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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.
divergenceThis 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)
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))
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.
<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))
curl ( F ) = ( 1 , 0 , 0 ) × ∂ ( ( 1 ) ) + ( 0 , 1 , 0 ) × ∂ ( ( 2 ) ) + ( 0 , 0 , 1 ) × ∂ ( ( 3 ) )
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
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