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

# # Introduction
# 
# In this section we learn to do the following:
# 
# * Import SymPy and set up pretty printing
# * Use mathematical operations like `sqrt` and `sin`
# * Make SymPy Symbols
# * Take derivatives of expressions
# * Simplify expressions

# ## Preamble

# Just like NumPy and Pandas replace functions like `sin`, `cos`, `exp`, and `log` to powerful numeric implementations, SymPy replaces `sin`, `cos`, `exp` and `log` with powerful mathematical implementations.

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from sympy import *
init_printing()  # Set up fancy printing


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import math
math.sqrt(2)


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sqrt(2)  # This `sqrt` comes from SymPy


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cos(0)


# ### Exercise
# 
# Use the function `acos` on `-1` to find when cosine equals `-1`.  Try this same function with the math library.  Do you get the same result?

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# Call acos on -1 to find where on the circle the x coordinate equals -1



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# Call `math.acos` on -1 to find the same result using the builtin math module.  
# Is the result the same?  
# What do you think `numpy.acos` give you?



# ## Symbols
# 
# Just like the NumPy `ndarray` or the Pandas `DataFrame`, SymPy has the `Symbol`, which represents a mathematical variable.
# 
# We create symbols using the function `symbols`.  Operations on these symbols don't do numeric work like with NumPy or Pandas, instead they build up mathematical expressions.

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x, y, z = symbols('x,y,z')
alpha, beta, gamma = symbols('alpha,beta,gamma')


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x + 1


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log(alpha ** beta) + gamma


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sin(x)**2 + cos(x)**2


# ### Exercise
# 
# Use `symbols` to create two variables, `mu` and `sigma`.  

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get_ipython().run_line_magic('pinfo', ", ? = symbols('?')")


# ### Exercise
# 
# Use `exp`, `sqrt`, and Python's arithmetic operators like `+, -, *, **` to create the standard bell curve with SymPy objects
# 
# $$ e^{\frac{(x - \mu)^2}{ \sigma^2}} $$

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exp(?)


# ## Derivatives
# 
# One of the most commonly requested operations in SymPy is the derivative.  To take the derivative of an expression use the `.diff` method

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(x**2).diff(x)


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sin(x).diff(x)


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(x**2 + x*y + y**2).diff(x)


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(x**2 + x*y + y**2).diff(y)


# ### Exercise
# 
# In the last section you made a normal distribution

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mu, sigma = symbols('mu,sigma')


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bell = exp((x - mu)**2 / sigma**2)
bell


# Take the derivative of this expression with respect to $x$

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get_ipython().run_line_magic('pinfo', '.diff(?)')


# ### Exercise
# 
# There are three symbols in that expression.  We normally are interested in the derivative with repspect to `x`, but we could just as easily ask for the derivative with respect to `sigma`.  Try this now

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# Derivative of bell curve with respect to sigma



# ### Exercise
# 
# The second derivative of an expression is just the derivative of the derivative.  Chain `.diff( )` calls to find the second and third derivatives of your expression.

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#  Find the second and third derivative of `bell`



# ## Functions
# 
# SymPy has a number of useful routines to manipulate expressions.  The most commonly used function is `simplify`.

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expr = sin(x)**2 + cos(x)**2
expr


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simplify(expr)


# ### Exercise
# 
# In the last section you found the third derivative of the bell curve

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bell.diff(x).diff(x).diff(x)


# You might notice that this expression has lots of shared structure.  We can factor out some terms to simplify this expression.  
# 
# Call `simplify` on this expression and observe the result.

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# Call simplify on the third derivative of the bell expression



# ## Sympify
# 
# The `sympify` function transforms Python objects (ints, floats, strings) into SymPy objects (Integers, Reals, Symbols). 
# 
# *note the difference between `sympify` and `simplify`.  These are not the same function.*

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sympify('r * cos(theta)^2')


# It's useful whenever you interact with the real world, or for quickly copy-pasting an expression from an external source.