I published a gist on how to get the gradient in any coordinate system, given the backward transformation to rectangular coordinates.
I was also trying to write the code to get the divergence, but I ran into problems with sympy's simplify( ) function. https://gist.github.com/Upabjojr/34f75ce115f1cf5a6afc This gist defines a function *get_grad( ... )* which returns the gradient, it requires as an input a lambda function (backwards transformation). Example: from_spherical = lambda r, theta, phi: (r*sin(theta)*cos(phi), r*sin(theta)*sin(phi), r*cos(theta)) grad = get_grad(from_spherical) Notice that the function *from_spherical* is simply the easiest way to define the spherical coordinates. I get the following output: ⎡ ∂_θ │sin(θ)│⋅∂ᵩ⎤ ⎢∂ᵣ, ───, ───────────⎥ ⎢ r 2 ⎥ ⎣ r⋅sin (θ) ⎦ SymPy doesn't know that in *theta*'s range of values its *sin(theta)* is always positive, this is the reason why it does not get simplified. I have only tried the spherical coordinates, but this should work with any coordinate transformation in three dimensions. It should not be too complicated to generalize this to any dimension. That's why I would like coordinate systems to be instances of a class, and not classes themselves. It is really uncomfortable to define a new class for every new coordinate system. On Thursday, November 27, 2014 7:17:49 AM UTC+1, Aaron Meurer wrote: > > On Tue, Nov 25, 2014 at 1:58 AM, Francesco Bonazzi > <franz....@gmail.com <javascript:>> wrote: > > > > > > On Wednesday, November 19, 2014 11:17:24 PM UTC+1, Jason Moore wrote: > >> > >> Sachin Joglekar developed the vector package this past summer. I was > his > >> GSoC mentor on the project. You can check out this implemented of the > >> Cartesian coordinate system: > >> > >> https://github.com/sympy/sympy/blob/master/sympy/vector/coordsysrect.py > > > > > > In sympy.diffgeom coordinate systems are defined as instances of the > > CoordSystem class, once more coordinate systems are defined, the > > transformation rules are set using the connect_to method: > > > > In [1]: from sympy.diffgeom import * > > > > In [2]: r, theta, phi = symbols('r theta phi', positive=True) > > > > In [4]: M = Manifold('M', 3) > > > > In [5]: P = Patch('P', M) > > > > In [7]: rect = CoordSystem('rect', P, ['x', 'y', 'z']) > > > > In [8]: spherical = CoordSystem('spherical', P, ['r', 'theta', 'phi']) > > > > In [9]: spherical.connect_to(rect, [r, theta, phi], > [r*sin(phi)*cos(theta), > > r*sin(phi)*sin(theta), r*cos(phi)], inverse=False) > > > > > > I put inverse=False in the last line, because SymPy's solvers are unable > to > > reverse the spherical-to-rect transformation. > > > > Manifold and Patch are some boilerplate, Manifold specifies the space > you're > > considering, while Patch is a connected subspace of the manifold. Patch > is > > used because on some manifolds there does not exist a homeomorphism to a > > coordinate system (e.g. on the sphere). > > > > I think that some algorithmic code could be shared between the vector > and > > the diffgeom modules. > > +1 to that, especially if we can avoid hard-coding things and just let > solve() compute things (similar to how sympy.stats works). > > Aaron Meurer > > > > > -- > > You received this message because you are subscribed to the Google > Groups > > "sympy" group. > > To unsubscribe from this group and stop receiving emails from it, send > an > > email to sympy+un...@googlegroups.com <javascript:>. > > To post to this group, send email to sy...@googlegroups.com > <javascript:>. > > Visit this group at http://groups.google.com/group/sympy. > > To view this discussion on the web visit > > > https://groups.google.com/d/msgid/sympy/7341823f-1172-4ecb-b322-f52f115d44f4%40googlegroups.com. > > > > > > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/14ce54ae-897d-41bd-9a35-36e9a0b11822%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
