aarjaneiro/planetmath huggingface dataset:
tensor algebra

tensor algebra

Let

R

be a

commutative ring

, and

M

an

R

-module.
The

tensor algebra

𝒯

⁢

(

M

)

=

⊕

n

=

0

∞

𝒯

n

⁢

(

M

)

is the graded

R

-

algebra

with

n

t

⁢

h

graded

component

simply the

n

t

⁢

h

tensor power:

𝒯

n

⁢

(

M

)

=

M

⊗

n

=

M

⊗

⋯

⊗

M

⏞

n

⁢

times

,

n

=

1

,

2

,

…

,

and

𝒯

0

⁢

(

M

)

=

R

.
The

multiplication

m

:

𝒯

⁢

(

M

)

×

𝒯

⁢

(

M

)

→

𝒯

⁢

(

M

)

is given
by the usual

tensor product

:

m

⁢

(

a

,

b

)

=

a

⊗

b

,

a

∈

M

⊗

n

,

b

∈

M

⊗

m

.

Remark 1.

One can generalize the above

definition

to
cover the case where the ground ring

R

is non-commutative by
requiring that the module

M

is a

bimodule

with

R

acting on both
the left and the right.

Remark 2.

From the point of view of

category theory

, one
can describe the tensor algebra construction as a

functor

𝒯

from the

category

of

R

-module to the category of

R

-algebras that
is left-adjoint to the

forgetful functor

ℱ

from algebras to
modules. Thus, for

M

an

R

-module and

S

an

R

-algebra, every

module homomorphism

M

→

ℱ

⁢

(

S

)

extends to a unique algebra
homomorphism

𝒯

⁢

(

M

)

→

S

.

Title

tensor algebra

Canonical

name

TensorAlgebra

Date of creation

2013-03-22 13:17:21

Last modified on

2013-03-22 13:17:21

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

13

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A69

Related topic

FreeAssociativeAlgebra

Defines

tensor power

Generated on Fri Feb 9 18:31:15 2018 by

LaTeXML
tensor product of subspaces of vector spaces

tensor product of subspaces of vector spaces

Proposition

.

Let

V

,

W

be

vector spaces

over a field

k

. Moreover let

A

⊆

V

,

B

⊆

W

be

subspaces

. Then

V

⊗

B

∩

A

⊗

W

=

A

⊗

B

.

Proof.

The

inclusion

,,

⊇

” is obvious. We will show the inclusion ,,

⊆

”.

Let

{

e

i

}

i

∈

I

and

{

e

j

′

}

j

∈

P

be bases of

A

and

B

respectively. Moreover let

{

e

i

}

i

∈

I

′

be a

completion

of given basis of

A

to the basis of

V

, i.e.

{

e

i

}

i

∈

I

∪

I

′

is a basis of

V

. Analogously let

{

e

j

′

}

j

∈

P

′

be a completion of a basis of

B

to the basis of

W

. Then each

element

q

∈

V

⊗

W

can be uniquely written in a form

q

=

∑

i

∈

I

,

j

∈

P

α

i

,

j

⁢

e

i

⊗

e

j

′

+

∑

i

∈

I

′

,

j

∈

P

β

i

,

j

⁢

e

i

⊗

e

j

′

+

+

∑

i

∈

I

,

j

∈

P

′

δ

i

,

j

⁢

e

i

⊗

e

j

′

+

∑

i

∈

I

′

,

j

∈

P

′

γ

i

,

j

⁢

e

i

⊗

e

j

′

.

Assume that

q

∈

V

⊗

B

∩

A

⊗

W

.

Let

i

∈

I

′

and

j

∈

P

′

. Consider the following

linear map

:

f

i

:

V

→

k

such that

f

i

⁢

(

e

t

)

=

1

if

i

=

t

and

f

i

⁢

(

e

t

)

=

0

if

i

≠

t

. Analogously we define

g

j

:

W

→

k

. Then we combine these two

mappings

into one, i.e.

f

i

⊗

g

j

:

V

⊗

W

→

k

;

(

f

i

⊗

g

j

)

⁢

(

v

⊗

w

)

=

f

i

⁢

(

v

)

⁢

g

j

⁢

(

w

)

.

Furthermore we have

(

f

i

⊗

g

j

)

⁢

(

q

)

=

γ

i

,

j

.

Note that since

q

∈

V

⊗

B

, then

(

f

i

⊗

g

j

)

⁢

(

q

)

=

0

and thus

γ

i

,

j

=

0

.

Similarly we obtain that all

β

i

,

j

and

δ

i

,

j

are equal to

0

. Thus

q

=

∑

i

∈

I

,

j

∈

P

α

i

,

j

⁢

e

i

⊗

e

j

′

∈

A

⊗

B

,

which

completes

the proof.

□

Title

tensor product of subspaces of vector spaces

Canonical name

TensorProductOfSubspacesOfVectorSpaces

Date of creation

2013-03-22 18:49:16

Last modified on

2013-03-22 18:49:16

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 15A69

Generated on Fri Feb 9 18:31:37 2018 by

LaTeXML
bac-cab rule

bac-cab rule

The

bac-cab rule

states that for vectors

𝐀

,

𝐁

, and

𝐂

(that can be either real or complex) in

ℝ

3

, we have

𝐀

×

(

𝐁

×

𝐂

)

=

𝐁

⁢

(

𝐀

⋅

𝐂

)

-

𝐂

⁢

(

𝐀

⋅

𝐁

)

.

Here

×

is the

cross product

, and

⋅

is the real

inner product

.

Title

bac-cab rule

Canonical name

BaccabRule

Date of creation

2013-03-22 14:04:34

Last modified on

2013-03-22 14:04:34

Owner

williamschips (4069)

Last modified by

williamschips (4069)

Numerical id

6

Author

williamschips (4069)

Entry type

Result

Classification

msc 15A72

Classification

msc 15A90

Generated on Fri Feb 9 18:31:49 2018 by

LaTeXML
calculating the solid angle of disc

calculating the solid angle of disc

We determine the

solid angle

formed by a disc when one is looking at it on the

normal line

of its plane set to the center of it.

Let us look the disc from the origin and let the disc with radius

R

situate such that its plane is

parallel

to the

x

⁢

y

-plane and the center is on the

z

-axis at

(

0

,

0

,

h

)

with

h

>

0

.  Into the

Ω

=

-

∫

a

d

⁢

a

→

⋅

∇

⁡

1

r

=

∫

a

d

⁢

a

→

⋅

r

→

|

r

→

|

3

(1)

of the parent entry (

http://planetmath.org/SolidAngle

), we may substitute the

position vector

r

→

=

x

⁢

i

→

+

y

⁢

j

→

+

h

⁢

k

→

of the directed surface element

d

⁢

a

→

=

k

→

⁢

d

⁢

a

,

getting

Ω

=

∫

a

h

⁢

d

⁢

a

(

x

2

+

y

2

+

h

2

)

3

/

2

.

Now we can use a

annulus

(

http://planetmath.org/Annulus2

)-formed surface element

d

⁢

a

=

2

⁢

π

⁢

ϱ

⁢

d

⁢

ϱ

where

ϱ

2

=

x

2

+

y

2

, whence the

surface integral

may be calculated as

Ω

=

π

⁢

h

⁢

∫

0

R

2

⁢

ϱ

⁢

d

⁢

ϱ

(

ϱ

2

+

h

2

)

3

/

2

=

π

⁢

h

-

2

⁢

/

ϱ

=

0

R

⁡

1

ϱ

2

+

h

2

.

Thus we have the result

Ω

=

2

⁢

π

⁢

h

⁢

(

1

h

-

1

R

2

+

h

2

)

.

Title

calculating the solid angle of disc

Canonical name

CalculatingTheSolidAngleOfDisc

Date of creation

2013-03-22 18:19:36

Last modified on

2013-03-22 18:19:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Example

Classification

msc 15A72

Classification

msc 51M25

Related topic

SubstitutionNotation

Related topic

AngleOfViewOfALineSegment

Generated on Fri Feb 9 18:31:52 2018 by

LaTeXML
direction cosines

direction cosines

If the non-zero vector

r

→

=

x

⁢

i

→

+

y

⁢

j

→

+

z

⁢

k

→

of

ℝ

3

forms the angles

α

,

β

and

γ

with the

positive

directions

of

x

-axis,

y

-axis and

z

-axis, respectively, then the numbers

cos

⁡

α

,

cos

⁡

β

,

cos

⁡

γ

are the

direction cosines

of the vector. Any triple

l

,

m

,

n

of numbers, which are proportional (

http://planetmath.org/Variation

) to the direction cosines, are

direction numbers

of the vector.

If

r

=

x

2

+

y

2

+

z

2

is the of

r

→

, we see easily that

cos

⁡

α

=

x

r

,

cos

⁡

β

=

y

r

,

cos

⁡

γ

=

z

r

.

Conversely, the

components

of the vector on the

coordinate

axes may be obtained from

x

=

r

⁢

cos

⁡

α

,

y

=

r

⁢

cos

⁡

β

,

z

=

r

⁢

cos

⁡

γ

.

We also see that the direction cosines satisfy

cos

2

⁡

α

+

cos

2

⁡

β

+

cos

2

⁡

γ

=

1

.

Title

direction cosines

Canonical

name

DirectionCosines

Date of creation

2013-03-22 17:16:32

Last modified on

2013-03-22 17:16:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 15A72

Classification

msc 51N20

Related topic

MutualPositionsOfVectors

Related topic

EquationOfPlane

Defines

direction numbers

Generated on Fri Feb 9 18:31:54 2018 by

LaTeXML
distance of non-parallel lines

distance of non-parallel lines

As an application of the

vector product

(

http://planetmath.org/CrossProduct

) we derive the expression of the

d

between two non-parallel lines in

ℝ

3

.

Suppose that the

position vectors

of the points of the two non-parallel lines are expressed in parametric forms

r

→

=

a

→

+

s

⁢

u

→

and

r

→

=

b

→

+

t

⁢

v

→

,

where

s

and

t

are parameters.  A common vector of the lines is the

cross product

u

→

×

v

→

of the

direction vectors

of the lines, and it may be normed to a

unit vector

n

→

:=

u

→

×

v

→

|

u

→

×

v

→

|

by dividing it by its , which is distinct from 0 because of the non-parallelity.  The vectors

a

→

and

b

→

are the position vectors of certain points

A

and

B

on the lines, and thus their

difference

a

→

-

b

→

is the vector from

B

to

A

.  If we

project

a

→

-

b

→

on the

unit normal

n

→

, the obtained vector

d

→

:=

[

(

a

→

-

b

→

)

⋅

n

→

]

⁢

n

→

has the sought

d

=

|

(

a

→

-

b

→

)

⋅

n

→

|

,  i.e.

d

=

|

(

a

→

-

b

→

)

⋅

(

u

→

×

v

→

)

|

|

u

→

×

v

→

|

.

For illustrating that

d

is the minimal distance between points of the two lines we underline, that the construction of

d

guarantees that it connects two points on the lines and is

perpendicular

to both lines — thus any displacement of its

end point

makes it longer.

Notes.

The

numerator

is the

absolute value

of a

triple scalar product

.  If the lines intersect each other, then the connecting vector

a

→

-

b

→

is at

right angles

to the common

normal vector

n

→

of their plane and thus the

dot product

of these vectors vanishes, i.e. also

d

=

0

.  If the lines do not intersect, they are called

agonic lines

or

skew lines

;  then

d

>

0

.

Title

distance of non-parallel lines

Canonical name

DistanceOfNonparallelLines

Date of creation

2013-03-22 15:27:16

Last modified on

2013-03-22 15:27:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Derivation

Classification

msc 15A72

Synonym

distance of lines

Related topic

LineInSpace

Related topic

DistanceFromPointToALine

Related topic

EuclideanDistance

Related topic

AngleBetweenTwoLines

Defines

agonic lines

Defines

skew lines

Generated on Fri Feb 9 18:31:57 2018 by

LaTeXML
dual homomorphism

dual homomorphism

Definition.

Let

U

,

V

be

vector spaces

over a field

𝕂

, and

T

:

U

→

V

be a

homomorphism

(a

linear map

) between them.
Letting

U

*

,

V

*

denote the corresponding

dual
spaces

, we define the

dual homomorphism

T

*

:

V

*

→

U

*

,
to be the linear mapping with action

α

→

α

∘

T

,

α

∈

V

*

.

We can also characterize

T

*

as the

adjoint

of

T

relative
to the natural evaluation bracket between

linear forms

and vectors:

⟨

-

,

-

⟩

U

:

U

*

×

U

→

𝕂

,

⟨

α

,

u

⟩

=

α

⁢

(

u

)

,

α

∈

U

*

,

u

∈

U

.

To be more precise

T

*

is characterized by the condition

⟨

T

*

⁢

α

,

u

⟩

U

=

⟨

α

,

T

⁢

u

⟩

V

,

α

∈

V

*

,

u

∈

U

.

If

U

and

V

are finite dimensional, we can also characterize the
dualizing

operation

as the

composition

of the following

canonical
isomorphisms

:

Hom

(

U

,

V

)

⟶

≃

U

*

⊗

V

⟶

≃

(

V

*

)

*

⊗

U

*

⟶

≃

Hom

(

V

*

,

U

*

)

.

Category theory perspective.

The dualizing operation behaves
contravariantly with respect to composition, i.e.

(

S

∘

T

)

*

=

T

*

∘

S

*

,

for all vector space homomorphisms

S

,

T

with suitably matched domains. Furthermore, the dual of the

identity

homomorphism is the identity homomorphism of the dual space.
Thus, using the

language

of

category theory

, the dualizing operation
can be characterized as the homomorphism action of the contravariant,
dual-space

functor

.

Relation to the matrix transpose.

The above properties closely
mirror the algebraic properties of the matrix transpose operation.
Indeed,

T

*

is sometimes referred to as the

transpose

of

T

,
because at the level of matrices the dual homomorphism is calculated
by taking the transpose.

To be more precise, suppose that

U

and

V

are

finite-dimensional

,
and let

M

∈

Mat

n

,

m

(

𝕂

)

be the matrix of

T

relative to
some fixed bases of

U

and

V

. Then, the dual homomorphism

T

*

is represented as the transposed matrix

M

t

∈

Mat

m

,

n

(

𝕂

)

relative to the corresponding dual bases of

U

*

,

V

*

.

Title

dual homomorphism

Canonical name

DualHomomorphism

Date of creation

2013-03-22 12:29:33

Last modified on

2013-03-22 12:29:33

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A72

Classification

msc 15A04

Synonym

adjoint homomorphism

Synonym

adjoint

Related topic

LinearTransformation

Related topic

DualSpace

Related topic

DoubleDualEmbedding

Generated on Fri Feb 9 18:32:01 2018 by

LaTeXML
mutual positions of vectors

mutual positions of vectors

In this entry, we work within a

Euclidean space

E

.

1.

Two non-zero

Euclidean vectors

a

→

and

b

→

are said to be

parallel

, denoted by

a

→

∥

b

→

,  iff there exists a real number

k

such that

a

→

=

k

⁢

b

→

.

Since both

a

→

and

b

→

are non-zero,

k

≠

0

.  So

∥

is a

binary relation

on on

E

∖

{

0

→

}

and called the

parallelism

.  If

k

>

0

,  then

a

and

b

are said to be in the

same

direction

, and we denote this by

a

→

⇈

b

→

;  if

k

<

0

,  then

a

and

b

are said to be in the

opposite

or

contrary directions

, and we denote this by

a

→

↓

↑

b

→

.

Remarks

–

Actually, the parallelism is an

equivalence relation

on

E

∖

{

0

→

}

.  If the

zero vector

0

→

were allowed along, then the

relation

were not

symmetric

(

0

→

=

0

⁢

b

→

but not necessarily

b

→

=

k

⁢

0

→

).

–

When two vectors

a

→

and

b

→

are not parallel to one another, written

a

→

∦

b

→

,  they are said to be

diverging

.

2.

Two Euclidean vectors

a

→

and

b

→

are

perpendicular

, denoted by

a

→

⟂

b

→

,  iff

a

→

⋅

b

→

=

0

,

i.e. iff their

scalar product

vanishes.  Then

a

→

and

b

→

are

normal vectors

of each other.

Remarks

–

We may say that

0

→

is perpendicular to all vectors, because its direction is and because

0

→

⋅

b

→

=

0

.

–

Perpendicularity is not an equivalence relation in the set of all vectors of the space in question, since it is neither

reflexive

nor

transitive

.

3.

The angle

θ

between two non-zero vectors

a

→

and

b

→

is obtained from

cos

⁡

θ

=

a

→

⋅

b

→

|

a

→

|

⁢

|

b

→

|

.

The angle is chosen so that

0

≦

θ

≦

π

.

Title

mutual positions of vectors

Canonical name

MutualPositionsOfVectors

Date of creation

2013-03-22 14:36:24

Last modified on

2013-03-22 14:36:24

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

25

Author

pahio (2872)

Entry type

Definition

Classification

msc 15A72

Related topic

AngleBetweenTwoLines

Related topic

DirectionCosines

Related topic

OrthogonalVectors

Related topic

PerpendicularityInEuclideanPlane

Related topic

MedianOfTrapezoid

Related topic

TriangleMidSegmentTheorem

Related topic

CommonPointOfTriangleMedians

Related topic

FluxOfVectorField

Related topic

NormalOfPlane

Defines

parallel

Defines

parallelism

Defines

perpendicular

Defines

perpendicularity

Defines

diverging

Defines

normal vector

Generated on Fri Feb 9 18:32:22 2018 by

LaTeXML
position vector

position vector

In the space

ℝ

3

, the vector

r

→

:=

(

x

,

y

,

z

)

=

x

⁢

i

→

+

y

⁢

j

→

+

z

⁢

k

→

directed from the origin to a point

(

x

,

y

,

z

)

is the

position vector

of this point. When the point is ,

r

→

a

vector field

and its

r

:=

x

2

+

y

2

+

z

2

a scalar .

The

•

∇

⋅

r

→

=

3

•

∇

×

r

→

=

0

→

•

∇

⁡

r

=

r

→

r

=

r

→

0

•

∇

⁡

1

r

=

-

r

→

r

3

=

-

r

→

0

r

2

•

∇

2

⁡

1

r

=

0

are valid, where

r

→

0

is the

unit vector

having the

direction

of

r

→

.

If

c

→

is a vector,

U

→

:

ℝ

3

→

ℝ

3

a

vector function

and

f

:

ℝ

→

ℝ

is a twice

differentiable function

, then the formulae

•

∇

⁡

(

c

→

⋅

r

→

)

=

c

→

•

∇

⋅

(

c

→

×

r

→

)

=

0

•

(

U

→

⋅

∇

)

⁢

r

→

=

U

→

•

(

U

→

×

∇

)

⋅

r

→

=

0

•

(

U

→

×

∇

)

×

r

→

=

-

2

⁢

U

→

•

∇

⁡

f

⁢

(

r

)

=

f

′

⁢

(

r

)

⁢

r

→

0

•

∇

2

⁡

f

⁢

(

r

)

=

f

′′

⁢

(

r

)

+

2

r

⁢

f

′

⁢

(

r

)

hold.

References

1

K. Väisälä:

Vektorianalyysi

.  Werner Söderström Osakeyhtiö, Helsinki (1961).

Title

position vector

Canonical

name

PositionVector

Date of creation

2013-03-22 15:25:15

Last modified on

2013-03-22 15:25:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Definition

Classification

msc 15A72

Synonym

radius vector

Related topic

ExampleOfCurvatureSpaceCurve

Related topic

DyadProduct

Related topic

TiltCurve

Generated on Fri Feb 9 18:32:25 2018 by

LaTeXML
right-handed system of vectors

right-handed system of vectors

Three ordered non-coplanar vectors

u

→

,

v

→

and

w

→

, which have a common , are said to form a

right-handed

or

dextral system

(Latin

dexter

= ), if a right-threaded screw rotated through an angle less than

180

o

from

u

→

to

v

→

will advance in the

direction

of

w

→

.  For instance, the usual basis vectors

i

→

,

j

→

and

k

→

form a dextral system.

Figure 1:

A right-handed system of vectors

Title

right-handed system of vectors

Canonical name

RighthandedSystemOfVectors

Date of creation

2013-03-22 14:59:52

Last modified on

2013-03-22 14:59:52

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

8

Author

juanman (12619)

Entry type

Definition

Classification

msc 15A72

Synonym

dextral system of vectors

Related topic

Helix

Defines

right-handed

Generated on Fri Feb 9 18:32:28 2018 by

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rotational invariance of cross product

rotational invariance of cross product

Theorem

Let

R

be a rotational

3

×

3

matrix, i.e., a

real
matrix

with

det

⁡

𝐑

=

1

and

𝐑

-

1

=

𝐑

T

.
Then for all vectors

𝐮

,

𝐯

in

ℝ

3

,

𝐑

⋅

(

𝐮

×

𝐯

)

=

(

𝐑

⋅

𝐮

)

×

(

𝐑

⋅

𝐯

)

.

Proof.

Let us first fix some right hand oriented

orthonormal basis

in

ℝ

3

.
Further, let

{

u

1

,

u

2

,

u

3

}

and

{

v

1

,

v

2

,

v

3

}

be the

components

of

u

and

v

in that basis. Also, in the chosen basis, we denote the entries
of

R

by

R

i

⁢

j

. Since

R

is rotational, we have

R

i

⁢

j

⁢

R

k

⁢

j

=

δ

i

⁢

k

where

δ

i

⁢

k

is the

Kronecker delta

symbol. Here we use the

Einstein summation convention

.
Thus, in the previous expression, on the

left hand side

,

j

should be summed
over

1

,

2

,

3

. We shall use the

Levi-Civita permutation symbol

ε

to write the

cross product

.
Then the

i

:th coordinate of

𝐮

×

𝐯

equals

(

𝐮

×

𝐯

)

i

=

ε

i

⁢

j

⁢

k

⁢

u

j

⁢

v

k

.
For the

k

th component of

(

𝐑

⋅

𝐮

)

×

(

𝐑

⋅

𝐯

)

we
then have

(

(

𝐑

⋅

𝐮

)

×

(

𝐑

⋅

𝐯

)

)

k

=

ε

i

⁢

m

⁢

k

⁢

R

i

⁢

j

⁢

R

m

⁢

n

⁢

u

j

⁢

v

n

=

ε

i

⁢

m

⁢

l

⁢

δ

k

⁢

l

⁢

R

i

⁢

j

⁢

R

m

⁢

n

⁢

u

j

⁢

v

n

=

ε

i

⁢

m

⁢

l

⁢

R

k

⁢

r

⁢

R

l

⁢

r

⁢

R

i

⁢

j

⁢

R

m

⁢

n

⁢

u

j

⁢

v

n

=

ε

j

⁢

n

⁢

r

⁢

det

⁡

𝐑

⁢

R

k

⁢

r

⁢

u

j

⁢

v

n

.

The last line follows since

ε

i

⁢

j

⁢

k

⁢

R

i

⁢

m

⁢

R

j

⁢

n

⁢

R

k

⁢

r

=

ε

m

⁢

n

⁢

r

⁢

ε

i

⁢

j

⁢

k

⁢

R

i

⁢

1

⁢

R

j

⁢

2

⁢

R

k

⁢

3

=

ε

m

⁢

n

⁢

r

⁢

det

⁡

𝐑

.
Since

det

⁡

𝐑

=

1

, it follows that

(

(

𝐑

⋅

𝐮

)

×

(

𝐑

⋅

𝐯

)

)

k

=

R

k

⁢

r

⁢

ε

j

⁢

n

⁢

r

⁢

u

j

⁢

v

n

=

R

k

⁢

r

⁢

(

𝐮

×

𝐯

)

r

=

(

𝐑

⋅

𝐮

×

𝐯

)

k

as claimed.

□

Title

rotational invariance of cross product

Canonical name

RotationalInvarianceOfCrossProduct

Date of creation

2013-03-22 13:33:53

Last modified on

2013-03-22 13:33:53

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Theorem

Classification

msc 15A72

Classification

msc 15A90

Generated on Fri Feb 9 18:32:31 2018 by

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scalar factor transfer rules

scalar factor transfer rules

The different kinds of products between two

Euclidean vectors

may an additional scalar as factor in either vector factor

u

→

,

v

→

.  Then such a scalar

r

can be transferred from a vector to the other vector and to the whole product.  This is true for

scalar product

,

u

→

⋅

(

r

⁢

v

→

)

=

(

r

⁢

u

→

)

⋅

v

→

=

r

⁢

(

u

→

⋅

v

→

)

,

for

vector product

,

u

→

×

(

r

⁢

v

→

)

=

(

r

⁢

u

→

)

×

v

→

=

r

⁢

(

u

→

×

v

→

)

,

and also for

dyad product

,

u

→

⁢

(

r

⁢

v

→

)

=

(

r

⁢

u

→

)

⁢

v

→

=

r

⁢

(

u

→

⁢

v

→

)

.

Title

scalar factor transfer rules

Canonical name

ScalarFactorTransferRules

Date of creation

2013-03-22 15:26:41

Last modified on

2013-03-22 15:26:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Topic

Classification

msc 15A72

Synonym

transfer rules of scalar factor

Generated on Fri Feb 9 18:32:34 2018 by

LaTeXML
tensor density

tensor density

0.1

Heuristic definition

A

tensor density

is a quantity whose transformation law under

change of basis

involves the

determinant

of the transformation matrix (as opposed to a tensor, whose transformation law does not involve the determinant).

0.2

Linear Theory

For any

real number

p

, we may define a representation

ρ

p

of the group

G

⁢

L

⁢

(

ℝ

k

)

on the

vector space

of

tensor arrays

of rank

m

,

n

as follows:

(

ρ

p

⁢

(

M

)

⁢

T

)

j

1

,

…

⁢

j

m

i

1

,

…

,

i

n

=

(

det

(

M

)

)

p

⁢

M

l

1

i

1

⁢

⋯

⁢

M

l

n

i

n

⁢

(

M

-

1

)

k

1

j

1

⁢

⋯

⁢

(

M

-

1

)

k

m

j

m

⁢

T

j

1

,

…

⁢

j

m

i

1

,

…

,

i

n

A

tensor density

T

of rank

m

,

n

and weight

p

is an

element of

the vector space on which this representation acts.

Note that if the weight equals zero, the concept of tensor density reduces to that of a tensor.

0.3

Examples

The simplest example of such a quantity is a scalar

density

. Under a change of basis

y

i

=

M

j

i

⁢

x

j

, a scalar density transforms as follows:

ρ

p

⁢

(

S

)

=

(

det

(

M

)

)

p

⁢

S

An important example of a tensor density is the

Levi-Civita permutation symbol

. It is a density of weight

1

because, under a change of coordinates,

(

ρ

1

⁢

ϵ

)

j

1

,

…

⁢

j

m

=

(

det

(

M

)

)

⁢

(

M

-

1

)

k

1

j

1

⁢

⋯

⁢

(

M

-

1

)

k

m

j

m

⁢

ϵ

j

1

,

…

⁢

j

m

i

1

,

…

,

i

n

=

ϵ

k

1

,

…

⁢

k

m

0.4

Tensor Densities on Manifolds

As with tensors, it is possible to define tensor density fields on manifolds. On each

coordinate

neighborhood

, the density field is given by a tensor array of

functions

. When two neighborhoods overlap, the tensor arrays are related by the change of variable formula

T

j

1

,

…

⁢

j

m

i

1

,

…

,

i

n

⁢

(

x

)

=

(

det

(

M

)

)

p

⁢

M

l

1

i

1

⁢

⋯

⁢

M

l

n

i

n

⁢

(

M

-

1

)

k

1

j

1

⁢

⋯

⁢

(

M

-

1

)

k

m

j

m

⁢

T

j

1

,

…

⁢

j

m

i

1

,

…

,

i

n

⁢

(

y

)

where

M

is the Jacobian matrix of the change of variables.

Title

tensor density

Canonical name

TensorDensity

Date of creation

2013-03-22 14:55:18

Last modified on

2013-03-22 14:55:18

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 15A72

Synonym

density

Related topic

tensor

Generated on Fri Feb 9 18:32:58 2018 by

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