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# OpenMath Content Dictionary: calculus1

Canonical URL:
http://www.openmath.org/cd/calculus1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
calculus1.ocd
CD as XML Encoded OpenMath:
calculus1.omcd
Defines:
defint, diff, int, nthdiff, partialdiff, partialdiffdegree
Date:
2009-04-01
Version:
5
Review Date:
2014-04-01
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.
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.
society at http://www.openmath.org.


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.

## diff

Role:
application
Description:

This symbol is used to express ordinary differentiation of a unary function. The single argument is the unary function.

Commented Mathematical property (CMP):
diff(lambda y:a(y) + b(y))(x) = diff(lambda y:a(y))(x) + diff(lambda y:b(y))(x)
Formal Mathematical property (FMP):
$\left(\frac{d}{dy}\left(a\left(y\right)+b\left(y\right)\right)\right)\left(x\right)=\left(\frac{d}{dy}\left(a\left(y\right)\right)\right)\left(x\right)+\left(\frac{d}{dy}\left(b\left(y\right)\right)\right)\left(x\right)$
Commented Mathematical property (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)
Formal Mathematical property (FMP):
$\left(\frac{d}{dy}\left(a\left(y\right)b\left(y\right)\right)\right)\left(x\right)=\left(\frac{d}{dy}\left(a\left(y\right)\right)\right)\left(x\right)b\left(x\right)+a\left(x\right)\left(\frac{d}{dy}\left(b\left(y\right)\right)\right)\left(x\right)$
Example:
This represents the equation: derivative(x + 1.0) = 1.0
$\left(\frac{d}{dx}\left(x+1.0\right)\right)\left(y\right)=1.0$
Signatures:
sts

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## nthdiff

Role:
application
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.

Formal Mathematical property (FMP):
${D}^{0}\left(f\right)=f$
Formal Mathematical property (FMP):
${D}^{n+1}\left(f\right)=D\left({D}^{n}\left(f\right)\right)$
Signatures:
sts

 [Next: partialdiff] [Previous: diff] [Top]

## partialdiff

Role:
application
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.

Example:
An example to represent the equation: \partial^2{xyz}/ \partial{x}\partial{z} = y
$\left(\frac{{\partial }^{2}}{\partial x\partial z}\left(xyz\right)\right)\left(x,y,z\right)=y$
Signatures:
sts

 [Next: partialdiffdegree] [Previous: nthdiff] [Top]

## partialdiffdegree

Role:
application
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.

Example:
An example to represent the equation: \partial^2{xyz}/ \partial{x}\partial{z} = y
$\frac{{\partial }^{2}}{\partial x\partial z}\left(xyz\right)=y$
Signatures:
sts

 [Next: int] [Previous: partialdiff] [Top]

## int

Role:
application
Description:

This symbol is used to represent indefinite integration of unary functions. The argument is the unary function.

Commented Mathematical property (CMP):
application of integrate followed by differentiate is the identity function, that is: diff(lambda y:integral(lambda z:f(z))(y)) = f
Formal Mathematical property (FMP):
$\frac{d}{dy}\left(\left(\int f\left(z\right)dz\right)\left(y\right)\right)=f$
Example:
An example which represents the equation: integral(x +-> sin(x)) w.r.t. x = x +-> -cos(x)
$\int \mathrm{sin}\left(x\right)dx=\lambda x.-\mathrm{cos}\left(x\right)$
Signatures:
sts

 [Next: defint] [Previous: partialdiffdegree] [Top]

## defint

Role:
application
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.

Commented Mathematical property (CMP):
for all a,b | integral from a to b = -integral from b to a
Formal Mathematical property (FMP):
$\forall a,b.\underset{a}{\overset{b}{\int }}f\left(x\right)dx=-\underset{b}{\overset{a}{\int }}f\left(x\right)dx$
Commented Mathematical property (CMP):
for all a < b < c | integral over [a,c] = integral over [a,b] + integral over [b,c]
Formal Mathematical property (FMP):
$\forall a,b,c.a
Example:
An example to represent the definite integration of sin(x) between the points -1.0 and 1.0.
$\underset{-1.0}{\overset{1.0}{\int }}\mathrm{sin}\left(x\right)dx$
Example:
An example to represent the definite integration of f(x), for x in the set C:
${\int }_{C}f\left(x\right)dx$
Signatures:
sts

 [First: diff] [Previous: int] [Top]

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