OpenMath

OpenMath is an extensible standard for representing the semantics of mathematical objects.

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OpenMath Content Dictionary: tensor1

Canonical URL:
http://www.openmath.org/cd/.ocd
CD File:
tensor1.ocd
CD as XML Encoded OpenMath:
tensor1.omcd
Defines:
Cartesian, Kronecker_tensor, Levi-Civita, basis_selector, contra_index, covar_index, metric_tensor, tensor_selector, tuple, tuple_selector, unit_Cartesian
Date:
2010-08-25
Version:
1 (Revision 2)
Review Date:
2017-12-31
Status:
experimental
    This document is distributed in the hope that it will be useful, 
    but WITHOUT ANY WARRANTY; without even the implied warranty of 
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
    
    The copyright holder grants you permission to redistribute this 
    document freely as a verbatim copy. Furthermore, the copyright
    holder permits you to develop any derived work from this document
    provided that the following conditions are met.
    a) The derived work acknowledges the fact that it is derived from
    this document, and maintains a prominent reference in the 
    work to the original source.
    b) The fact that the derived work is not the original OpenMath 
    document is stated prominently in the derived work.  Moreover if
    both this document and the derived work are Content Dictionaries
    then the derived work must include a different CDName element,
    chosen so that it cannot be confused with any works adopted by
    the OpenMath Society.  In particular, if there is a Content 
    Dictionary Group whose name is, for example, `math' containing
    Content Dictionaries named `math1', `math2' etc., then you should 
    not name a derived Content Dictionary `mathN' where N is an integer.
    However you are free to name it `private_mathN' or some such.  This
    is because the names `mathN' may be used by the OpenMath Society
    for future extensions.
    c) The derived work is distributed under terms that allow the
    compilation of derived works, but keep paragraphs a) and b)
    intact.  The simplest way to do this is to distribute the derived
    work under the OpenMath license, but this is not a requirement.
    If you have questions about this license please contact the OpenMath
    society at http://www.openmath.org.
    
    Author: Joseph B. Collins (2010), Naval Research Laboratory, Washington, DC.
    Copyright Notice:  This is a work of the U.S. Government and is not
    subject to copyright protection in the United States. Foreign copyrights
    may apply.
  
  Author: J B Collins

This CD defines content markup symbols which may be used to represent tensor formulae,particularly with index notation.

tuple

Role:
application
Description:

This symbol is an n-ary symbol, returning an n-tuple of the arguments. The number of arguments, n, is a non-negative integer. The elements of the n-tuple are ordered as the arguments are ordered. Elements of a tuple may have the same type and value as each other, or not. An n-tuple, unlike a list, is generally not mutable.

Commented Mathematical property (CMP):
(X, Y, Z) = tuple(X, Y, Z)
Signatures:
sts

[Next: tuple_selector] [Last: Levi-Civita] [Top]

tuple_selector

Role:
application
Description:

This symbol takes 2 arguments, a tuple and a natural number index, and returns the tuple component indicated by the index value.

Commented Mathematical property (CMP):
tuple_selector(tuple(A, B), 1) = A
Signatures:
sts

[Next: Cartesian] [Previous: tuple] [Top]

Cartesian

Role:
application
Description:

This symbol takes one argument, a natural number, and returns the Cartesian coordinate, of a right handed Cartesian coordinate frame, corresponding to the value of the argument. These coordinates are commonly named X, Y, and Z in three dimensions, though X, Y, and Z are non-exclusively used for this and other purposes.

Commented Mathematical property (CMP):
tuple(X, Y, Z) = tuple(Cartesian(1), Cartesian(2), Cartesian(3))
Commented Mathematical property (CMP):
partialdiff(Cartesian(i), Cartesian(j)) = Kronecker_tensor(i,j)
Signatures:
sts

[Next: unit_Cartesian] [Previous: tuple_selector] [Top]

unit_Cartesian

Role:
application
Description:

This symbol takes one argument, a natural number, and returns the Cartesian basis element, of a right handed Cartesian coordinate frame, corresponding to the value of the argument. The unit_Cartesian basis elements are each constant with respect to position in the space and define an orthonormal vector space basis.

Commented Mathematical property (CMP):
tuple(e_x, e_y, e_z) = tuple(unit_Cartesian(1), unit_Cartesian(2), unit_Cartesian(3))
Commented Mathematical property (CMP):
scalar_product(unit_Cartesian(i), unit_Cartesian(j)) = Kronecker_tensor(i,j)
Commented Mathematical property (CMP):
partial(unit_Cartesian(i), Cartesian(j)) = 0
Signatures:
sts

[Next: contra_index] [Previous: Cartesian] [Top]

contra_index

Role:
application
Description:

This symbol takes a natural number as its argument and returns a contravariant index.

Signatures:
sts

[Next: covar_index] [Previous: unit_Cartesian] [Top]

covar_index

Role:
application
Description:

This symbol takes a natural number as its argument and returns a covariant index.

Signatures:
sts

[Next: basis_selector] [Previous: contra_index] [Top]

basis_selector

Role:
application
Description:

This symbol takes 2 arguments, a tuple of basis elements and a covar_index or a contra_index, and returns the basis element indicated by the index value.

Commented Mathematical property (CMP):
unit_Cartesian(1) = basis_selector(tuple(unit_Cartesian(1), unit_Cartesian(2), unit_Cartesian(3)), covar_index(1))
Formal Mathematical property (FMP):
unit_Cartesian ⁡ ( 1 ) = basis_selector ⁡ ( tuple ⁡ ( unit_Cartesian ⁡ ( 1 ) , unit_Cartesian ⁡ ( 2 ) , unit_Cartesian ⁡ ( 3 ) ) , covar_index ⁡ ( 1 ) )
Signatures:
sts

[Next: tensor_selector] [Previous: covar_index] [Top]

tensor_selector

Role:
application
Description:

This symbol takes 3 arguments: a tensor, a basis, and a tuple of contravariant and/or covariant indexes. It returns the indexed tensor component in the given basis.

Signatures:
sts

[Next: Kronecker_tensor] [Previous: basis_selector] [Top]

Kronecker_tensor

Role:
constant
Description:

This symbol represents the Kronecker tensor or Kronecker delta.

Commented Mathematical property (CMP):
tensor_selector(Kronecker_tensor, tuple(unit_Cartesian(1), unit_Cartesian(2), unit_Cartesian(3)), contra_index(i), covar_index(i)) = 1 OR tensor_selector(Kronecker_tensor, tuple(unit_Cartesian(1), unit_Cartesian(2), unit_Cartesian(3)), contra_index(i), covar_index(j)) = 0 AND (i != j)
Formal Mathematical property (FMP):
1 = tensor_selector ⁡ ( Kronecker_tensor , tuple ⁡ ( unit_Cartesian ⁡ ( 1 ) , unit_Cartesian ⁡ ( 2 ) , unit_Cartesian ⁡ ( 3 ) ) , contra_index ⁡ ( i ) , covar_index ⁡ ( i ) )
Signatures:
sts

[Next: metric_tensor] [Previous: tensor_selector] [Top]

metric_tensor

Role:
constant
Description:

This symbol represents the metric tensor, typically depicted using a lower case g. The metric tensor is a nondegenerate, symmetric bilinear form. It defines the ideas of leng th and angle in a metric space, the most common example being the Euclidean metric. The square of a differential length, ds*ds, is given by the bilinear product of the coordinate differentials, dx^i, with the metric tensor.

Commented Mathematical property (CMP):
ds^2 = sum_i sum_j (dx^i g_i_j dx^j)
Signatures:
sts

[Next: Levi-Civita] [Previous: Kronecker_tensor] [Top]

Levi-Civita

Role:
constant
Description:

This symbol represents the Levi-Civita alternating pseudo-tensor or permutation symbol. It's definition depends on the number of dimensions, d, of the space: it has as many indexes as there are dimensions in the space. It is totally antisymmetric, its value being: 1 for an even permutation of unequally valued indexes (e.g., (1,2,...,d)); -1 for an odd permutation of unequally valued indexes, and; 0 whenever two indexes take the same value.

Commented Mathematical property (CMP):
epsilon(i,j,k) = (i-j)(j-k)(k-i)/2, i,j,k in {1,2,3}
Signatures:
sts

[First: tuple] [Previous: metric_tensor] [Top]