I need to solve this problem - details below.
Below is my Jupyter notebook exported to Python code.
I have this problem.
I have no idea how to call SymPy's idiff to find the
mixed second order derivative (that is the (d^2)z/dxdy).
https://calculus.subwiki.org/wiki/Second-order_mixed_partial_derivative
#!/usr/bin/env python
# coding: utf-8
# ### Differentiating an implicit function using SymPy
# In[1]:
import sympy as sp
# In[2]:
sp.__version__
# In[3]:
sp.init_printing(use_latex='mathjax') # use pretty mathjax output
# In[4]:
sp.var('x y z')
### We define some implicit function z = f(x,y)
### via the following equation F(x,y,z) = 0
### which is the same as the equation
### cos^2(x) + cos^2(y) + cos^2(z) = 1
F = (sp.cos(x))**2 + (sp.cos(y))**2 + (sp.cos(z))**2 - 1
### OK, our goal now is to find the second order
### partial derivatives of z=f(x,y) w.r.t.
### 1) x,x
### 2) x,y
### 3) y,y
### How do we do this?!
### Seems 1) and 3) are doable but 2) is not doable.
# #### So now we are looking to find these
# #### second order partial derivatives:
# #### $d^2z / dx^2$, $d^2z / dy^2$, $d^2z / dxdy$
# #### How do we do it?!
# In[5]:
f1 = sp.idiff( F, z, x )
f1
# In[6]:
f2 = sp.idiff( F, z, y )
f2
# In[7]:
sp.simplify(sp.idiff( F, z, x, 2 ))
# In[8]:
sp.simplify(sp.idiff( F, z, y, 2 ))
# In[ ]: