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

# $$u_t + \nabla \cdot \mathbf{F} = 0$$

# $$g(t) = u(x(t), y(t), t) \Longrightarrow 0 \equiv g'(t) = u_x x'(t) + u_y y'(t) + u_t$$

# $$\mathbf{F} = u \left[ \begin{array}{c} \beta_0 \\ \beta_1 \end{array}\right]
# \Longrightarrow \nabla \cdot \mathbf{F} = \left(\beta_0 u\right)_x + 
# \left(\beta_1 u\right)_y \stackrel{?}{=} x'(t) u_x + y'(t) u_y$$

# $$x'(t) = \beta_0(y), \; y'(t) = \beta_1(x)$$

# $$\left[ \begin{array}{c} x'(t) \\ y'(t) \end{array}\right] = \left[ \begin{array}{c} \beta_0(y) \\ \beta_1(x) \end{array}\right]$$

# $$\left[ \begin{array}{c} x'(t) \\ y'(t) \end{array}\right] = \left[ \begin{array}{c} a \\ b \end{array}\right] \Longrightarrow \left[ \begin{array}{c} x(t) \\ y(t) \end{array}\right] = \left[ \begin{array}{c} x(0) \\ y(0) \end{array}\right] + t \left[ \begin{array}{c} a \\ b \end{array}\right]$$

# In[2]:


get_ipython().run_line_magic('matplotlib', 'inline')
from JSAnimation import IPython_display
import matplotlib.animation

from for_slides import example1

args, kwargs = example1()
matplotlib.animation.FuncAnimation(*args, **kwargs)


# $$\left[ \begin{array}{c} x'(t) \\ y'(t) \end{array}\right] = \left[ \begin{array}{c} -y \\ x \end{array}\right] \Longrightarrow \left[ \begin{array}{c} x(t) \\ y(t) \end{array}\right] = \left[ \begin{array}{c c} \cos t & - \sin t \\ \sin t & \cos t \end{array}\right] \left[ \begin{array}{c} x(0) \\ y(0) \end{array}\right]$$

# In[3]:


from for_slides import example2

args, kwargs = example2()
matplotlib.animation.FuncAnimation(*args, **kwargs)


# In[4]:


from for_slides import example4

example4()


# In[5]:


from for_slides import example5

example5()


# $$\left.u\right|_{T_1} = f_1, \quad \left.u\right|_{T_6} = f_6, \quad \ldots$$

# $u "=" v$ when $$\int_{T} u \, \varphi \, dV = \int_T v \, \varphi \, dV$$

# # Galerkin projection

# $$u = \sum_j u_j \varphi_j$$

# $$\int_{T} u \, \varphi_i \, dV = \int_{T \cap T_1} f_1 \, \varphi_i \, dV + \int_{T \cap T_6} f_6 \, \varphi_i \, dV + \cdots$$

# In[6]:


from for_slides import example6

example6()


# $$T \cap T_{28} = Q = D_1 \cup D_2$$

# $$\int_{T \cap T_{28}} f_{28} \, \varphi_i \, dV = \int_{D_1} f_{28} \, \varphi_i \, dV + \int_{D_2} f_{28} \, \varphi_i \, dV$$

# # DG or CG?

# In[7]:


from for_slides import example6a

example6a()


# In[8]:


from for_slides import example7

example7()


# In[9]:


from for_slides import example8

example8()


# In[10]:


from for_slides import example9

example9()


# # Thanks!

# In[1]:


# Custom styling, borrowed from https://github.com/ellisonbg/talk-2013-scipy

from IPython import display


# Make plots centered. H/T to http://stackoverflow.com/a/27168595/1068170
STYLE = """\
<style>
div.output_area, .ui-wrapper {
  margin-left: auto !important;
  margin-right: auto !important;
}
</style>
"""


SLIDES_STYLE = """
<style>

h1.bigtitle {
    margin: 4cm 1cm 4cm 1cm;
    font-size: 300%;
}

h1.title { 
    font-size: 250%;
}

.rendered_html h1 {
    margin: 0.25em 0em 0.5em;
    color: #015C9C;
    text-align: center;
    line-height: 1.2; 
    page-break-before: always;
}
</style>
"""

STYLE = STYLE + '\n' + SLIDES_STYLE

display.display(display.HTML(STYLE))