Okay, the lectures go out of order of the book, so we have a little mismatching here. Right now, he is going over I think the method of characteristics to solve the transport equation.

Lecture 2-4 in here #

We start with the transport equation: $$ u_t + b \cdot D u = 0 $$ $$u(x,0) = u_0(x)$$ We will denote $u_0$ as $\bar{u}$ as the lecture notates. where $b$ is a constant vector in $\mathbb{R}^n$ and $u$ is a function of $x \in \mathbb{R}^n$ and $t \geq 0$.

Definitions: Hypersurface: A hypersurface in $\mathbb{R}^n$ is a subset of $\mathbb{R}^n$ that locally resembles an $(n-1)$-dimensional manifold. It can be defined as the set of points that satisfy a single equation of the form $f(x_1, x_2, \ldots, x_n) = 0$, where $f$ is a smooth function. Hypersurfaces can be thought of as “slices” or “surfaces” in higher-dimensional spaces.

Okay, thank you AI. Essentially, it’s a generalization of a hyperplane. We had the hyperplane ${t=0}$ and now we have hypersurfaces that may not be flat like a hyperplane.

According to Appendix C.1 of Partial Differential Equations (Evans), a $C^1$ hypersurface is an $(n-1)$-dimensional geometric surface embedded in $\mathbb{R}^n$ that is “smooth” in the sense that it is once continuously differentiable.

Continuously Differentiable: A function $f$ is continuously differentiable if it has a derivative at every point in its domain and the derivative is a continuous function.

Lu = Linear operator applied to u = $\sum_{i=1}^n b_i \frac{\partial u}{\partial x_i}$

We have that $Lu = 0$ in $\Omega$ open in $\mathbb{R}^N$

where N = n + 1 and n+1 is time and space.

As well, $b_i \in C^1(\Omega)$ for all i from 1 to N. and now $(b_2 \dots b_N) = b$

The notation is supposed to be a generalization of our previous, but I don’t know. TODO.

Previously, we have $\Sigma = {t=0}$ $\Omega = {t > 0}$ or the hyperplane and the half space.

Now, we have $\Sigma \subset \mathbb{R}^N$ be a $C^1$ hypersurface (for simplicity, without boundary)

$\bar{u} \in C^1(\Sigma)$