In [1]:

    

%pylab inline


    

    




Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].
For more information, type 'help(pylab)'.

In [2]:

    

from sympy import init_session
init_session()


    

    




Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].
For more information, type 'help(pylab)'.
IPython console for SymPy 0.7.5 (Python 2.7.5-64-bit) (ground types: python)

These commands were executed:
>>> from __future__ import division
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)

Documentation can be found at http://www.sympy.org

In [25]:

    

phi = (cos(pi*x) + cos(pi*y) + cos(pi*z))*(1-x)*(1-y)*(1-z)
phi


    

    
Out[25]:





$$\left(- x + 1\right) \left(- y + 1\right) \left(- z + 1\right) \left(\cos{\left (\pi x \right )} + \cos{\left (\pi y \right )} + \cos{\left (\pi z \right )}\right)$$

In [16]:

    

f = diff(phi, x, 2) + diff(phi, y, 2) + diff(phi, z, 2)
f


    

    
Out[16]:





$$\pi \left(x - 1\right) \left(y - 1\right) \left(\pi \left(z - 1\right) \cos{\left (\pi z \right )} + 2 \sin{\left (\pi z \right )}\right) + \pi \left(x - 1\right) \left(z - 1\right) \left(\pi \left(y - 1\right) \cos{\left (\pi y \right )} + 2 \sin{\left (\pi y \right )}\right) + \pi \left(y - 1\right) \left(z - 1\right) \left(\pi \left(x - 1\right) \cos{\left (\pi x \right )} + 2 \sin{\left (\pi x \right )}\right)$$

In [24]:

    

print(f)


    

    




pi*(x - 1)*(y - 1)*(pi*(z - 1)*cos(pi*z) + 2*sin(pi*z)) + pi*(x - 1)*(z - 1)*(pi*(y - 1)*cos(pi*y) + 2*sin(pi*y)) + pi*(y - 1)*(z - 1)*(pi*(x - 1)*cos(pi*x) + 2*sin(pi*x))

In [27]:

    

xl = phi.subs(x, 0)
xl


    

    
Out[27]:





$$\left(- y + 1\right) \left(- z + 1\right) \left(\cos{\left (\pi y \right )} + \cos{\left (\pi z \right )} + 1\right)$$

In [28]:

    

print(xl)


    

    




(-y + 1)*(-z + 1)*(cos(pi*y) + cos(pi*z) + 1)

In [19]:

    

phi.subs(x, 1)


    

    
Out[19]:





$$0$$

In [29]:

    

yl = phi.subs(y, 0)
yl


    

    
Out[29]:





$$\left(- x + 1\right) \left(- z + 1\right) \left(\cos{\left (\pi x \right )} + \cos{\left (\pi z \right )} + 1\right)$$

In [30]:

    

print(yl)


    

    




(-x + 1)*(-z + 1)*(cos(pi*x) + cos(pi*z) + 1)

In [21]:

    

phi.subs(y, 1)


    

    
Out[21]:





$$0$$

In [31]:

    

zl = phi.subs(z, 0)
zl


    

    
Out[31]:





$$\left(- x + 1\right) \left(- y + 1\right) \left(\cos{\left (\pi x \right )} + \cos{\left (\pi y \right )} + 1\right)$$

In [32]:

    

print(zl)


    

    




(-x + 1)*(-y + 1)*(cos(pi*x) + cos(pi*y) + 1)

In [23]:

    

phi.subs(z, 1)


    

    
Out[23]:





$$0$$

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