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

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from sympy import *
init_printing()
x, y, z = symbols('x,y,z')


# ## Solve
# 
# Equation solving is both a common need also a common building block for more complicated symbolic algorithms.  
# 
# Here we introduce the `solve` function

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solve(x**2 - 4, x)


# Solve takes two arguments, an equation like $x^2 - 4$ and a variable on which we want to solve, like $x$.  
# 
# Solve returns the values of the variable, $x$, for which the equation, $x^2 - 4$ equals 0.

# ### Exercise
# 
# What would the following code produce?  Are you sure?

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solve(x**2 - 9 == 0, x)


# ## Symbolic use of `solve`
# 
# Results of `solve` don't need to be numeric, like `[-2, 2]`.  We can use solve to perform algebraic manipulations.  For example if we know a simple equation for the area of a square
# 
#     area = height * width
#     
# we can solve this equation for any of the variables.  For example how would we solve this system for the `height`, given the `area` and `width`?

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height, width, area = symbols('height,width,area')
solve(area - height*width, height)


# Note that we would have liked to have written
# 
#     solve(area == height * width, height)
#     
# But the `==` gotcha bites us.  Instead we remember that `solve` expects an expression that is equal to zero, so we rewrite the equation
# 
#     area = height * width
#     
# into the equation
# 
#     0 = height * width - area
#     
# and that is what we give to solve.

# ### Exercise
# 
# Compute the radius of a sphere, given the volume.  Reminder, the volume of a sphere of radius `r` is given by
# 
# $$ V = \frac{4}{3}\pi r^3 $$

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# Solve for the radius of a sphere, given the volume



# You will probably get several solutions, this is fine.  The first one is probably the one that you want.

# ## Substitution
# 
# We often want to substitute in one expression for another.  For this we use the subs method

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x**2


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# Replace x with y
(x**2).subs({x: y})


# ### Exercise
# 
# Subsitute $x$ for $sin(x)$ in the equation $x^2 + 2\cdot x + 1$

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# Replace x with sin(x)



# ## Subs + Solve
# 
# We can use subs and solve together to plug the solution of one equation into another

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# Solve for the height of a rectangle given area and width

soln = solve(area - height*width, height)[0]
soln


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# Define perimeter of rectangle in terms of height and width

perimeter = 2*(height + width)


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# Substitute the solution for height into the expression for perimeter

perimeter.subs({height: soln})


# ### Exercise
# 
# In the last section you solved for the radius of a sphere given its volume

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V, r = symbols('V,r', real=True)
4*pi/3 * r**3


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solve(V - 4*pi/3 * r**3, r)[0]


# Now lets compute the surface area of a sphere in terms of the volume.  Recall that the surface area of a sphere is given by
# 
# $$ 4 \pi r^2 $$

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(?).subs(?)


# Does the expression look right?  How would you expect the surface area to scale with respect to the volume?  What is the exponent on $V$?

# ## Plotting
# 
# SymPy can plot expressions easily using the `plot` function.  By default this links against matplotlib.

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get_ipython().run_line_magic('matplotlib', 'inline')


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plot(x**2)


# ### Exercise
# 
# In the last exercise you derived a relationship between the volume of a sphere and the surface area.  Plot this relationship using `plot`.

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plot(?)


# ## Low dependencies
# 
# You may know that SymPy tries to be a very low-dependency project.  Our user base is very broad.  Some entertaining aspects result.  For example, `textplot`.

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textplot(x**2, -3, 3)


# ### Exercise
# 
# Play with `textplot` and enjoy :)