#17445: Missing documentation of derivative operator/notation
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Reporter: schymans | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.5
Component: symbolics | Resolution:
Keywords: | Merged in:
Authors: schymans | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by nbruin):
OK, I do not know where it should go in the documentation for best
visibility (I'd think in "diff" somewhere), but the explanation should be
along the lines of:
Partial derivatives are represented in sage using differential operators,
referencing the position of the variable with respect to which the partial
derivative is taken. This means that for a function f in $r+1$ variables
we have
\[D[i_1,\ldots,i_n](f)(x_0,\ldots,x_r) = \left.\frac{\partial
f(t_0,\ldots,t_r)}{\partial t_{i_1}\cdots \partial t_{i_n}}
\right|_{t_0=x_0,\ldots,t_r=x_r}\]
An advantage of this notation is that it is clear which derivative is
taken, regardless of the names of the variables. For instance, if we have
{{{
sage: var("x,y,t")
sage: f(x,y)=function('f',x,y)
sage: g=f(x,y).diff(x,y); g
D[0, 1](f)(x, y)
sage: g.subs(x=1,y=1)
D[0, 1](f)(1, 1)
sage: g.subs(x=t,y=t+1)
D[0, 1](f)(t, t+1)
}}}
Note that in the last two lines are completely unambiguous using operator
notation, whereas Leibnitz notation would require the use of some
arbitrary explicit choice of auxiliary variable names.
--
Ticket URL: <http://trac.sagemath.org/ticket/17445#comment:1>
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