I'm tring to do some variational calculus in Python with sympy and I have runned to a problem, and I do not quite understand the origin. I am using the Anaconda suite, Jupyter Notebook and IPython version 5.3.0, with python 3.6.
If I writte
from sympy import Funtion, symbols, integrate
from IPython.display import display
t, s = symbols('t s')
f = Function('f')
c = Function('c')
compute the following expressions (they will be the elements of the part that fails)
display(f(t, s))
display(c(t, s))
display(f(c(t, s)))
display(f(c(t, s), c(t, s).diff(t)))
display(integrate( f(t, s), t))
display(integrate( f(c(t, s)), t))
display(integrate( f(c(t, s).diff(t)), t))
display(integrate( f(c(t, s),c(s,t) ), t))
the expected (latex) expressions appear. I see that it can handle derivation and integration of multiple variables, but when I try to put some things togheter, as in
display(integrate( f(c(t, s), c(t, s).diff(t)), t))
the following error is raised
Can't calculate 1st derivative wrt Derivative(_x1, t).
On the other hand,
display(integrate( f(c(t,s), c(t,s).diff(t)), s))
displays the correct output. Any ideas how one might fix this?
Thank you in advance
EDIT : .doit() hasn't helped.
The error is preceeded by
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-21-66abd1a98aff> in <module>()
11 display(integrate( f(c(t,s).diff(t)), t))
12 display(integrate( f(c(t,s), c(s,t)), t))
---> 13 display(integrate( f(c(t,s), c(t,s).diff(t)), t))
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\integrals.py in integrate(*args, **kwargs)
1278 if isinstance(integral, Integral):
1279 return integral.doit(deep=False, meijerg=meijerg, conds=conds,
-> 1280 risch=risch, manual=manual)
1281 else:
1282 return integral
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\integrals.py in doit(self, **hints)
484 function, xab[0],
485 meijerg=meijerg1, risch=risch, manual=manual,
--> 486 conds=conds)
487 if antideriv is None and meijerg1 is True:
488 ret = try_meijerg(function, xab)
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\integrals.py in _eval_integral(self, f, x, meijerg, risch, manual, conds)
885 try:
886 if conds == 'piecewise':
--> 887 h = heurisch_wrapper(g, x, hints=[])
888 else:
889 h = heurisch(g, x, hints=[])
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch_wrapper(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations)
128
129 res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
--> 130 unnecessary_permutations)
131 if not isinstance(res, Basic):
132 return res
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations)
672 else:
673 if retries >= 0:
--> 674 result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations)
675
676 if result is not None:
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations)
672 else:
673 if retries >= 0:
--> 674 result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations)
675
676 if result is not None:
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations)
672 else:
673 if retries >= 0:
--> 674 result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations)
675
676 if result is not None:
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in heurisch(f, x, rewrite, hints, mappings, retries, degree_offset, unnecessary_permutations)
451 mapping = list(mapping)
452 mapping = mapping + unnecessary_permutations
--> 453 diffs = [ _substitute(dcache.get_diff(g)) for g in terms ]
454 denoms = [ g.as_numer_denom()[1] for g in diffs ]
455 if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V):
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in <listcomp>(.0)
451 mapping = list(mapping)
452 mapping = mapping + unnecessary_permutations
--> 453 diffs = [ _substitute(dcache.get_diff(g)) for g in terms ]
454 denoms = [ g.as_numer_denom()[1] for g in diffs ]
455 if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V):
....\Anaconda2\envs\Py3\lib\site-packages\sympy\integrals\heurisch.py in _substitute(expr)
446
447 def _substitute(expr):
--> 448 return expr.subs(mapping)
449
450 for mapping in mappings:
....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\basic.py in subs(self, *args, **kwargs)
900 rv = self
901 for old, new in sequence:
--> 902 rv = rv._subs(old, new, **kwargs)
903 if not isinstance(rv, Basic):
904 break
....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\cache.py in wrapper(*args, **kwargs)
91 def wrapper(*args, **kwargs):
92 try:
---> 93 retval = cfunc(*args, **kwargs)
94 except TypeError:
95 retval = func(*args, **kwargs)
....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\basic.py in _subs(self, old, new, **hints)
1014 rv = self._eval_subs(old, new)
1015 if rv is None:
-> 1016 rv = fallback(self, old, new)
1017 return rv
1018
....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\basic.py in fallback(self, old, new)
986 if not hasattr(arg, '_eval_subs'):
987 continue
--> 988 arg = arg._subs(old, new, **hints)
989 if not _aresame(arg, args[i]):
990 hit = True
....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\cache.py in wrapper(*args, **kwargs)
91 def wrapper(*args, **kwargs):
92 try:
---> 93 retval = cfunc(*args, **kwargs)
94 except TypeError:
95 retval = func(*args, **kwargs)
....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\basic.py in _subs(self, old, new, **hints)
1012 return new
1013
-> 1014 rv = self._eval_subs(old, new)
1015 if rv is None:
1016 rv = fallback(self, old, new)
....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\function.py in _eval_subs(self, old, new)
1340 variables = self_vars_front + self_vars
1341 return Derivative(new, *variables)
-> 1342 return Derivative(*(x._subs(old, new) for x in self.args))
1343
1344 def _eval_lseries(self, x, logx):
....\Anaconda2\envs\Py3\lib\site-packages\sympy\core\function.py in __new__(cls, expr, *variables, **assumptions)
1068 ordinal = 'st' if last_digit == 1 else 'nd' if last_digit == 2 else 'rd' if last_digit == 3 else 'th'
1069 raise ValueError(filldedent('''
-> 1070 Can\'t calculate %s%s derivative wrt %s.''' % (count, ordinal, v)))
1071
1072 if all_zero and not count == 0:
ValueError:
Can't calculate 1st derivative wrt Derivative(_x1, t).
I would use Integral for displaying integrals, which only constructs them and does not attempt to compute them (unless your perform a doit()). The following works
display(Integral( f(c(t, s), c(t, s).diff(t)), t))