计算科学 - 如何在 SymPy 中获得二阶张量本身的行列式和逆的导数？ - 吾爱随笔录

我有一个二阶张量，我需要计算它的行列式和逆本身的导数。方程如下：

\frac{\partial d e t (F)}{\partial F_{i j}} = d e t (F) F_{j i}^{- 1}

$\frac{\partial \, det(\mathbf{F})}{\partial F_{ij}} = det(\mathbf{F}) \, F^{-1}_{ji}$

\frac{\partial F_{k l}^{- 1}}{\partial F_{i j}} = - F_{k i}^{- 1} F_{j l}^{- 1}

$\frac{\partial \, F_{kl}^{-1}}{\partial F_{ij}} = - F^{-1}_{ki} \, F^{-1}_{jl}$

这些是矩阵食谱中的方程 (49) 和 (60) 。

我已经使用 SymPy 大约一周了。我可以得到F wrt 本身的导数，但不知道如何得到它的行列式和逆的导数。作为参考，我将代码和输出与错误消息一起发布。

Python代码

from sympy import *

i = tensor.Idx('i',3)
j = tensor.Idx('j',3)
k = tensor.Idx('k',3)
l = tensor.Idx('l',3)

F = MatrixSymbol('F', 3, 3)

print("Derivative of F wrt F")
print("---------------------")
print(diff(F[k,l], F[i,j]))
print("\n")


J = det(F)
print("Derivative of det(F) wrt F")
print("--------------------------")
print(diff(J, F[i,j]))
print("\n")

print("Derivative of inv(F) wrt F")
print("--------------------------")
Finv = Inverse(F)
print(Finv[i,j])
print(diff(Finv[k,l], F[i,j]))
print("\n")

输出

Derivative of F wrt F
---------------------------
KroneckerDelta(i, k)*KroneckerDelta(j, l)

Derivative of det(F) wrt F
-------------------------------
Derivative(Determinant(F), F[i, j])

Derivative of inv(F) wrt F
-------------------------------
Traceback (most recent call last):
  File "matdiffinverse.py", line 25, in <module>
    print(Finv[i,j])
  File "/usr/lib/python3/dist-   packages/sympy/matrices/expressions/matexpr.py", line 248, in __getitem__
return self._entry(i, j)
  File "/usr/lib/python3/dist-   packages/sympy/matrices/expressions/matpow.py", line 46, in _entry
    raise NotImplementedError(("(%d, %d) entry" % (int(i), int(j)))
  File "/usr/lib/python3/dist-packages/sympy/core/expr.py", line 207, in __int__
    raise TypeError("can't convert symbols to int")
TypeError: can't convert symbols to int