👁 OpenMath

http://www.openmath.org/cd/veccalc1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

veccalc1.ocd

CD as XML Encoded OpenMath:

veccalc1.omcd

Defines:

Laplacian, curl, divergence, grad

Date:

2004-03-30

Version:

3 (Revision 1)

Review Date:

2006-03-30

Status:

official

 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
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 Author: OpenMath Consortium
 SourceURL: https://github.com/OpenMath/CDs
 

This CD contains symbols to represent functions which are concerned with vector calculus.

Role:

application

Description:

This symbol is used to represent the divergence function. It takes one argument which should be a vector of scalar valued functions, intended to represent a vector valued function and returns a scalar value. It should satisfy the defining relation: divergence(F) = \partial(F_(x_1))/\partial(x_1) + ... + \partial(F_(x_n))/\partial(x_n)

Commented Mathematical property (CMP):

divergence(F) = \partial(F_(x_1))/\partial(x_1) + ... + \partial(F_(x_n))/\partial(x_n)

Signatures:

sts

Role:

application

Description:

This symbol is used to represent the grad function. It takes one argument which should be a scalar valued function and returns a vector of functions. It should satisfy the defining relation: grad(F) = (\partial(F)/\partial(x_1), ... ,\partial(F)/partial(x_n))

Commented Mathematical property (CMP):

grad(F) = (\partial(F)/\partial(x_1), ... ,\partial(F)/partial(x_n))

Signatures:

sts

Role:

application

Description:

This symbol is used to represent the curl function. It takes one argument which should be a vector of scalar valued functions, intended to represent a vector valued function and returns a vector of functions. It should satisfy the defining relation: curl(F) = i X \partial(F)/\partial(x) + j X \partial(F)/\partial(y) + j X \partial(F)/\partial(Z) where i,j,k are the unit vectors corresponding to the x,y,z axes respectively and the multiplication X is cross multiplication.

Commented Mathematical property (CMP):

curl(F) = i X \partial(F)/\partial(x) + j X \partial(F)/\partial(y) + j X \partial(F)/\partial(Z)

Formal Mathematical property (FMP):

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
 <OMA>
 <OMS cd="relation1" name="eq"/>
 <OMA>
 <OMS cd="veccalc1" name="curl"/>
 <OMV name="F"/>
 </OMA>
 <OMA>
 <OMS cd="arith1" name="plus"/>
 <OMA>
 <OMS cd="linalg1" name="vectorproduct"/>
	<OMA>
	 <OMS cd="linalg2" name="vector"/>
	 <OMI> 1 </OMI>
	 <OMI> 0 </OMI>
	 <OMI> 0 </OMI>
	</OMA>
	<OMA>
	 <OMS cd="calculus1" name="partialdiff"/>
	 <OMA>
	 <OMS cd="list1" name="list"/>
	 <OMI> 1 </OMI>
	 </OMA>
	 <OMV name="F"/>
	</OMA>
 </OMA>
 <OMA>
 <OMS cd="linalg1" name="vectorproduct"/>
	<OMA>
	 <OMS cd="linalg2" name="vector"/>
	 <OMI> 0 </OMI>
	 <OMI> 1 </OMI>
	 <OMI> 0 </OMI>
	</OMA>
	<OMA>
	 <OMS cd="calculus1" name="partialdiff"/>
	 <OMA>
	 <OMS cd="list1" name="list"/>
	 <OMI> 2 </OMI>
	 </OMA>
	 <OMV name="F"/>
	</OMA>
 </OMA>
 <OMA>
 <OMS cd="linalg1" name="vectorproduct"/>
	<OMA>
	 <OMS cd="linalg2" name="vector"/>
	 <OMI> 0 </OMI>
	 <OMI> 0 </OMI>
	 <OMI> 1 </OMI>
	</OMA>
	<OMA>
	 <OMS cd="calculus1" name="partialdiff"/>
	 <OMA>
	 <OMS cd="list1" name="list"/>
	 <OMI> 3 </OMI>
	 </OMA>
	 <OMV name="F"/>
	</OMA>
 </OMA>
 </OMA>
 </OMA>
</OMOBJ>

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
 <apply><csymbol cd="veccalc1">curl</csymbol><ci>F</ci></apply>
 <apply><csymbol cd="arith1">plus</csymbol>
 <apply><csymbol cd="linalg1">vectorproduct</csymbol>
 <apply><csymbol cd="linalg2">vector</csymbol>
 <cn type="integer">1</cn>
 <cn type="integer">0</cn>
 <cn type="integer">0</cn>
 </apply>
 <apply><csymbol cd="calculus1">partialdiff</csymbol>
 <apply><csymbol cd="list1">list</csymbol><cn type="integer">1</cn></apply>
 <ci>F</ci>
 </apply>
 </apply>
 <apply><csymbol cd="linalg1">vectorproduct</csymbol>
 <apply><csymbol cd="linalg2">vector</csymbol>
 <cn type="integer">0</cn>
 <cn type="integer">1</cn>
 <cn type="integer">0</cn>
 </apply>
 <apply><csymbol cd="calculus1">partialdiff</csymbol>
 <apply><csymbol cd="list1">list</csymbol><cn type="integer">2</cn></apply>
 <ci>F</ci>
 </apply>
 </apply>
 <apply><csymbol cd="linalg1">vectorproduct</csymbol>
 <apply><csymbol cd="linalg2">vector</csymbol>
 <cn type="integer">0</cn>
 <cn type="integer">0</cn>
 <cn type="integer">1</cn>
 </apply>
 <apply><csymbol cd="calculus1">partialdiff</csymbol>
 <apply><csymbol cd="list1">list</csymbol><cn type="integer">3</cn></apply>
 <ci>F</ci>
 </apply>
 </apply>
 </apply>
 </apply>
</math>

eq (curl ( F) , plus (vectorproduct (vector ( 1 , 0 , 0 ) , partialdiff (list ( 1 ) , F) ) , vectorproduct (vector ( 0 , 1 , 0 ) , partialdiff (list ( 2 ) , F) ) , vectorproduct (vector ( 0 , 0 , 1 ) , partialdiff (list ( 3 ) , F) ) ) )

veccalc1.curl($F) = linalg1.vectorproduct(linalg2.vector(1, 0, 0), calculus1.partialdiff([1], $F)) + linalg1.vectorproduct(linalg2.vector(0, 1, 0), calculus1.partialdiff([2], $F)) + linalg1.vectorproduct(linalg2.vector(0, 0, 1), calculus1.partialdiff([3], $F))

Signatures:

sts

Role:

application

Description:

This symbol is used to represent the laplacian function. It takes one argument which should be a vector of scalar valued functions, intended to represent a vector valued function and returns a vector of functions. It should satisfy the defining relation: laplacian(F) = \partial^2(F)/\partial(x_1)^2 + ... + \partial^2(F)/\partial(x_n)^2

Commented Mathematical property (CMP):

laplacian(F) = \partial^2(F)/\partial(x_1)^2 + ... + \partial^2(F)/\partial(x_n)^2

Signatures:

sts