在Sympy中集成衍生產品的功能

Question

我想用sympy在Python中做一些變分微積分，並且我遇到了一個問題，並且我不太明白原點。我使用的是Anaconda套件，Jupyter Notebook和IPython版本5.3.0，以及python 3.6。

如果我被軟件寫

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

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

計算下面的表達式（它們將是失敗的部分的元素）的預期的（膠乳）表達式出現

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)) 

。我看到，它可以處理推導和多變量的整合，但是當我嘗試把一些事情togheter，如

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

引發以下錯誤

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

在另一方面，

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

顯示正確的輸出。任何想法如何解決這個問題？

謝謝你提前

編輯：.doit（）沒有幫助。

錯誤是由

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

Answer 1

preceeded我會用Integral顯示積分，只有構建他們不嘗試計算它們（除非你執行doit()）。以下作品

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