I'm having problems defining a metric using sympy's diffgeom package, where the metric depends on a function f(x,y).
I get the error ValueError: Can't calculate 1st derivative wrt x.
import sympy as sym
from sympy.diffgeom import Manifold, Patch, CoordSystem
m = Manifold("M",2)
patch = Patch("P",m)
cartesian = CoordSystem("cartesian", patch, ["x", "y"])
x, y = cartesian.coord_functions()
f = sym.Function('f')
xi = sym.Symbol('xi')
g = sym.Matrix([
[ xi + sym.diff(f,x)**2 , sym.diff(f,x) * sym.diff(f,y) ],
[ sym.diff(f,x) * sym.diff(f,y) , xi + sym.diff(f,y)**2 ]
])
I have a feeling it's because of how x and y are defined, but I haven't been able to figure it out.
Indeed, differentiation with respect to these x and y (objects of class sympy.diffgeom.diffgeom.BaseScalarField) is not supported. You can see this by accessing the internal property _diff_wrt which indicates if something can be a thing with respect to which to differentiate.
>>> x._diff_wrt
False
Do those derivative (with respect to a scalar field) make mathematical sense here? I'm not sure.
An additional issue is that SymPy does not differentiate functions, so
f = Function('f')
diff(f, x)
is always an error. SymPy can differentiate expressions, e.g., diff(f(x, y), x).
Aside: diff can be used as a method of an expression, which in your case would result in shorter code, like f(x, y).diff(x).