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calculus1.ocd
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<CD xmlns="http://www.openmath.org/OpenMathCD">
<CDComment>
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.
</CDComment>
<CDName>calculus1</CDName>
<CDBase>http://www.openmath.org/cd</CDBase>
<CDURL>http://www.openmath.org/cd/calculus1.ocd</CDURL>
<CDReviewDate>2014-04-01</CDReviewDate>
<CDStatus>official</CDStatus>
<CDDate>2009-04-01</CDDate>
<CDVersion>5</CDVersion>
<CDRevision>0</CDRevision>
<CDComment>
Author: OpenMath Consortium
SourceURL: https://github.com/OpenMath/CDs
</CDComment>
<Description>
This CD is intended to be compatible with the calculus operations
in Content MathML.
Integration is just for the univariate case and is either definite
or indefinite.
</Description>
<CDDefinition>
<Name>diff</Name>
<Role>application</Role>
<Description>
This symbol is used to express ordinary differentiation of a unary
function. The single argument is the unary function.
</Description>
<CMP>
diff(lambda y:a(y) + b(y))(x) = diff(lambda y:a(y))(x) + diff(lambda y:b(y))(x)
</CMP>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA>
<OMV name="a"/>
<OMV name="y"/>
</OMA>
<OMA>
<OMV name="b"/>
<OMV name="y"/>
</OMA>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
</OMA>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMV name="a"/>
<OMV name="y"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
</OMA>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMV name="b"/>
<OMV name="y"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<CMP>
diff(lambda y:a(y) * b(y))(x) = diff(lambda y:a(y))(x) * b(x) +
a(x) * diff(lambda y:b(y))(x)
</CMP>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMV name="a"/>
<OMV name="y"/>
</OMA>
<OMA>
<OMV name="b"/>
<OMV name="y"/>
</OMA>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
</OMA>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMV name="a"/>
<OMV name="y"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
</OMA>
<OMA>
<OMV name="b"/>
<OMV name="x"/>
</OMA>
</OMA>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMV name="a"/>
<OMV name="x"/>
</OMA>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMV name="b"/>
<OMV name="y"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
</OMA>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
<Example>
This represents the equation:
derivative(x + 1.0) = 1.0
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMV name="x"/>
<OMF dec="1.0"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="y"/>
</OMA>
<OMF dec="1.0"/>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name>nthdiff</Name>
<Role>application</Role>
<Description>
This symbol is used to express the nth-iterated ordinary differentiation of
a unary function. The first argument is n, and the second the unary function.
</Description>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMS name="nthdiff" cd="calculus1"/>
<OMS name="zero" cd="alg1"/>
<OMV name="f"/>
</OMA>
<OMV name="f"/>
</OMA>
</OMOBJ>
</FMP>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS name="eq" cd="relation1"/>
<OMA>
<OMS name="nthdiff" cd="calculus1"/>
<OMA>
<OMS name="plus" cd="arith1"/>
<OMV name="n"/>
<OMS name="one" cd="alg1"/>
</OMA>
<OMV name="f"/>
</OMA>
<OMA>
<OMS name="diff" cd="calculus1"/>
<OMA>
<OMS name="nthdiff" cd="calculus1"/>
<OMV name="n"/>
<OMV name="f"/>
</OMA>
</OMA>
</OMA>
</OMOBJ>
</FMP>
</CDDefinition>
<CDDefinition>
<Name>partialdiff</Name>
<Role>application</Role>
<Description>
This symbol is used to express partial differentiation of a function
of more than one variable. It has two arguments, the first is a list of
integers which index the variables of the function, the second is the function.
</Description>
<Example>
An example to represent the equation:
\partial^2{xyz}/ \partial{x}\partial{z} = y
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="partialdiff"/>
<OMA>
<OMS cd="list1" name="list"/>
<OMI>1</OMI>
<OMI>3</OMI>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
<OMV name="y"/>
<OMV name="z"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="x"/>
<OMV name="y"/>
<OMV name="z"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
<OMV name="y"/>
<OMV name="z"/>
</OMA>
<OMV name="y"/>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name>partialdiffdegree</Name>
<Role>application</Role>
<Description>
This symbol is used to express partial differentiation of a function
of more than one variable. It has three arguments, the first is a list of
integers which give the degrees by which the function is differentiated
by the corresponding variable. The second is the total degree (which should
therefore be the sum of the values in the first list, but may be
given symbolically). The third is the function.
</Description>
<Example>
An example to represent the equation:
\partial^2{xyz}/ \partial{x}\partial{z} = y
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="partialdiffdegree"/>
<OMA>
<OMS cd="list1" name="list"/>
<OMI>1</OMI>
<OMI>0</OMI>
<OMI>1</OMI>
</OMA>
<OMI>2</OMI>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
<OMV name="y"/>
<OMV name="z"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="times"/>
<OMV name="x"/>
<OMV name="y"/>
<OMV name="z"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
<OMV name="y"/>
<OMV name="z"/>
</OMA>
<OMV name="y"/>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name>int</Name>
<Role>application</Role>
<Description>
This symbol is used to represent indefinite integration of unary functions.
The argument is the unary function.
</Description>
<CMP>
application of integrate followed by differentiate is the identity
function, that is:
diff(lambda y:integral(lambda z:f(z))(y)) = f
</CMP>
<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="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMA>
<OMS cd="calculus1" name="int"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="z"/>
</OMBVAR>
<OMA>
<OMV name="f"/>
<OMV name="z"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="y"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="f"/>
</OMA>
</OMOBJ>
</FMP>
<Example>
An example which represents the equation:
integral(x +-> sin(x)) w.r.t. x = x +-> -cos(x)
<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="calculus1" name="int"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS cd="transc1" name="sin"/>
<OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="unary_minus"/>
<OMA>
<OMS cd="transc1" name="cos"/>
<OMV name="x"/>
</OMA>
</OMA>
</OMBIND>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
<CDDefinition>
<Name>defint</Name>
<Role>application</Role>
<Description>
This symbol is used to represent definite integration of unary
functions. It takes two arguments; the first being the range (e.g. a
set) of integration, and the second the function.
</Description>
<CMP>for all a,b | integral from a to b = -integral from b to a</CMP>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="a"/>
<OMV name="b"/>
</OMBVAR>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="calculus1" name="defint"/>
<OMA>
<OMS cd="interval1" name="ordered_interval"/>
<OMV name="a"/>
<OMV name="b"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
<OMA>
<OMS cd="arith1" name="unary_minus"/>
<OMA>
<OMS cd="calculus1" name="defint"/>
<OMA>
<OMS cd="interval1" name="ordered_interval"/>
<OMV name="b"/>
<OMV name="a"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMOBJ>
</FMP>
<CMP>
for all a < b < c | integral over [a,c] = integral over [a,b] +
integral over [b,c]
</CMP>
<FMP>
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMBIND>
<OMS cd="quant1" name="forall"/>
<OMBVAR>
<OMV name="a"/>
<OMV name="b"/>
<OMV name="c"/>
</OMBVAR>
<OMA>
<OMS cd="logic1" name="implies"/>
<OMA>
<OMS cd="logic1" name="and"/>
<OMA>
<OMS cd="relation1" name="lt"/>
<OMV name="a"/>
<OMV name="b"/>
</OMA>
<OMA>
<OMS cd="relation1" name="lt"/>
<OMV name="b"/>
<OMV name="c"/>
</OMA>
</OMA>
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMS cd="calculus1" name="defint"/>
<OMA>
<OMS cd="interval1" name="ordered_interval"/>
<OMV name="a"/>
<OMV name="c"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA>
<OMS cd="calculus1" name="defint"/>
<OMA>
<OMS cd="interval1" name="ordered_interval"/>
<OMV name="a"/>
<OMV name="b"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
<OMA>
<OMS cd="calculus1" name="defint"/>
<OMA>
<OMS cd="interval1" name="ordered_interval"/>
<OMV name="b"/>
<OMV name="c"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
</OMA>
</OMA>
</OMA>
</OMBIND>
</OMOBJ>
</FMP>
<Example>
An example to represent the definite integration of sin(x) between the points
-1.0 and 1.0.
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="calculus1" name="defint"/>
<OMA>
<OMS cd="interval1" name="ordered_interval"/>
<OMF dec="-1.0"/>
<OMF dec="1.0"/>
</OMA>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS cd="transc1" name="sin"/>
<OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
</OMOBJ>
</Example>
<Example>
An example to represent the definite integration of f(x), for x in the
set C:
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS name="defint" cd="calculus1"/>
<OMV name="C"/>
<OMBIND>
<OMS name="lambda" cd="fns1"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMV name="f"/>
<OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
</OMOBJ>
</Example>
</CDDefinition>
</CD>