2012/5/25 krastanov.ste...@gmail.com <krastanov.ste...@gmail.com>:
>>> An example from the (upcoming) differential geometry module:
>>>
>>> p is a point with x=a and y=b
>>>
>>> rect is the Cartesian coordinate system
>>> rect.x (y) are ScalarFields taking a point and returning the x (y)
>>> coordinate
>>> polar is the polar coordinate system with r and theta as basis fields
>>>
>>> rect.d_dx is the unit vector along x
>>> rect.d_dy, polar.d_dr, polar.d_dtheta are the other unit vectors
>>>
>>> now the examples
>>>
>>> Creating scalar fields without the need for special scalar field
>>> subclass of Expr:
>>>
>>> rect.x(p) = a
>>> (rect.x+rect.y**2) = a+b**2
>>>
>>> Creating vector fields (i.e. directional derivatives) without the need
>>> for subclasses of Expr:
>>>
>>> (polar.d_dr+polar.d_theta) ( rect.x + 3 ) (p) = cos(theta(p)) +
>>> r(p)*sin(theta(p))
>>> # I was to lazy to calculate r(p) and theta(p)
>>>
>>> This can be done with a helper function (e.g. `apply(field, point)`)
>>> but the current approach seems nicer.
>>
>> If you do that, you won't have any simple way of recognising
>> ScalarFields, or things-like-rect.d_dx (what do you call them?), which
>> seems rather inconvenient. Worse than that, objects like (rect.x *
>> rect.d_dx) cannot possibly work satisfactorily.
>
> Why would I want to recognize them? It is all about interface, not
> what class they are.
You always end up needing to recognise what kind of object you're
handling, for some reason or other. Making it impossible in principle
is a bad idea that will come back to bite you.
>
> The other example ( x * d_dx ) is nonsensical mathematically. If a
> user has a formalism in which it makes sense, he is free to use it,
> that is all.
IIUC, this is the "dequantised" version of X P, which is a perfectly
well-defined quantum operator. "Obviously", (x * d_dx)(f) should
return x * f'(x), but I see no way to make it work in your design.
> Off topic: d_dx is the unit vector along x. It needs better name. (in
> latex it is \frac{\part}{\part x})
I'm rather confused by your explanations. You seem to alternate between
describing it as a simple vector (but in which space?) and a differential
operator. These seem to be completely different things to me.
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