OpenMath Content Dictionary: calculus1

Canonical URL:
http://www.openmath.org/cd/calculus1.ocd
CD File:
calculus1.ocd
CD as XML Encoded OpenMath:
calculus1.omcd
Defines:
defint, diff, int, partialdiff
Date:
2001-03-12
Version:
2
Review Date:
2003-04-01
Status:
official
Uses CD:
arith1, fns1, interval1, list1, transc1, relation1, quant1, logic1

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

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):
<OMOBJ>
  <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>

eq (diff (lambda [ y ] . (plus ( a ( y) , b ( y) ) ) ) ( x) , plus (diff (lambda [ y ] . ( a ( y) ) ) ( x) , diff (lambda [ y ] . ( b ( y) ) ) ( x) ) )

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):
<OMOBJ>
  <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>

eq (diff (lambda [ y ] . (times ( a ( y) , b ( y) ) ) ) ( x) , plus (times (diff (lambda [ y ] . ( a ( y) ) ) ( x) , b ( x) ) , times ( a ( x) , diff (lambda [ y ] . ( b ( y) ) ) ( x) ) ) )

Example:
This represents the equation: derivative(x + 1.0) = 1.0
<OMOBJ>
  <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>

eq (diff (lambda [ x ] . (plus ( x, 1.0 ) ) ) ( y) , 1.0 )

Signatures:
sts


[Next: partialdiff] [Last: defint] [Top]

partialdiff

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
<OMOBJ>
<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="y"/>
  </OMA>
  <OMV name="y"/>
</OMA>
</OMOBJ>

eq (partialdiff (list ( 1 , 3 ) , lambda [ x y z ] . (times ( x, y, z) ) ) ( y) , y)

Signatures:
sts


[Next: int] [Previous: diff] [Top]

int

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: lambda x:diff(lambda y:integral(lambda z:f(z))(y))(x) = lambda x:f(x)
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="x"/>
      </OMBVAR>
      <OMA>
        <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="x"/>
      </OMA>
    </OMBIND>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="x"/>
      </OMBVAR>
      <OMA>
        <OMV name="f"/>
	<OMV name="x"/>
      </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

eq (lambda [ x ] . (diff (lambda [ y ] . (int (lambda [ z ] . ( f ( z) ) ) ( y) ) ) ( x) ) , lambda [ x ] . ( f ( x) ) )

Example:
An example which represents the equation: integral(x +-> sin(x)) w.r.t. x = x +-> -cos(x)
<OMOBJ>
  <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>

eq (int (lambda [ x ] . (sin ( x) ) ) , lambda [ x ] . (unary_minus (cos ( x) ) ) )

Signatures:
sts


[Next: defint] [Previous: partialdiff] [Top]

defint

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):
<OMOBJ>
  <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="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="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>

forall [ a b ] . (eq (defint (interval ( a, b) , lambda [ x ] . ( f ( x) ) ) , unary_minus (defint (interval ( b, a) , lambda [ x ] . ( f ( x) ) ) ) ) )

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):
<OMOBJ>
  <OMBIND>
    <OMS cd="quant1" name="forall"/>
    <OMBVAR>
      <OMV name="a"/>
      <OMV name="b"/>
      <OMV name="c"/>
    </OMBVAR>
    <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>
        <OMS cd="relation1" name="eq"/>
	<OMA>
	  <OMS cd="calculus1" name="defint"/>
	  <OMA>
	    <OMS cd="interval1" name="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="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="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>

forall [ a b c ] . (and (lt ( a, b) , lt ( b, c) , eq (defint (interval ( a, c) , lambda [ x ] . ( f ( x) ) ) , plus (defint (interval ( a, b) , lambda [ x ] . ( f ( x) ) ) , defint (interval ( b, c) , lambda [ x ] . ( f ( x) ) ) ) ) ) )

Example:
An example to represent the definite integration of sin(x) between the points -1.0 and 1.0.
<OMOBJ>
  <OMA>
    <OMS cd="calculus1" name="defint"/>
    <OMA>
      <OMS cd="interval1" name="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>

defint (interval ( -1.0 , 1.0 ) , lambda [ x ] . (sin ( x) ) )

Example:
An example to represent the definite integration of f(x), for x in the set C:
<OMOBJ>
  <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>

defint ( C, lambda [ x ] . ( f ( x) ) )

Signatures:
sts


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