#!/usr/bin/env python
# coding: utf-8

# # Pullback with large symbolic expressions
# ## exp version
# 
# This worksheet is relative the `ask.sagemath` question [*Pullback computation hanging*](https://ask.sagemath.org/question/40852/pullback-computation-hanging/)

# In[1]:


version()


# In[2]:


get_ipython().run_line_magic('display', 'latex')


# In[3]:


M = Manifold(3, 'M')
X.<x,y,z> = M.chart()

N = Manifold(3, 'N')
XN.<a,b1,b2> = N.chart()

omega = N.diff_form(2)
omega[0,1] = 2*b2/a^3
omega[0,2] = -2*b1/a^3
omega[1,2] = -2/a^2
omega.display()


# We enforce the use of the $\exp$ representation of $\cosh$ and $\sinh$:

# In[4]:


cosh_e(x) = (exp(x) + exp(-x))/2
sinh_e(x) = (exp(x) - exp(-x))/2


# In[5]:


r = sqrt(x^2+y^2+z^2)
t = var('t', domain='real')
STSa = r^(1/2)*(r*cosh_e(2*r*t) - z*sinh_e(2*r*t))^(-1/2)
STSb1 = (x*sinh_e(2*r*t)/r)*STSa
STSb2 = (y*sinh_e(2*r*t)/r)*STSa

STS = M.diffeomorphism(N, [STSa, STSb1, STSb2])
STS.display()


# In[6]:


get_ipython().run_line_magic('time', 'STS.jacobian_matrix()[0,0]')


# In[7]:


get_ipython().run_line_magic('time', 's = STS.pullback(omega)')
s


# The number of characters in each of the non-vanishing components of $s$:

# In[8]:


[len(str(s[ind])) for ind in [(0,1), (0,2), (1,2)]] 


# The first 1000 characters of $s_{01}$:

# In[9]:


print(str(s[0,1])[:1000])


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