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. calculus1 http://www.openmath.org/cd http://www.openmath.org/cd/calculus1.ocd 2006-03-30 official 2004-06-01 4 1 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 application This symbol is used to express ordinary differentiation of a unary function. The single argument is the unary function. diff(lambda y:a(y) + b(y))(x) = diff(lambda y:a(y))(x) + diff(lambda y:b(y))(x) diff(lambda y:a(y) * b(y))(x) = diff(lambda y:a(y))(x) * b(x) + a(x) * diff(lambda y:b(y))(x) This represents the equation: derivative(x + 1.0) = 1.0 nthdiff application 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. partialdiff application 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. An example to represent the equation: \partial^2{xyz}/ \partial{x}\partial{z} = y 1 3 int application This symbol is used to represent indefinite integration of unary functions. The argument is the unary function. application of integrate followed by differentiate is the identity function, that is: diff(lambda y:integral(lambda z:f(z))(y)) = f An example which represents the equation: integral(x +-> sin(x)) w.r.t. x = x +-> -cos(x) defint application 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. for all a,b | integral from a to b = -integral from b to a for all a < b < c | integral over [a,c] = integral over [a,b] + integral over [b,c] An example to represent the definite integration of sin(x) between the points -1.0 and 1.0. An example to represent the definite integration of f(x), for x in the set C: