Sympy中导数的积分函数

Question

我正在尝试用Python和sympy做一些变分演算，我遇到了一个问题，我不太明白它的起源。我使用的是Anaconda套件、Jupyter笔记本和IPython版本5.3.0和python3.6。在

如果我写

from sympy import Funtion, symbols, integrate
from IPython.display import display

t, s = symbols('t s')
f = Function('f')
c = Function('c')

计算以下表达式（它们将是失败部分的元素）

^{pr2}$

出现预期的（乳胶）表达式。我看到它可以处理多个变量的求导和积分，但是当我试图把一些东西放在一起时，比如

display(integrate( f(c(t, s), c(t, s).diff(t)), t))

引发以下错误

Can't calculate 1st derivative wrt Derivative(_x1, t).

另一方面

display(integrate( f(c(t,s), c(t,s).diff(t)), s))

显示正确的输出。有什么办法解决这个问题吗？在

提前谢谢你

编辑：.doit（）没有帮助。在

错误由

---------------------------------------------------------------------------
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).

在

Answer 1

我将使用Integral来显示积分，它只构造积分，不尝试计算积分（除非执行doit()）。以下工作

display(Integral( f(c(t, s), c(t, s).diff(t)), t))