$$u_t + \nabla \cdot \mathbf{F} = 0$$
$$g(t) = u(x(t), y(t), t) \Longrightarrow 0 \equiv g'(t) = u_x x'(t) + u_y y'(t) + u_t$$
$$\mathbf{F} = u \left[ \begin{array}{c} \beta_0 \\ \beta_1 \end{array}\right] \Longrightarrow \nabla \cdot \mathbf{F} = \left(\beta_0 u\right)_x + \left(\beta_1 u\right)_y \stackrel{?}{=} x'(t) u_x + y'(t) u_y$$
$$x'(t) = \beta_0(y), \; y'(t) = \beta_1(x)$$
$$\left[ \begin{array}{c} x'(t) \\ y'(t) \end{array}\right] = \left[ \begin{array}{c} \beta_0(y) \\ \beta_1(x) \end{array}\right]$$
$$\left[ \begin{array}{c} x'(t) \\ y'(t) \end{array}\right] = \left[ \begin{array}{c} a \\ b \end{array}\right] \Longrightarrow \left[ \begin{array}{c} x(t) \\ y(t) \end{array}\right] = \left[ \begin{array}{c} x(0) \\ y(0) \end{array}\right] + t \left[ \begin{array}{c} a \\ b \end{array}\right]$$
In [2]:
%matplotlib inline
from JSAnimation import IPython_display
import matplotlib.animation

from for_slides import example1

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

Out[2]:

Once Loop Reflect
$$\left[ \begin{array}{c} x'(t) \\ y'(t) \end{array}\right] = \left[ \begin{array}{c} -y \\ x \end{array}\right] \Longrightarrow \left[ \begin{array}{c} x(t) \\ y(t) \end{array}\right] = \left[ \begin{array}{c c} \cos t & - \sin t \\ \sin t & \cos t \end{array}\right] \left[ \begin{array}{c} x(0) \\ y(0) \end{array}\right]$$
In [3]:
from for_slides import example2

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

Out[3]:

Once Loop Reflect