SymPy

The SymPy package is useful for symbolic algebra, much like the commercial software Mathematica.

We won't make much use of SymPy during the boot camp, but it is definitely useful to know about for mathematics courses.



In [1]:

    
import sympy as sp
sp.init_printing()

Symbols and Expressions

We'll define x and y to be sympy symbols, then do some symbolic algebra.



In [2]:

    
x, y = sp.symbols("x y")



In [3]:

    
expression = (x+y)**4



In [4]:

    
expression









    Out[4]:





$$\left(x + y\right)^{4}$$



In [5]:

    
sp.expand(expression)









    Out[5]:





$$x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}$$



In [6]:

    
expression = 8*x**2 + 26*x*y + 15*y**2



In [7]:

    
expression









    Out[7]:





$$8 x^{2} + 26 x y + 15 y^{2}$$



In [8]:

    
sp.factor(expression)









    Out[8]:





$$\left(2 x + 5 y\right) \left(4 x + 3 y\right)$$



In [9]:

    
expression - 20 *x*y - 14*y**2









    Out[9]:





$$8 x^{2} + 6 x y + y^{2}$$



In [10]:

    
sp.factor(expression - 20*x*y - 14*y**2)









    Out[10]:





$$\left(2 x + y\right) \left(4 x + y\right)$$



In [16]:

    
expression









    Out[16]:





$$8 x^{2} + 26 x y + 15 y^{2}$$



In [12]:

    
f = sp.lambdify((x,y), expression, 'numpy')



In [17]:

    
f(3,4)









    Out[17]:





$$624$$



In [18]:

    
8 * 3**2 + 26 * 3 * 4 + 15 * 4**2









    Out[18]:





$$624$$

You can use sympy to perform symbolic integration or differentiation.



In [26]:

    
expression = 5*x**2 * sp.sin(3*x**3)



In [27]:

    
expression









    Out[27]:





$$5 x^{2} \sin{\left (3 x^{3} \right )}$$



In [28]:

    
expression.diff(x)









    Out[28]:





$$45 x^{4} \cos{\left (3 x^{3} \right )} + 10 x \sin{\left (3 x^{3} \right )}$$



In [29]:

    
expression = sp.cos(x)



In [30]:

    
expression.integrate(x)









    Out[30]:





$$\sin{\left (x \right )}$$



In [34]:

    
expression.integrate((x, 0, sp.pi / 2))









    Out[34]:





$$1$$

You can also create unevalated integrals or derivatives. These can later be evaluated with their doit methods.



In [35]:

    
deriv = sp.Derivative(expression)



In [36]:

    
deriv









    Out[36]:





$$\frac{d}{d x} \cos{\left (x \right )}$$



In [37]:

    
deriv.doit()









    Out[37]:





$$- \sin{\left (x \right )}$$



In [42]:

    
inte = sp.Integral(expression, (x, 0, sp.pi / 2))



In [43]:

    
inte









    Out[43]:





$$\int_{0}^{\frac{\pi}{2}} \cos{\left (x \right )}\, dx$$



In [44]:

    
inte.doit()









    Out[44]:





$$1$$



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