Partial Derivatives in sympy



In [1]:

    
import sympy



In [2]:

    
x, u = sympy.symbols('x u', real=True)



In [3]:

    
U = sympy.Function('U')(x,u)



In [4]:

    
U









    Out[4]:





U(x, u)

The case of a(n arbitrary) point transformation

cf. Introduction to Differential Invariants, Chapter 2 Lie Transformations pp. 16



In [12]:

    
x = sympy.Symbol('x',real=True)



In [13]:

    
y = sympy.Function('y')(x)



In [14]:

    
U = sympy.Function('U')(x,y)
X = sympy.Function('X')(x,y)
Y = sympy.Function('Y')(X)



In [15]:

    
sympy.pprint(sympy.diff(U,x))









    



d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    
──(y(x))⋅⎜───(U(x, ξ₂))⎟│        + ⎜───(U(ξ₁, y(x)))⎟│    
dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x



In [18]:

    
sympy.pprint( sympy.diff(Y,x))









    



⎛d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    ⎞ ⎛ d        ⎞│    
⎜──(y(x))⋅⎜───(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁, y(x)))⎟│    ⎟⋅⎜───(Y(ξ₁))⎟│    
⎝dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x⎠ ⎝dξ₁       ⎠│ξ₁=X

         
         
(x, y(x))



In [39]:

    
sympy.pprint( sympy.diff(Y,x).args[0] )









    



d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    
──(y(x))⋅⎜───(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁, y(x)))⎟│    
dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x



In [40]:

    
sympy.pprint( sympy.diff(U,x)/ sympy.diff(Y,x).args[0])









    



d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    
──(y(x))⋅⎜───(U(x, ξ₂))⎟│        + ⎜───(U(ξ₁, y(x)))⎟│    
dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x
──────────────────────────────────────────────────────────
d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    
──(y(x))⋅⎜───(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁, y(x)))⎟│    
dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x

For $Y''(X)$,



In [51]:

    
YprimeX = sympy.diff(U,x)/sympy.diff(Y,x).args[0]
sympy.pprint( sympy.diff(YprimeX,x).simplify() )









    



                                                               ⎛⎛            2
  ⎛d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    ⎞ ⎜⎜d          d 
- ⎜──(y(x))⋅⎜───(U(x, ξ₂))⎟│        + ⎜───(U(ξ₁, y(x)))⎟│    ⎟⋅⎜⎜──(y(x))⋅────
  ⎝dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x⎠ ⎜⎜dx           
                                                               ⎝⎝         dy(x
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              
                                                                              
                                                                              

                     2               ⎞                     ⎛   2            ⎞│
                    d                ⎟ d          d        ⎜  ∂             ⎟│
──(X(x, y(x))) + ────────(X(x, y(x)))⎟⋅──(y(x)) + ──(y(x))⋅⎜──────(X(x, ξ₃))⎟│
 2               dx dy(x)            ⎟ dx         dx       ⎝∂ξ₃ ∂x          ⎠│
)                                    ⎠                                        
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              
                                                                              
                                                                              

            2                 2                              ⎞                
           d                 d        ⎛ ∂           ⎞│       ⎟   ⎛d        ⎛ ∂
        + ───(X(x, y(x))) + ───(y(x))⋅⎜───(X(x, ξ₂))⎟│       ⎟ + ⎜──(y(x))⋅⎜──
ξ₃=y(x)     2                 2       ⎝∂ξ₂          ⎠│ξ₂=y(x)⎟   ⎝dx       ⎝∂ξ
          dx                dx                               ⎠                
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                ⎛d        ⎛ ∂           ⎞│          ⎛ ∂       
                                ⎜──(y(x))⋅⎜───(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁,
                                ⎝dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁      

                                                ⎛⎛            2               
           ⎞│          ⎛ ∂              ⎞│    ⎞ ⎜⎜d          d                
─(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁, y(x)))⎟│    ⎟⋅⎜⎜──(y(x))⋅──────(U(x, y(x))) 
₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x⎠ ⎜⎜dx            2             
                                                ⎝⎝         dy(x)              
──────────────────────────────────────────────────────────────────────────────
              2                                                               
       ⎞│    ⎞                                                                
 y(x)))⎟│    ⎟                                                                
       ⎠│ξ₁=x⎠                                                                

      2               ⎞                     ⎛   2            ⎞│            2  
     d                ⎟ d          d        ⎜  ∂             ⎟│           d   
+ ────────(U(x, y(x)))⎟⋅──(y(x)) + ──(y(x))⋅⎜──────(U(x, ξ₃))⎟│        + ───(U
  dx dy(x)            ⎟ dx         dx       ⎝∂ξ₃ ∂x          ⎠│ξ₃=y(x)     2  
                      ⎠                                                  dx   
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              
                                                                              
                                                                              

               2                              ⎞
              d        ⎛ ∂           ⎞│       ⎟
(x, y(x))) + ───(y(x))⋅⎜───(U(x, ξ₂))⎟│       ⎟
               2       ⎝∂ξ₂          ⎠│ξ₂=y(x)⎟
             dx                               ⎠
───────────────────────────────────────────────



In [32]:

    
sympy.factor_list( sympy.diff(Y,x)) # EY 20160522 I don't know how to simply obtain the factors of an expression 
# EY 20160522 update resolved: look at above and look at this page; it explains all: 
# http://docs.sympy.org/dev/tutorial/manipulation.html









    Out[32]:





(1,
 [(Subs(Derivative(Y(_xi_1), _xi_1), (_xi_1,), (X(x, y(x)),)), 1),
  (Derivative(y(x), x)*Subs(Derivative(X(x, _xi_2), _xi_2), (_xi_2,), (y(x),)) + Subs(Derivative(X(_xi_1, y(x)), _xi_1), (_xi_1,), (x,)),
   1)])



In [53]:

    
t, x, u, u_1, x_t, u_t, u_1t = sympy.symbols('t x u u_1 x_t u_t u_1t', real=True)
X = -u_1
U = u - x*u_1
U_1 = x

cf. How to do total derivatives: http://robotfantastic.org/total-derivatives-in-sympy.html



In [61]:

    
from sympy import Derivative, diff, expr
def difftotal(expr, diffby, diffmap):
    """Take the total derivative with respect to a variable.

    Example:

        theta, t, theta_dot = symbols("theta t theta_dot")
        difftotal(cos(theta), t, {theta: theta_dot})

    returns

        -theta_dot*sin(theta)
    """
    # Replace all symbols in the diffmap by a functional form
    fnexpr = expr.subs({s:s(diffby) for s in diffmap})
    # Do the differentiation
    diffexpr = diff(fnexpr, diffby)
    # Replace the Derivatives with the variables in diffmap
    derivmap = {Derivative(v(diffby), diffby):dv 
                for v,dv in diffmap.iteritems()}
    finaldiff = diffexpr.subs(derivmap)
    # Replace the functional forms with their original form
    return finaldiff.subs({s(diffby):s for s in diffmap})



In [63]:

    
difftotal( U,t,{x:x_t, u:u_t, u_1:u_1t}) + (-U_1)* (-u_1t)









    Out[63]:





-u_1*x_t + u_t

This transformation is the Legendre transformation

cf. 4. Exercises Chapter 2 Lie Transformations Introduction to Differential Invariants.

Consider transformation $(x,u)=(x,u(x)) \to (X,U)=(X(x,u),U(x,u))=(u,x)$. Let $Y=Y(X)$. $Y(X) \in \Gamma(\mathbb{R}^1 \times \mathbb{R}^1)$, i.e. $Y(X)$ is a section. So $Y(X) = Y(X(x,u)) = U(x,u)$. And so in this case, $Y(X(x,u))=Y(u)=U(x,u) = x$



In [65]:

    
x = sympy.Symbol('x',real=True)
u = sympy.Function('u')(x)
U = x
X = u
Y = sympy.Function('Y')(X)



In [66]:

    
sympy.pprint( sympy.diff(Y,x))









    



d        ⎛ d        ⎞│       
──(u(x))⋅⎜───(Y(ξ₁))⎟│       
dx       ⎝dξ₁       ⎠│ξ₁=u(x)



In [67]:

    
sympy.pprint(sympy.diff(U,x))

And so $Y'(X)$ is



In [68]:

    
sympy.pprint( 1/ sympy.diff(Y,x).args[0])









    



   1    
────────
d       
──(u(x))
dx

And so $Y''(X)$ is



In [70]:

    
sympy.pprint( sympy.diff( 1/ sympy.diff(Y,x).args[0], x))









    



   2       
  d        
-───(u(x)) 
   2       
 dx        
───────────
          2
⎛d       ⎞ 
⎜──(u(x))⎟ 
⎝dx      ⎠

cf. (2) from 4. Exercises, Chapter 2 Lie Transformations pp. 20

Recall an arbitrary point transformation:



In [71]:

    
x = sympy.Symbol('x',real=True)



In [72]:

    
y = sympy.Function('y')(x)



In [73]:

    
U = sympy.Function('U')(x,y)
X = sympy.Function('X')(x,y)
Y = sympy.Function('Y')(X)



In [74]:

    
sympy.pprint(sympy.diff(U,x))









    



d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    
──(y(x))⋅⎜───(U(x, ξ₂))⎟│        + ⎜───(U(ξ₁, y(x)))⎟│    
dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x



In [75]:

    
sympy.pprint( sympy.diff(Y,x))









    



⎛d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    ⎞ ⎛ d        ⎞│    
⎜──(y(x))⋅⎜───(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁, y(x)))⎟│    ⎟⋅⎜───(Y(ξ₁))⎟│    
⎝dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x⎠ ⎝dξ₁       ⎠│ξ₁=X

         
         
(x, y(x))



In [76]:

    
sympy.pprint( sympy.diff(Y,x).args[0] )









    



d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    
──(y(x))⋅⎜───(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁, y(x)))⎟│    
dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x



In [77]:

    
sympy.pprint( sympy.diff(U,x)/ sympy.diff(Y,x).args[0])









    



d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    
──(y(x))⋅⎜───(U(x, ξ₂))⎟│        + ⎜───(U(ξ₁, y(x)))⎟│    
dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x
──────────────────────────────────────────────────────────
d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    
──(y(x))⋅⎜───(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁, y(x)))⎟│    
dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x

For $Y''(X)$,



In [79]:

    
YprimeX = sympy.diff(U,x)/sympy.diff(Y,x).args[0]
Yprime2X = sympy.diff(YprimeX,x)
sympy.pprint( Yprime2X.simplify() )









    



                                                               ⎛⎛            2
  ⎛d        ⎛ ∂           ⎞│          ⎛ ∂              ⎞│    ⎞ ⎜⎜d          d 
- ⎜──(y(x))⋅⎜───(U(x, ξ₂))⎟│        + ⎜───(U(ξ₁, y(x)))⎟│    ⎟⋅⎜⎜──(y(x))⋅────
  ⎝dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x⎠ ⎜⎜dx           
                                                               ⎝⎝         dy(x
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              
                                                                              
                                                                              

                     2               ⎞                     ⎛   2            ⎞│
                    d                ⎟ d          d        ⎜  ∂             ⎟│
──(X(x, y(x))) + ────────(X(x, y(x)))⎟⋅──(y(x)) + ──(y(x))⋅⎜──────(X(x, ξ₃))⎟│
 2               dx dy(x)            ⎟ dx         dx       ⎝∂ξ₃ ∂x          ⎠│
)                                    ⎠                                        
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              
                                                                              
                                                                              

            2                 2                              ⎞                
           d                 d        ⎛ ∂           ⎞│       ⎟   ⎛d        ⎛ ∂
        + ───(X(x, y(x))) + ───(y(x))⋅⎜───(X(x, ξ₂))⎟│       ⎟ + ⎜──(y(x))⋅⎜──
ξ₃=y(x)     2                 2       ⎝∂ξ₂          ⎠│ξ₂=y(x)⎟   ⎝dx       ⎝∂ξ
          dx                dx                               ⎠                
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                ⎛d        ⎛ ∂           ⎞│          ⎛ ∂       
                                ⎜──(y(x))⋅⎜───(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁,
                                ⎝dx       ⎝∂ξ₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁      

                                                ⎛⎛            2               
           ⎞│          ⎛ ∂              ⎞│    ⎞ ⎜⎜d          d                
─(X(x, ξ₂))⎟│        + ⎜───(X(ξ₁, y(x)))⎟│    ⎟⋅⎜⎜──(y(x))⋅──────(U(x, y(x))) 
₂          ⎠│ξ₂=y(x)   ⎝∂ξ₁             ⎠│ξ₁=x⎠ ⎜⎜dx            2             
                                                ⎝⎝         dy(x)              
──────────────────────────────────────────────────────────────────────────────
              2                                                               
       ⎞│    ⎞                                                                
 y(x)))⎟│    ⎟                                                                
       ⎠│ξ₁=x⎠                                                                

      2               ⎞                     ⎛   2            ⎞│            2  
     d                ⎟ d          d        ⎜  ∂             ⎟│           d   
+ ────────(U(x, y(x)))⎟⋅──(y(x)) + ──(y(x))⋅⎜──────(U(x, ξ₃))⎟│        + ───(U
  dx dy(x)            ⎟ dx         dx       ⎝∂ξ₃ ∂x          ⎠│ξ₃=y(x)     2  
                      ⎠                                                  dx   
──────────────────────────────────────────────────────────────────────────────
                                                                              
                                                                              
                                                                              
                                                                              

               2                              ⎞
              d        ⎛ ∂           ⎞│       ⎟
(x, y(x))) + ───(y(x))⋅⎜───(U(x, ξ₂))⎟│       ⎟
               2       ⎝∂ξ₂          ⎠│ξ₂=y(x)⎟
             dx                               ⎠
───────────────────────────────────────────────



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