Comment #19 on issue 1620 by [email protected]: Allow derivatives of unknown
functions evaluated at a point
http://code.google.com/p/sympy/issues/detail?id=1620
Aaron wrote:
We can still make an operator, D, which acts like D[1, 2](f)(x, y), but
also allow the syntax D(f, orders=(1, 2))(x, y), and it would satisfy the
invariant.
Do I right understand that you talk about mathematical operators?
If we consider the derivative as a mathematical diff operator than this
operator contain (and define) all of necessary into itself (orders and
differentiation variables) and only then apply on some expression (function
in particular).
Those diff operators can be added and multiplicated as expressions.
E.g. (notations are rough)
>>> dx = Derivative(x) # create operator
>>> d[f] # apply operator to function
f'(x) # some convinient notation
>>> Derivative(x)*Derivative(x)
Derivative(x, 2) # operator order 2
>>> dy2 = Derivative(y, 2) # create operator order 2
>>> dx + dy2
d/dx + d^2/dy^2
>>> (dx + dy2)[f + x + sin(y)] # apply complex operator to expression
f_x(x) + 1 + f_yy(y) - sin(y) # may be some other convenient notation
>>> exp(Derivative(y, 2, 0))[f] # create operator order 2 at point y=0
f(0) + f'(0) + f''(0)/2! + ... (Taylor series)
# nested function mechanism
>>> Derivative(x, x0=a)[f(g)]
Derivative(x, x0=g(a))[f] * Derivative(x, x0=a)[g]
>>> print _
f'(g(a)) * g'(a) # notation is very rough, because it implyes that g' is
different by x.
I want to say that if we consider operator, then it act on the function (on
the function which is a terminal of usage and may be with unknown
arguments).
Sofor this case I would replace `D[1, 2](f)(x, y)` or `D(f, orders=(1,
2))(x, y)` by
`D[1, 2](x, y) (f)` or `D(orders=(1, 2), varibales=(x, y), values=(x0, y0))
(f)` respectably in your examples.
But if we know the function's arguments(apply diff to function on its
argument), then we can consider your case Diff[1,2] f (x, y) syntax.
All above only a remark (but it might help us to look at things in new)
Also the f(x0) ("value of function at point x0") can be considered also as
functional operator (applied to function).
http://reference.wolfram.com/mathematica/tutorial/FunctionalOperationsOverview.html
In general, I still tend to work-out those complex issues (related with
function, abstract function and derivatives) with the help of wiki page -
with requirements, proposals, variants of use-cases and possible plans of
realization and. Sorry, It is very hard for me to read all (tangled)
comments in various pages and various commits without vision. At least as
some intermediate conclusion (result).
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