particular,

D

0

⁢

f

=

0

.

2.

If

f

is twice

differentiable

and

v

,

w

∈

ℝ

n

, then

D

v

⁢

D

w

⁢

f

⁢

(

u

)

=

∂

2

∂

⁡

s

⁢

∂

⁡

t

⁢

f

⁢

(

u

+

s

⁢

v

+

t

⁢

w

)

|

s

=

0

,

=

v

T

⋅

Hess

⁡

f

⁢

(

u

)

⋅

w

,

where

Hess

is the

Hessian matrix

of

f

.

Example

For example, if

f

⁢

(

x

y

z

)

=

x

2

+

3

⁢

y

2

⁢

z

, and we wanted to find the derivative at the point

𝐚

=

(

1

2

3

)

in the direction

v

→

=

[

1

1

1

]

, our equation would be

lim

h

→

0

⁡

1

h

⁢

(

(

1

+

h

)

2

+

3

⁢

(

2

+

h

)

2

⁢

(

3

+

h

)

-

37

)

=

lim

h

→

0

⁡

1

h

⁢

(

3

⁢

h

3

+

37

⁢

h

2

+

50

⁢

h

)

=

lim

h

→

0

⁡

3

⁢

h

2

+

37

⁢

h

+

50

=

50

Title

directional derivative

Canonical

name

DirectionalDerivative

Date of creation

2013-03-22 11:58:37

Last modified on

2013-03-22 11:58:37

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

15

Author

matte (1858)

Entry type

Definition

Classification

msc 26B12

Classification

msc 26B10

Synonym

derivative with respect to a vector

Synonym

partial derivative

with respect to a vector

Related topic

PartialDerivative

Related topic

Derivative

Related topic

DerivativeNotation

Related topic

JacobianMatrix

Related topic

Gradient

Related topic

FixedPointsOfNormalFunctions

Related topic

HessianMatrix

Generated on Fri Feb 9 19:57:18 2018 by

LaTeXML
directional derivative, derivation of

directional derivative, derivation of

Let

f

⁢

(

x

,

y

)

be a

function

where

x

=

x

⁢

(

t

)

and

y

=

y

⁢

(

t

)

. Let

r

→

=

a

⁢

𝐢

^

+

b

⁢

𝐣

^

be the vector in the desired

direction

. The line through this vector is given parametrically by:

x

=

x

0

+

a

⁢

t

;

y

=

y

0

+

b

⁢

t

The

derivative

of

f

with respect to

t

is as follows:

∂

⁡

f

∂

⁡

t

=

∂

⁡

f

∂

⁡

x

⁢

d

⁢

x

d

⁢

t

+

∂

⁡

f

∂

⁡

y

⁢

d

⁢

y

d

⁢

t

But from the equation of the line, we know that

d

⁢

x

d

⁢

t

=

a

and

d

⁢

y

d

⁢

t

=

b

so the derivative becomes:

∂

⁡

f

∂

⁡

t

=

∂

⁡

f

∂

⁡

x

⁢

a

+

∂

⁡

f

∂

⁡

y

⁢

b

=

∇

⁡

f

⋅

r

→

Title

directional derivative

,

derivation

of

Canonical name

DirectionalDerivativeDerivationOf

Date of creation

2013-03-22 15:25:22

Last modified on

2013-03-22 15:25:22

Owner

apmc (9183)

Last modified by

apmc (9183)

Numerical id

7

Author

apmc (9183)

Entry type

Derivation

Classification

msc 26B12

Classification

msc 26B10

Generated on Fri Feb 9 19:57:20 2018 by

LaTeXML
divergence

divergence

Basic Definition.

Let

x

,

y

,

z

be a system of

Cartesian coordinates

on

3

-dimensional

Euclidean space

, and let

𝐢

,

𝐣

,

𝐤

be the
corresponding basis of

unit vectors

. The

divergence

of a

continuously
differentiable

vector field

𝐅

=

F

1

⁢

𝐢

+

F

2

⁢

𝐣

+

F

3

⁢

𝐤

,

is defined to be the

function

div

⁡

𝐅

=

∂

⁡

F

1

∂

⁡

x

+

∂

⁡

F

2

∂

⁡

y

+

∂

⁡

F

3

∂

⁡

z

.

Another common notation for the divergence
is

∇

⋅

𝐅

(see

gradient

), a convenient mnemonic.

Physical interpretation.

In physical , the divergence of a vector field
is the extent to which the vector field flow behaves like a source or
a sink at a given point. Indeed, an alternative, but

logically
equivalent

definition, gives the divergence as the

derivative

of the
net flow of the vector field across the

surface

of a small sphere
relative to the

surface area

of the sphere. To wit,

(

div

⁡

𝐅

)

⁢

(

p

)

=

lim

r

→

0

⁡

∫

S

(

𝐅

⋅

𝐍

)

⁢

𝑑

S

/

(

4

⁢

π

⁢

r

2

)

,

where

S

denotes the sphere of radius

r

about a point

p

∈

ℝ

3

, and the

integral

is a

surface integral

taken with
respect to

𝐍

, the normal to that sphere.

The non-infinitesimal

interpretation

of divergence is given by Gauss’s

Theorem

. This theorem is a conservation law, stating that the volume total of
all sinks and sources, i.e. the volume integral of the divergence, is
equal to the net flow across the volume’s

boundary

. In symbols,

∫

V

div

⁡

𝐅

⁢

d

⁢

V

=

∫

S

(

𝐅

⋅

𝐍

)

⁢

𝑑

S

,

where

V

⊂

ℝ

3

is a

compact

region with a smooth boundary, and

S

=

∂

⁡

V

is that boundary

oriented

by outward-pointing normals.
We note that Gauss’s theorem follows from the more general Stokes’
Theorem, which itself generalizes the

fundamental theorem of calculus

.

In light of the physical interpretation, a vector field with

constant

zero divergence is called

incompressible

– in this case, no
flow can occur across any
surface.

General definition.

The notion of divergence has meaning in the more general setting of
Riemannian geometry. To that end, let

𝐕

be a vector field on a

Riemannian manifold

. The

covariant derivative

of

𝐕

is a type

(

1

,

1

)

tensor field

. We define the

divergence

of

𝐕

to be the
trace of that field. In terms of

coordinates

(see tensor and

Einstein
summation convention

), we have

div

𝐕

=

V

i

.

;

i

Title

divergence

Canonical name

Divergence

Date of creation

2013-03-22 12:55:08

Last modified on

2013-03-22 12:55:08

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

11

Author

rmilson (146)

Entry type

Definition

Classification

msc 26B12

Related topic

SourcesAndSinksOfVectorField

Defines

incompressible

Defines

divergence theorem

Defines

Gauss’s theorem

Generated on Fri Feb 9 19:57:22 2018 by

LaTeXML
Euler’s theorem on homogeneous functions

Euler’s theorem on homogeneous functions

Theorem 1

(Euler).

Let

f

⁢

(

x

1

,

…

,

x

k

)

be a smooth

homogeneous function of degree

n

. That is,

f

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

=

t

n

⁢

f

⁢

(

x

1

,

…

,

x

k

)

.

(*)

Then the following

identity

holds

x

1

⁢

∂

⁡

f

∂

⁡

x

1

+

⋯

+

x

k

⁢

∂

⁡

f

∂

⁡

x

k

=

n

⁢

f

.

Proof.

By homogeneity, the

relation

(

(*)

‣

1

) holds for all

t

. Taking the t-derivative of both sides, we establish that the following identity holds for all

t

:

x

1

⁢

∂

⁡

f

∂

⁡

x

1

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

+

⋯

+

x

k

⁢

∂

⁡

f

∂

⁡

x

k

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

k

)

=

n

⁢

t

n

-

1

⁢

f

⁢

(

x

1

,

…

,

x

k

)

.

To obtain the result of the

theorem

, it suffices to set

t

=

1

in the previous

formula

.
∎

Sometimes the

differential operator

x

1

⁢

∂

∂

⁡

x

1

+

⋯

+

x

k

⁢

∂

∂

⁡

x

k

is called the

Euler operator

. An

equivalent

way to state the theorem is to say that homogeneous functions are eigenfunctions of the Euler operator, with the degree of homogeneity as the

eigenvalue

.

Title

Euler’s theorem on homogeneous functions

Canonical name

EulersTheoremOnHomogeneousFunctions

Date of creation

2013-03-22 15:18:58

Last modified on

2013-03-22 15:18:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 26B12

Classification

msc 26A06

Classification

msc 15-00

Defines

Euler operator

Generated on Fri Feb 9 19:57:25 2018 by

LaTeXML
example of tests for local extrema in Lagrange multiplier method

example of tests for local extrema in Lagrange multiplier method

Let

n

∈

ℕ

+

and

c

∈

ℝ

. We want to find the local

extrema

of the

function

f

:

ℝ

n

→

ℝ

,

x

↦

∑

1

≤

i

<

j

≤

n

x

i

⁢

x

j

subject to the condition

g

=

0

, where

g

:

ℝ

n

→

ℝ

,

x

↦

∑

1

≤

i

≤

n

x

i

-

c

.

The first and

second order

partial derivatives

are for all

i

,

j

∈

{

1

,

…

,

n

}

∂

i

⁡

f

⁢

(

x

)

=

∑

k

≠

i

x

k

,

∂

i

⁡

g

⁢

(

x

)

=

1

,

∂

i

⁡

∂

j

⁡

f

⁢

(

x

)

=

1

-

δ

i

,

j

,

∂

i

⁡

∂

j

⁡

g

⁢

(

x

)

=

0

,

where

δ

i

,

j

is the Kroenecker-delta. Thus the necessary condition

f

′

⁢

(

x

)

=

λ

⁢

g

′

⁢

(

x

)

together with

g

⁢

(

x

)

=

0

gives the system of equations

∑

j

≠

i

x

j

=

λ

,

i

∈

{

1

,

…

,

n

}

,

∑

1

≤

j

≤

n

x

j

=

c

.

By summing the first

n

equations and then substituting in the last we get

(

n

-

1

)

⁢

c

=

n

⁢

λ

,

x

i

=

∑

1

≤

j

≤

n

x

j

-

∑

j

≠

i

x

j

=

c

-

λ

=

c

n

,

i

∈

{

1

,

…

,

n

}

.

Thus there is only one point, where

local extremum

is possible. We apply the test in the parent entry to the matrix

D

2

⁢

(

f

-

λ

⁢

g

)

⁢

(

x

)

=

[

1

-

δ

i

,

j

]

i

,

j

=

1

n

=

n

⁢

P

-

I

,

where

P

is the matrix containing

1

/

n

in all entries, and

I

is the

identity matrix

.

P

is a rank one

projection

. Therefore the

second derivative

has

spectrum

σ

⁢

(

n

⁢

P

-

I

)

=

{

n

-

1

,

-

1

}

, where

-

1

has

multiplicity

n

-

1

, and

n

-

1

has multiplicity

1

. Thus the second derivative of

f

-

λ

⁢

g

is indefinit, so it has no local extrema. However the

nullspace

of

g

′

⁢

(

x

)

is precisely the nullspace of

P

, thus the second derivative is strictly negative on the

tangent space

T

x

⁢

(

M

)

, so the vector

(

c

/

n

,

…

,

c

/

n

)

is a

local maximum

of

f

subject to

g

=

0

.

Title

example of

tests for local extrema in Lagrange multiplier method

Canonical name

ExampleOfTestsForLocalExtremaInLagrangeMultiplierMethod

Date of creation

2013-03-22 19:12:19

Last modified on

2013-03-22 19:12:19

Owner

scineram (4030)

Last modified by

scineram (4030)

Numerical id

9

Author

scineram (4030)

Entry type

Example

Classification

msc 26B12

Classification

msc 49K35

Classification

msc 49-00

Generated on Fri Feb 9 19:57:28 2018 by

LaTeXML
example of vector potential

example of vector potential

If the

solenoidal

vector

U

→

=

U

→

⁢

(

x

,

y

,

z

)

is a

homogeneous function of degree

λ

(

≠

-

2

),  then it has the

vector potential

A

→

=

1

λ

+

2

⁢

U

→

×

r

→

,

(1)

where

r

→

=

x

⁢

i

→

+

y

⁢

j

→

+

z

⁢

k

→

is the

position vector

.

Proof.

Using the entry

nabla acting on products

, we first may write

∇

×

(

1

λ

+

2

⁢

U

→

×

r

→

)

=

1

λ

+

2

⁢

[

(

r

→

⋅

∇

)

⁢

U

→

-

(

U

→

⋅

∇

)

⁢

r

→

-

(

∇

⋅

U

→

)

⁢

r

→

+

(

∇

⋅

r

→

)

⁢

U

→

]

.

In the brackets the first product is, according to Euler’s theorem on

homogeneous functions

, equal to

λ

⁢

U

→

.  The second product can be written as

U

x

⁢

∂

⁡

r

→

∂

⁡

x

+

U

y

⁢

∂

⁡

r

→

∂

⁡

y

+

U

z

⁢

∂

⁡

r

→

∂

⁡

z

, which is

U

x

⁢

i

→

+

U

y

⁢

j

→

+

U

z

⁢

k

→

, i.e.

U

→

.  The third product is, due to the sodenoidalness, equal to

0

⁢

r

→

=

0

→

.  The last product equals to

3

⁢

U

→

(see the first formula (

http://planetmath.org/PositionVector

) for position vector).  Thus we get the result

∇

×

(

1

λ

+

2

⁢

U

→

×

r

→

)

=

1

λ

+

2

⁢

[

λ

⁢

U

→

-

U

→

-

0

→

+

3

⁢

U

→

]

=

U

→

.

This means that

U

→

has the vector potential (1).

Title

example of vector potential

Canonical name

ExampleOfVectorPotential

Date of creation

2013-03-22 15:42:56

Last modified on

2013-03-22 15:42:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Example

Classification

msc 26B12

Generated on Fri Feb 9 19:57:30 2018 by

LaTeXML
extremum

extremum

Extrema

are minima and maxima. The forms of these are

extremum

,

minimum

, and

maximum

.

Extrema may be “global” or “local”. A

global minimum

of a

function

f

is the lowest value that

f

ever achieves. If you imagine the function as a surface, then a global minimum is the lowest point on that surface. Formally, it is said that

f

:

U

→

V

has a

global minimum

at

x

if

∀

u

∈

U

,

f

⁢

(

x

)

≤

f

⁢

(

u

)

.

A

local minimum

of a function

f

is a point

x

which has less value than all points “next to” it. If you imagine the function as a surface, then a local minimum is the of a “valley” or “bowl” in the surface somewhere. Formally, it is said that

f

:

U

→

V

has a

local minimum

at

x

if

∃

a

neighborhood

N

of

x

such that

∀

y

∈

N

,

f

⁢

(

x

)

≤

f

⁢

(

y

)

.

If you flip the

≤

signs above to

≥

, you get the

definitions

of global and local maxima.

A “strict local minima” or “strict local maxima” means that nearby points are

strictly

less than or strictly

greater

than the

critical point

, rather than

≤

or

≥

. For instance, a strict local minima at

x

has a neighborhood

N

such that

∀

y

∈

N

,

(

f

⁢

(

x

)

<

f

⁢

(

y

)

⁢

or

⁢

y

=

x

)

.

A

saddle point

is a critical point which is not a

local extremum

.

A related concept is

plateau

.

Finding minima or maxima is an important task which is part of the of optimization.
This task is also important in Physics where the minima correspond to equilibria.

Title

extremum

Canonical

name

Extremum

Date of creation

2013-03-22 12:59:44

Last modified on

2013-03-22 12:59:44

Owner

bshanks (153)

Last modified by

bshanks (153)

Numerical id

20

Author

bshanks (153)

Entry type

Definition

Classification

msc 26B12

Synonym

extrema

Related topic

Plateau

Related topic

RelationsBetweenHessianMatrixAndLocalExtrema

Related topic

LeastAndGreatestValueOfFunction

Related topic

SpeediestInclinedPlane

Defines

global minima

Defines

global maxima

Defines

local minima

Defines

local maxima

Defines

global minimum

Defines

global maximum

Defines

local minimum

Defines

local maximum

Defines

strict local minima

Defines

strict local maxima

Defines

strict local minimum

Defines

strict local maximum

Defines

saddle point

Generated on Fri Feb 9 19:57:36 2018 by

LaTeXML
extremum points of function of several variables

extremum points of function of several variables

The points where a

function

of two or more real variables attains its

extremum

values are found in the set containing the points where all

first order

partial derivatives vanish

, the points where one or more of those

derivatives

does not exist, and the points where the function itself is

discontinuous

.

Example 1.

The function

f

⁢

(

x

,

y

)

=

x

2

+

y

2

+

1

from

ℝ

2

to

ℝ

has a (global)

minimum

point

(

0

,

0

)

,  where its

partial derivatives

∂

⁡

f

∂

⁡

x

=

2

⁢

x

and

∂

⁡

f

∂

⁡

y

=

2

⁢

y

both equal to zero.

Example 2.

Also the function

g

⁢

(

x

,

y

)

=

x

2

+

y

2

from

ℝ

2

to

ℝ

has a (global) minimum in

(

0

,

0

)

,  where neither of its partial derivatives

∂

⁡

g

∂

⁡

x

and

∂

⁡

g

∂

⁡

y

exist.

Example 3.

The function

f

⁢

(

x

,

y

,

z

)

=

x

2

+

y

2

+

z

2

from

ℝ

3

to

ℝ

has an

absolute minimum

point

(

0

,

0

,

0

)

,  since

∇

⁡

f

=

2

⁢

x

⁢

𝐢

+

2

⁢

y

⁢

𝐣

+

2

⁢

z

⁢

𝐤

=

𝟎

⟹

x

=

y

=

z

=

0

,

∂

2

⁡

f

∂

⁡

x

2

=

∂

2

⁡

f

∂

⁡

y

2

=

∂

2

⁡

f

∂

⁡

z

2

=

2

>

0

, and

f

⁢

(

0

,

0

,

0

)

≤

f

⁢

(

x

,

y

,

z

)

for all

(

x

,

y

,

z

)

∈

ℝ

3

.

Title

extremum points of function of several variables

Canonical name

ExtremumPointsOfFunctionOfSeveralVariables

Date of creation

2013-03-22 17:23:57

Last modified on

2013-03-22 17:23:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26B12

Related topic

VanishingOfGradientInDomain

Generated on Fri Feb 9 19:57:38 2018 by

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Fubini’s theorem

Fubini’s theorem

Fubini’s

theorem

Let

I

⊂

ℝ

N

and

J

⊂

ℝ

M

be compact

intervals

, and let

f

:

I

×

J

→

ℝ

K

be a

Riemann integrable

function such that, for each

x

∈

I

the

integral

F

⁢

(

x

)

:=

∫

J

f

⁢

(

x

,

y

)

⁢

𝑑

μ

J

⁢

(

y

)

exists. Then

F

:

I

→

ℝ

K

is Riemann integrable, and

∫

I

F

=

∫

I

×

J

f

.

This theorem effectively states that, given a function of

N

variables

, you may

integrate

it one variable at a time, and that the order of integration does not affect the result.

Example

Let

I

:=

[

0

,

π

/

2

]

×

[

0

,

π

/

2

]

, and let

f

:

I

→

ℝ

,

x

↦

sin

⁡

(

x

)

⁢

cos

⁡

(

y

)

be a function.
Then

∫

I

f

=

∬

[

0

,

π

/

2

]

×

[

0

,

π

/

2

]

sin

⁡

(

x

)

⁢

cos

⁡

(

y

)

=

∫

0

π

/

2

(

∫

0

π

/

2

sin

⁡

(

x

)

⁢

cos

⁡

(

y

)

⁢

𝑑

y

)

⁢

𝑑

x

=

∫

0

π

/

2

sin

(

x

)

(

1

-

0

)

d

x

=

(

0

-

-

1

)

=

1

.

Note that it is often simpler (and no less correct) to write

∫⋯∫

I

f

as

∫

I

f

.

Title

Fubini’s theorem

Canonical

name

FubinisTheorem

Date of creation

2013-03-22 13:39:13

Last modified on

2013-03-22 13:39:13

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 26B12

Related topic

TonellisTheorem

Related topic

FubinisTheoremForTheLebesgueIntegral

Related topic

IntegrationUnderIntegralSign

Generated on Fri Feb 9 19:57:40 2018 by

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Generalised N-dimensional Riemann Sum

Generalised N-dimensional Riemann Sum

Let

I

=

[

a

1

,

b

1

]

×

⋯

×

[

a

N

,

b

N

]

be an

N

-cell in

ℝ

N

. For each

j

=

1

,

…

,

N

, let

a

j

=

t

j

,

0

<

…

<

t

j

,

N

=

b

j

be a

partition

P

j

of

[

a

j

,

b

j

]

. We define a partition

P

of

I

as

P

:=

P

1

×

⋯

×

P

N

Each partition

P

of

I

generates a

subdivision

of

I

(denoted by

(

I

ν

)

ν

) of the form

I

ν

=

[

t

1

,

j

,

t

1

,

j

+

1

]

×

⋯

×

[

t

N

,

k

,

t

N

,

k

+

1

]

Let

f

:

U

→

ℝ

M

be such that

I

⊂

U

, and let

(

I

ν

)

ν

be the corresponding subdivision of a partition

P

of

I

. For each

ν

, choose

x

ν

∈

I

ν

. Define

S

⁢

(

f

,

P

)

:=

∑

ν

f

⁢

(

x

ν

)

⁢

μ

⁢

(

I

⁢

ν

)

As the

Riemann sum

of

f

corresponding to the partition

P

.

A partition

Q

of

I

is called a

refinement

of

P

if

P

⊂

Q

.

Title

Generalised N-dimensional Riemann Sum

Canonical name

GeneralisedNdimensionalRiemannSum

Date of creation

2013-03-22 13:37:40

Last modified on

2013-03-22 13:37:40

Owner

vernondalhart (2191)

Last modified by

vernondalhart (2191)

Numerical id

4

Author

vernondalhart (2191)

Entry type

Definition

Classification

msc 26B12

Generated on Fri Feb 9 19:57:42 2018 by

LaTeXML
Generalized N-dimensional Riemann Integral

Generalized N-dimensional Riemann Integral

Let

I

=

[

a

1

,

b

1

]

×

⋯

×

[

a

N

,

b

N

]

⊂

ℝ

N

be a

compact

interval

, and let

f

:

I

→

ℝ

M

be a function. Let

ϵ

>

0

. If there exists a

y

∈

ℝ

M

and a

partition

P

ϵ

of

I

such that for each

refinement

P

of

P

ϵ

(and corresponding

Riemann Sum

S

⁢

(

f

,

P

)

),

∥

S

⁢

(

f

,

P

)

-

y

∥

<

ϵ

Then we say that

f

is

Riemann integrable

over

I

, that

y

is the Riemann integral of

f

over

I

, and we write

∫

I

f

:=

∫

I

f

⁢

𝑑

μ

:=

y

Note also that it is possible to extend this definition to more arbitrary sets; for any bounded set

D

, one can find a compact interval

I

such that

D

⊂

I

, and define a function

f

~

:

I

→

ℝ

M

x

↦

{

f

⁢

(

x

)

,

x

∈

D

0

,

x

∉

D

in which case we define

∫

D

f

:=

∫

I

f

~

Title

Generalized N-dimensional Riemann Integral

Canonical name

GeneralizedNdimensionalRiemannIntegral

Date of creation

2013-03-22 13:37:43

Last modified on

2013-03-22 13:37:43

Owner

vernondalhart (2191)

Last modified by

vernondalhart (2191)

Numerical id

6

Author

vernondalhart (2191)

Entry type

Definition

Classification

msc 26B12

Generated on Fri Feb 9 19:57:44 2018 by

LaTeXML
gradient in curvilinear coordinates

gradient in curvilinear coordinates

We give the formulas for the

gradient

expressed in various

curvilinear coordinate

systems.
We also show the

metric tensors

g

i

⁢

j

so
that the reader may verify the results
by working from the basic formulas for the gradient.

Contents:

1

Cylindrical coordinate system

2

Polar coordinate system

3

Spherical coordinate system

1

Cylindrical coordinate system

In the
cylindrical system of

coordinates

(

r

,

θ

,

z

)

we have

g

i

⁢

j

=

(

1

0

0

0

r

2

0

0

0

1

)

.

So that

∇

⁡

f

=

∂

⁡

f

∂

⁡

r

⁢

𝐞

r

+

1

r

⁢

∂

⁡

f

∂

⁡

θ

⁢

𝐞

θ

+

∂

⁡

f

∂

⁡

z

⁢

𝐤

,

where

𝐞

r

=

∂

∂

⁡

r

=

x

r

⁢

𝐢

+

y

r

⁢

𝐣

𝐞

θ

=

1

r

⁢

∂

∂

⁡

θ

=

-

y

r

⁢

𝐢

+

x

r

⁢

𝐣

are the unit vectors in the direction of increase of

r

and

θ

. Of course,

𝐢

,

𝐣

,

𝐤

denote the unit vectors along the positive

x

,

y

,

z

axes respectively.

The notations

∂

/

∂

⁡

r

,

∂

/

∂

⁡

θ

, etc.,
denote the

tangent vectors

corresponding to

infinitesimal

changes in

r

,

θ

, etc. respectively.
Concretely, in terms of

Cartesian coordinates

,

∂

/

∂

⁡

r

is the vector

𝐢

⁢

∂

⁡

x

/

∂

⁡

r

+

𝐣

⁢

∂

⁡

y

/

∂

⁡

r

+

𝐤

⁢

∂

⁡

z

/

∂

⁡

r

.
And similarly for the other variables. (There is a deep reason for using the seemingly strange notation:
see

Leibniz notation for vector fields

for details.)

2

Polar coordinate system

This is just the special case of the

cylindrical coordinate system

where we chop off the

z

coordinate.
Thus

∇

⁡

f

=

∂

⁡

f

∂

⁡

r

⁢

𝐞

r

+

1

r

⁢

∂

⁡

f

∂

⁡

θ

⁢

𝐞

θ

.

3

Spherical coordinate system

To stave off confusion, note that this is the “mathematicians’ ” convention for the

spherical coordinate
system

(

ρ

,

ϕ

,

θ

)

.
That is,

ϕ

is the co-latitude angle, and

θ

is the longitudinal angle.

g

i

⁢

j

=

(

1

0

0

0

ρ

2

0

0

0

ρ

2

⁢

sin

2

⁡

ϕ

)

.

∇

⁡

f

=

∂

⁡

f

∂

⁡

ρ

⁢

𝐞

ρ

+

1

ρ

⁢

∂

⁡

f

∂

⁡

ϕ

⁢

𝐞

ϕ

+

1

ρ

⁢

sin

⁡

ϕ

⁢

∂

⁡

f

∂

⁡

θ

⁢

𝐞

θ

,

where

𝐞

ρ

=

∂

∂

⁡

ρ

=

x

ρ

⁢

𝐢

+

y

ρ

⁢

𝐣

+

z

ρ

⁢

𝐤

𝐞

ϕ

=

1

ρ

⁢

∂

∂

⁡

ϕ

=

z

⁢

x

r

⁢

ρ

⁢

𝐢

+

z

⁢

y

r

⁢

ρ

⁢

𝐣

-

r

ρ

⁢

𝐤

𝐞

θ

=

1

ρ

⁢

sin

⁡

θ

⁢

∂

∂

⁡

θ

=

-

y

r

⁢

𝐢

+

x

r

⁢

𝐣

are the unit vectors in the direction of increase of

ρ

,

ϕ

,

θ

, respectively.

Title

gradient in curvilinear coordinates

Canonical name

GradientInCurvilinearCoordinates

Date of creation

2013-03-22 15:27:32

Last modified on

2013-03-22 15:27:32

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

5

Author

stevecheng (10074)

Entry type

Result

Classification

msc 26B12

Classification

msc 26B10

Related topic

gradient

Related topic

Gradient

Generated on Fri Feb 9 19:57:48 2018 by

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