Mike Shulman's profile - activity

Ohh... is that true even if I evaluate the first example and then edit it to change it into the second? (Not a very helpful error message, in that case...)

Following the sagemanifolds tutorial I can make $\mathbb{R}^2$ with rectangular and polar coordinates:

R2 = Manifold(2, 'R2', r'\mathbb{R}^2', start_index=1)
Rect.<x,y> = R2.chart()
U = R2.open_subset('U', coord_def={Rect : (y != 0, x < 0)})
RectU = Rect.restrict(U)
Polar.<r,th> = U.chart(r'r:(0,+oo) th:(0,2*pi):\theta')

This uses the range $(0,2\pi)$ for $\theta$, excluding the positive $x$-axis. But if I try to use instead the range $(-\pi,\pi)$ for $\theta$, excluding the negative $x$-axis:

R2 = Manifold(2, 'R2', r'\mathbb{R}^2', start_index=1)
Rect.<x,y> = R2.chart()
U = R2.open_subset('U', coord_def={Rect : (y != 0, x > 0)})
RectU = Rect.restrict(U)
Polar.<r,th> = U.chart(r'r:(0,+oo) th:(-pi,pi):\theta')

SageMathCloud gives me "ValueError: Assumption is redundant". But strangely, SageMathCell doesn't complain at all. What is the problem?

Okay, I guess I see what the developers were thinking, even if I don't agree with it. (Wouldn't "coproduct" be a more appropriate term?)

I'm trying to understand this. Is the expression tree really necessary? It seems like a wrapper around a callable symbolic expression that knows how to reset the variables as needed might be sufficient for what I want. For instance, could you also overload function application, addition, etc.?

Hmm... if it were really consistent about behaving this way, then I would expect `f+h` to be a type error, since you can't add elements of different rings.

Thanks! Why do you refer to defining mathematical functions as "adding more semantics"?

Is there a way to define a symbolic function that can (e.g.) be differentiated, but doesn't remember the name of its input variable(s)? For instance, consider:

sage: f(x) = x^2
sage: g(x) = x^2
sage: h(t) = t^2

Mathematically, f, g, and h, should all be the same function. However, Sage doesn't think so:

sage: f+g
x |--> 2*x^2
sage: f+h
(t, x) |--> t^2 + x^2

I guess that this is happening because a "function" defined with f(x)=x^2 is actually just a symbolic expression equipped with an ordering on its variables, rather than what a mathematician would call a "function". Is there a way to define an actual mathematical function?