aarjaneiro/planetmath huggingface dataset:
gradient theorem

gradient theorem

If

u

=

u

⁢

(

x

,

y

,

z

)

is

continuously differentiable function

in a simply connected domain (

http://planetmath.org/Domain2

)

D

of

ℝ

3

and

P

=

(

x

0

,

y

0

,

z

0

)

and

Q

=

(

x

1

,

y

1

,

z

1

)

lie in this domain, then

∫

P

Q

∇

⁡

u

⋅

d

⁢

s

→

=

u

⁢

(

x

1

,

y

1

,

z

1

)

-

u

⁢

(

x

0

,

y

0

,

z

0

)

(1)

where the

line integral

of the left hand side is taken along an arbitrary path in

D

.

The equation (1) is illustrated by the fact that

∇

⁡

u

⋅

d

⁢

s

→

=

∂

⁡

u

∂

⁡

x

⁢

d

⁢

x

+

∂

⁡

u

∂

⁡

y

⁢

d

⁢

y

+

∂

⁡

u

∂

⁡

z

⁢

d

⁢

z

is the

total differential

of

u

, and thus

∫

P

Q

∇

⁡

u

⋅

d

⁢

s

→

=

∫

P

Q

𝑑

u

.

Title

gradient theorem

Canonical name

GradientTheorem

Date of creation

2013-03-22 19:11:16

Last modified on

2013-03-22 19:11:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26B12

Synonym

fundamental theorem of line integrals

Related topic

LaminarField

Related topic

Gradient

Generated on Fri Feb 9 19:57:50 2018 by

LaTeXML
Helmholtz decomposition

Helmholtz decomposition

The

Helmholtz theorem

states that any vector

𝐅

may be decomposed into an

irrotational

(curl-free) and a

solenoidal

(divergence-free) part under certain conditions (given below). More precisely, it may be written in the form:

𝐅

=

-

∇

⁡

φ

+

∇

×

𝐀

(1)

where

φ

is a

scalar potential

and

𝐀

is a

vector potential

. By the definitions of scalar and vector potentials it follows that the first term on the right-hand side is irrotational and the second is solenoidal. The general conditions for this to be true are:

1.

The

divergence

of

𝐅

must

vanish at infinity

.

2.

The curl of

𝐅

must also vanish at infinity.

Title

Helmholtz decomposition

Canonical name

HelmholtzDecomposition

Date of creation

2013-03-22 17:59:40

Last modified on

2013-03-22 17:59:40

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

5

Author

invisiblerhino (19637)

Entry type

Definition

Classification

msc 26B12

Synonym

fundamental theorem of vector calcululs

Generated on Fri Feb 9 19:57:51 2018 by

LaTeXML
Helmholtz equation

Helmholtz equation

The

Helmholtz equation

is a

partial differential equation

which, in scalar form is

∇

2

⁡

f

+

k

2

⁢

f

=

0

,

or in vector form is

∇

2

⁡

𝐀

+

k

2

⁢

𝐀

=

0

,

where

∇

2

is the

Laplacian

.
The solutions of this equation represent the solution of the

wave equation

, which is of great interest in physics.

Consider a wave equation

∂

2

⁡

ψ

∂

⁡

t

2

=

c

2

⁢

∇

2

⁡

ψ

with wave speed

c

. If we look for time harmonic standing waves of frequency

ω

,

ψ

⁢

(

𝐱

,

t

)

=

e

-

j

⁢

ω

⁢

t

⁢

ϕ

⁢

(

𝐱

)

we find that

ϕ

⁢

(

x

)

satisfies the Helmholtz equation:

(

∇

2

+

k

2

)

⁢

ϕ

=

0

where

k

=

ω

/

c

is the wave number.

Usually the Helmholtz equation is solved by the

separation of variables

method, in Cartesian, spherical or

cylindrical coordinates

.

Title

Helmholtz equation

Canonical name

HelmholtzEquation

Date of creation

2013-03-22 13:09:09

Last modified on

2013-03-22 13:09:09

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 26B12

Classification

msc 35-00

Synonym

Helmholtz differential equation

Synonym

reduced wave equation

Related topic

WaveEquation

Related topic

PoissonsEquation

Generated on Fri Feb 9 19:57:53 2018 by

LaTeXML
Hessian form

Hessian form

Given a

smooth manifold

M

and

f

:

M

→

ℝ

being in

C

2

⁢

(

M

)

, if

x

is a

critical point

of

f

, that is

d

⁢

f

=

0

at

x

, then we can define a symmetric

2

-form

H

⁢

(

u

x

,

v

x

)

=

u

⁢

(

v

⁢

(

f

)

)

=

v

⁢

(

u

⁢

(

f

)

)

,

where

H

∈

T

*

⁣

⊗

2

and

u

and

v

are any vector fields that take the values

u

x

and

v

x

, respectively, at point

x

. Equality of the two defining

expressions

follows from the fact that

x

is a critical point of

f

, because then

[

u

,

v

]

⁢

(

f

)

=

d

⁢

f

⁢

(

[

u

,

v

]

)

=

0

, where

[

u

,

v

]

denotes the

Lie bracket

of the two vector fields. The form

H

is called the

Hessian form

.

In local coordinates, the Hessian form is given by

H

=

∂

2

⁡

f

∂

⁡

x

i

⁢

∂

⁡

x

j

⁢

d

⁢

x

i

⊗

d

⁢

x

j

.

Its components are those of the

Hessian matrix

in the same coordinates. The advantage of the above formulation is coordinate independence. However, the price is that the Hessian form is only defined at critical points. It does not define a

tensor field

as one would naïvely expect.

Using the Hessian form, it is possible to analyze the critical points of

f

(determine whether they are

local minima

, maxima, or

saddle points

) in a coordinate independent way.

Title

Hessian form

Canonical

name

HessianForm

Date of creation

2013-03-22 15:00:03

Last modified on

2013-03-22 15:00:03

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 26B12

Related topic

HessianMatrix

Related topic

RelationsBetweenHessianMatrixAndLocalExtrema

Related topic

TestsForLocalExtremaForLagrangeMultiplierMethod

Generated on Fri Feb 9 19:57:55 2018 by

LaTeXML
Hessian matrix

Hessian matrix

Let

x

∈

ℝ

n

and let

f

:

ℝ

n

→

ℝ

be a real-valued function having 2nd-order

partial derivatives

in an open set

U

containing

x

. The

Hessian matrix

of

f

is the matrix of second partial derivatives evaluated at

x

:

𝐇

⁢

(

x

)

:=

[

∂

2

⁡

f

∂

⁡

x

1

2

∂

2

⁡

f

∂

⁡

x

1

⁢

∂

⁡

x

2

…

∂

2

⁡

f

∂

⁡

x

1

⁢

∂

⁡

x

n

∂

2

⁡

f

∂

⁡

x

2

⁢

∂

⁡

x

1

∂

2

⁡

f

∂

⁡

x

2

2

…

∂

2

⁡

f

∂

⁡

x

2

⁢

∂

⁡

x

n

⋮

⋮

⋱

⋮

∂

2

⁡

f

∂

⁡

x

n

⁢

∂

⁡

x

1

∂

2

⁡

f

∂

⁡

x

n

⁢

∂

⁡

x

2

…

∂

2

⁡

f

∂

⁡

x

n

2

]

.

(1)

If

f

is in

C

2

⁢

(

U

)

,

𝐇

⁢

(

x

)

is

symmetric

(

http://planetmath.org/SymmetricMatrix

) because of the

equality of mixed partials

. Note that

𝐇

⁢

(

x

)

=

𝐉

⁢

(

∇

⁡

f

)

, the

Jacobian

of the

gradient

of

f

.

Given a vector

𝒗

∈

ℝ

n

, the

Hessian

of

f

at

𝒗

is:

𝐇

⁢

(

x

)

⁢

(

𝒗

)

:=

1

2

⁢

𝒗

⁢

𝐇

⁢

(

x

)

⁢

𝒗

T

.

(2)

Here we view

𝒗

as a

1

by

n

matrix so that

𝒗

T

is the

transpose

of

𝒗

.

Remark

. The Hessian of

f

at

𝒗

is a

quadratic form

, since

𝐇

⁢

(

x

)

⁢

(

r

⁢

𝒗

)

=

r

2

⁢

𝐇

⁢

(

x

)

⁢

(

𝒗

)

for any

r

∈

ℝ

.

If

f

is further assumed to be in

C

2

⁢

(

U

)

, and

x

is a

critical point

of

f

such that

𝐇

⁢

(

x

)

is

positive definite

(

http://planetmath.org/PositiveDefinite

), then

x

is a

strict local minimum

of

f

.

This is not difficult to show. Since

𝐇

⁢

(

x

)

is positive definite (

http://planetmath.org/PositiveDefinite

), the

Rayleigh-Ritz theorem

shows that there is a

c

>

0

such that for all

h

∈

ℝ

n

,

h

T

⁢

𝐇

⁢

(

x

)

⁢

h

≥

2

⁢

c

⁢

∥

h

∥

2

. Thus by
Taylor’s theorem (

http://planetmath.org/TaylorPolynomialsInBanachSpaces

) ( form)

f

⁢

(

x

+

h

)

=

f

⁢

(

x

)

+

1

2

⁢

h

T

⁢

𝐇

⁢

(

x

)

⁢

h

+

o

⁢

(

∥

h

∥

2

)

≥

c

⁢

∥

h

∥

2

+

o

⁢

(

∥

h

∥

2

)

.

For small

∥

h

∥

the first on the the second, so that both sides are

positive

for small

∥

h

∥

.

Title

Hessian matrix

Canonical name

HessianMatrix

Date of creation

2013-03-22 12:59:41

Last modified on

2013-03-22 12:59:41

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

31

Author

cvalente (11260)

Entry type

Definition

Classification

msc 26B12

Related topic

Gradient

Related topic

PartialDerivative

Related topic

SymmetricMatrix

Related topic

ComplexHessianMatrix

Related topic

HessianForm

Related topic

DirectionalDerivative

Defines

Hessian

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

LaTeXML
irrotational field

irrotational field

Suppose

Ω

is an open set in

ℝ

3

, and

𝐕

is a

vector field

with

differentiable

real (or possibly complex) valued

component

functions.
If

∇

×

𝐕

=

0

, then

𝐕

is called an

irrotional vector field

,
or

curl free field

.

If

𝐔

and

𝐕

are

irrotational

, then

𝐔

×

𝐕

is

solenoidal

.

Title

irrotational field

Canonical

name

IrrotationalField

Date of creation

2013-03-22 13:09:05

Last modified on

2013-03-22 13:09:05

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 26B12

Synonym

irrotational vector field

Synonym

curl free field

Synonym

curl-free vector field

Related topic

Curl

Related topic

LaminarField

Generated on Fri Feb 9 19:57:59 2018 by

LaTeXML
iterated limit in ℝ^2

iterated limit in

ℝ

2

Let

f

be a

function

from a subset

S

of

ℝ

2

to

ℝ

and

(

a

,

b

)

an

accumulation point

of

S

. The limits

lim

x

→

a

⁡

(

lim

y

→

b

⁡

f

⁢

(

x

,

y

)

)

and

lim

y

→

b

⁡

(

lim

x

→

a

⁡

f

⁢

(

x

,

y

)

)

are called

iterated limits

.

Example 1.

If

f

⁢

(

x

,

y

)

:=

x

⁢

sin

⁡

1

x

+

y

x

+

y

,  then

•

lim

x

→

0

⁡

(

lim

y

→

0

⁡

f

⁢

(

x

,

y

)

)

=

lim

x

→

0

⁡

sin

⁡

1

x

does not exist

•

lim

y

→

0

⁡

(

lim

x

→

0

⁡

f

⁢

(

x

,

y

)

)

=

lim

y

→

0

⁡

1

=

1

•

the usual limit

lim

(

x

,

y

)

→

(

0

,

0

)

⁡

f

⁢

(

x

,

y

)

does not exist.

Example 2.

If

f

⁢

(

x

,

y

)

:=

x

2

x

2

+

y

2

,  then

•

lim

x

→

0

⁡

(

lim

y

→

0

⁡

f

⁢

(

x

,

y

)

)

=

lim

x

→

0

⁡

x

2

x

2

=

1

•

lim

y

→

0

⁡

(

lim

x

→

0

⁡

f

⁢

(

x

,

y

)

)

=

lim

y

→

0

⁡

0

=

0

•

the usual limit

lim

(

x

,

y

)

→

(

0

,

0

)

⁡

f

⁢

(

x

,

y

)

again does not exist, though both of the iterated limits do.

So far we have studied examples that present

discontinuity

at its point of accumulation. We now expose an illustrative example where such discontinuity can be avoided.

Example 3.

Consider the function

f

⁢

(

x

,

y

)

:=

x

⁢

sin

⁡

x

⁢

cosh

⁡

y

+

y

⁢

cos

⁡

x

⁢

sinh

⁡

y

x

2

+

y

2

;

then (we apply l’Hôpital’s rule (

http://planetmath.org/LHpitalsRule

) throughout)

•

lim

x

→

0

⁡

(

lim

y

→

0

⁡

f

⁢

(

x

,

y

)

)

=

lim

x

→

0

⁡

(

lim

y

→

0

⁡

x

⁢

sin

⁡

x

⁢

cosh

⁡

y

+

y

⁢

cos

⁡

x

⁢

sinh

⁡

y

x

2

+

y

2

)

=

lim

x

→

0

⁡

x

⁢

sin

⁡

x

x

2

=

lim

x

→

0

⁡

sin

⁡

x

x

=

lim

x

→

0

⁡

cos

⁡

x

=

1

•

lim

y

→

0

⁡

(

lim

x

→

0

⁡

f

⁢

(

x

,

y

)

)

=

lim

y

→

0

⁡

(

lim

x

→

0

⁡

x

⁢

sin

⁡

x

⁢

cosh

⁡

y

+

y

⁢

cos

⁡

x

⁢

sinh

⁡

y

x

2

+

y

2

)

=

lim

y

→

0

⁡

y

⁢

sinh

⁡

y

y

2

=

lim

y

→

0

⁡

sinh

⁡

y

y

=

lim

y

→

0

⁡

cosh

⁡

y

=

1

•

the usual limit

lim

(

x

,

y

)

→

(

0

,

0

)

⁡

f

⁢

(

x

,

y

)

exists in this case. An essential reason which assures the

continuity

of this function, arises from the fact that

f

⁢

(

x

,

y

)

≡

ℜ

⁡

(

sin

⁡

z

z

)

,

z

=

x

+

i

⁢

y

,  i.e. it is the

real part

of the

analytic function

w

:=

sin

⁡

z

z

having the

removable singularity

at

z

=

0

(see the entry

complex sine and cosine

).

Title

iterated limit in

ℝ

2

Canonical name

IteratedLimitInmathbbR2

Date of creation

2013-03-22 17:28:27

Last modified on

2013-03-22 17:28:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B12

Classification

msc 26A06

Defines

iterated limit

Generated on Fri Feb 9 19:58:02 2018 by

LaTeXML
Jordan content of an n-cell

Jordan content of an

n

-cell

Let

I

=

[

a

1

,

b

1

]

×

⋯

×

[

a

n

,

b

n

]

be an

n

-cell in

ℝ

n

. Then the

Jordan content

(denoted

μ

⁢

(

I

)

) of

I

is defined as

μ

⁢

(

I

)

:=

∏

j

=

1

n

(

b

j

-

a

j

)

.

Title

Jordan content of an

n

-cell

Canonical

name

JordanContentOfAnNcell

Date of creation

2013-03-22 13:37:37

Last modified on

2013-03-22 13:37:37

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 26B12

Generated on Fri Feb 9 19:58:03 2018 by

LaTeXML
lamellar field

lamellar field

A

vector field

F

→

=

F

→

⁢

(

x

,

y

,

z

)

,  defined in an open set

D

of

ℝ

3

, is

lamellar

if the condition

∇

×

F

→

=

0

→

is satisfied in every point

(

x

,

y

,

z

)

of

D

.

Here,

∇

×

F

→

is the curl or

rotor

of

F

→

.  The condition is

equivalent

with both of the following:

•

The

line integrals

∮

s

F

→

⋅

𝑑

s

→

taken around any contractible curve

s

vanish.

•

The vector field has a

u

=

u

⁢

(

x

,

y

,

z

)

which has

continuous

partial derivatives

and which is up to a unique in a simply connected domain; the scalar potential means that

F

→

=

∇

⁡

u

.

The scalar potential has the

expression

u

=

∫

P

0

P

F

→

⋅

𝑑

s

→

,

where the point

P

0

may be chosen freely,

P

=

(

x

,

y

,

z

)

.

Note.

In physics,

u

is in general replaced with

V

=

-

u

.  If the

F

→

is interpreted as a , then the potential

V

is equal to the work made by the when its point of application is displaced from

P

0

to

infinity

.

Title

lamellar field

Canonical

name

LamellarField

Date of creation

2013-03-22 14:43:44

Last modified on

2013-03-22 14:43:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B12

Synonym

lamellar

Synonym

irrotational

Synonym

conservative

Synonym

laminar

Related topic

CurlFreeField

Related topic

PoincareLemma

Related topic

VectorPotential

Related topic

GradientTheorem

Defines

scalar potential

Defines

potential

Defines

rotor

Generated on Fri Feb 9 19:58:05 2018 by

LaTeXML
Laplace equation

Laplace equation

The scalar form of

Laplace’s equation

is the

partial differential equation

∇

2

⁡

f

=

0

and the vector form is

∇

2

⁡

𝐀

=

0

,

where

∇

2

is the

Laplacian

. It is a special case of the

Helmholtz differential equation

with

k

=

0

.

A function

f

which satisfies Laplace’s equation is said to be

harmonic

. Since Laplace’s equation is linear, the superposition of any two solutions is also a solution.

Title

Laplace equation

Canonical

name

LaplaceEquation

Date of creation

2013-03-22 13:09:11

Last modified on

2013-03-22 13:09:11

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 26B12

Synonym

Laplace differential equation

Related topic

PoissonsEquation

Related topic

ExampleOfSolvingTheHeatEquation

Defines

harmonic

Generated on Fri Feb 9 19:58:07 2018 by

LaTeXML
least and greatest value of function

least and greatest value of function

Theorem

.

If the

real function

f

is

1.

continuous

on the

closed interval

[

a

,

b

]

and

2.

differentiable

on the

open interval

(

a

,

b

)

,

then the

function

has on the interval

[

a

,

b

]

a least value and a greatest value.  These are always got in the end of the interval or in the zero of the

derivative

.

Remark 1.

If the preconditions of the

theorem

are fulfilled by a function

f

, then one needs only to determine the values of

f

in the

end points

a

and

b

of the interval and in the zeros of the derivative

f

′

inside the interval; then the least and the greatest value are found among those values.

Remark 2.

Note that the theorem does not require anything of the derivative

f

′

in the points

a

and

b

; one needs not even the

right-sided derivative

in

a

or the left-sided derivative in

b

.  Thus e.g. the function

f

:

x

↦

1

-

x

2

,  fulfilling the conditions of the theorem on the interval

[

-

1

,

1

]

but not having such one-sided derivatives, gains its least value in the end-point

x

=

-

1

and its greatest value in the zero

x

=

0

of the derivative.

Remark 3.

The least value of a function is also called the

absolute minimum

and the greatest value the

absolute maximum

of the function.

Title

least and greatest value of function

Canonical name

LeastAndGreatestValueOfFunction

Date of creation

2013-03-22 15:38:57

Last modified on

2013-03-22 15:38:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26B12

Synonym

global extrema of real function

Related topic

Extremum

Related topic

LeastAndGreatestNumber

Related topic

FermatsTheoremStationaryPoints

Related topic

MinimalAndMaximalNumber

Defines

absolute minimum

Defines

absolute maximum

Generated on Fri Feb 9 19:58:09 2018 by

LaTeXML
minimal and maximal number

minimal and maximal number

Let’s consider a finite non-empty set

A

=

{

a

1

,

…

,

a

n

}

of real numbers or an infinite but compact (i.e.

bounded

and closed) set

A

of real numbers.  In both cases the set has a unique least number and a unique greatest number.

•

The least number of the set is denoted by

min

⁡

{

a

1

,

…

,

a

n

}

or

min

⁡

A

.

•

The greatest number of the set is denoted by

max

⁡

{

a

1

,

…

,

a

n

}

or

max

⁡

A

.

In both cases we have

min

⁡

A

=

inf

⁡

A

,

max

⁡

A

=

sup

⁡

A

,

min

⁡

A

≦

x

≦

max

⁡

A

∀

x

∈

A

,

where

inf

⁡

A

and

sup

⁡

A

are the infimum and supremum of the set

A

.

The

min

and

max

are

set functions

, i.e. they map
subsets of a certain set to

ℝ

.

The

min

and

max

have the following

distributive properties

with respect to

addition

:

min

⁡

{

a

1

,

…

,

a

n

}

+

b

=

min

⁡

{

a

1

+

b

,

…

,

a

n

+

b

}

max

⁡

{

a

1

,

…

,

a

n

}

+

b

=

max

⁡

{

a

1

+

b

,

…

,

a

n

+

b

}

The minimal and maximal number of a set of two real numbers obey
the formulae

min

⁡

{

a

,

b

}

=

a

+

b

2

-

|

a

-

b

|

2

,

max

⁡

{

a

,

b

}

=

a

+

b

2

+

|

a

-

b

|

2

,

max

⁡

{

a

,

b

}

-

min

⁡

{

a

,

b

}

=

|

a

-

b

|

,

max

⁡

{

a

,

b

}

+

min

⁡

{

a

,

b

}

=

a

+

b

,

max

⁡

{

a

,

-

a

}

=

|

a

|

Title

minimal and maximal number

Canonical name

MinimalAndMaximalNumber

Date of creation

2014-02-15 18:33:33

Last modified on

2014-02-15 18:33:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

25

Author

pahio (2872)

Entry type

Definition

Classification

msc 26B12

Classification

msc 03E04

Synonym

least and greatest number

Related topic

Infimum

Related topic

Supremum

Related topic

UltrametricTriangleInequality

Related topic

GrowthOfExponentialFunction

Related topic

EstimatingTheoremOfContourIntegral

Related topic

LeastAndGreatestValueOfFunction

Related topic

FuzzyLogic2

Related topic

ZerosAndPolesOfRationalFunction

Related topic

UniformConvergenceOnUnionInterval

Related topic

Interprime

Related topic

LehmerMean

Related topic

Ab

Defines

least number

Defines

greatest number

Defines

minimal number

Defines

maximal number

Defines

set function

Generated on Fri Feb 9 19:58:14 2018 by

LaTeXML
nabla acting on products

nabla acting on products

Let

f

,

g

be

differentiable

scalar fields and

u

→

,

v

→

differentiable

vector fields

in some domain of

ℝ

3

.  There are following formulae:

•

Gradient

of a product function

∇

⁡

(

f

⁢

g

)

=

(

∇

⁡

f

)

⁢

g

+

(

∇

⁡

g

)

⁢

f

•

Divergence

of a scalar-multiplied vector

∇

⋅

(

f

⁢

u

→

)

=

(

∇

⁡

f

)

⋅

u

→

+

(

∇

⋅

u

→

)

⁢

f

•

Curl of a scalar-multiplied vector

∇

×

(

f

⁢

u

→

)

=

(

∇

⁡

f

)

×

u

→

+

(

∇

×

u

→

)

⁢

f

•

Divergence of a

vector product

∇

⋅

(

u

→

×

v

→

)

=

(

∇

×

u

→

)

⋅

v

→

-

(

∇

×

v

→

)

⋅

u

→

•

Curl of a vector product

∇

×

(

u

→

×

v

→

)

=

(

v

→

⋅

∇

)

⁢

u

→

-

(

u

→

⋅

∇

)

⁢

v

→

-

(

∇

⋅

u

→

)

⁢

v

→

+

(

∇

⋅

v

→

)

⁢

u

→

•

Gradient of a

scalar product

∇

⁡

(

u

→

⋅

v

→

)

=

(

v

→

⋅

∇

)

⁢

u

→

+

(

u

→

⋅

∇

)

⁢

v

→

+

v

→

×

(

∇

×

u

→

)

+

u

→

×

(

∇

×

v

→

)

or, using dyads,

∇

⁡

(

u

→

⋅

v

→

)

=

(

∇

⁡

u

→

)

⋅

v

→

+

(

∇

⁡

v

→

)

⋅

u

→

•

Gradient of a vector product

∇

⁡

(

u

→

×

v

→

)

=

(

∇

⁡

u

→

)

×

v

→

-

(

∇

⁡

v

→

)

×

u

→

•

Divergence of a

dyad product

∇

⋅

(

u

→

⁢

v

→

)

=

(

∇

⋅

u

→

)

⁢

v

→

+

u

→

⋅

∇

⁡

v

→

•

Curl of a dyad product

∇

×

(

u

→

⁢

v

→

)

=

(

∇

×

u

→

)

⁢

v

→

-

u

→

×

∇

⁡

v

→

Explanations

1.

v

→

⋅

∇

means the operator

v

x

⁢

∂

∂

⁡

x

+

v

y

⁢

∂

∂

⁡

y

+

v

z

⁢

∂

∂

⁡

z

.

2.

The

gradient of a vector

w

→

is defined as the dyad

∇

⁡

w

→

:=

i

→

⁢

∂

⁡

w

→

∂

⁡

x

+

j

→

⁢

∂

⁡

w

→

∂

⁡

y

+

k

→

⁢

∂

⁡

w

→

∂

⁡

z

.

3.

The

divergence

and the

curl

of a dyad product are defined by the equation

∇

*

(

u

→

⁢

v

→

)

:=

i

→

*

∂

⁡

(

u

→

⁢

v

→

)

∂

⁡

x

+

j

→

*

∂

⁡

(

u

→

⁢

v

→

)

∂

⁡

y

+

k

→

*

∂

⁡

(

u

→

⁢

v

→

)

∂

⁡

z

,  where the asterisks are dots or crosses and the

partial derivatives

of the dyad product the expression

∂

⁡

(

u

→

⁢

v

→

)

∂

⁡

x

=

∂

⁡

u

→

∂

⁡

x

⁢

v

→

+

u

→

⁢

∂

⁡

v

→

∂

⁡

x

and so on.

Title

nabla acting on products

Canonical name

NablaActingOnProducts

Date of creation

2013-03-22 15:27:05

Last modified on

2013-03-22 15:27:05

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Topic

Classification

msc 26B12

Classification

msc 26B10

Related topic

Nabla

Related topic

NablaNabla

Defines

gradient of vector

Defines

divergence of dyad product

Defines

curl of dyad product

Generated on Fri Feb 9 19:58:18 2018 by

LaTeXML