I am struggling with a concept introduced in a third year undergraduate course on "Mathematical Methods"; specifically at the very beginning of the chapter on PDEs. Contour Lines are introduced, followed later by Data Curves.
Contour Lines are defined as follows:
Consider a linear, homogeneous PDE
$$a(x,y)u_x+b(x,y)u_y=0$$ with $a^2 + b^2 \neq 0$.
This may be solved by considering the solutions of
$$\frac{dy}{dx} = \frac{b}{a}(x,y) \hspace{20mm} (*)$$
for any initial point $(x_0,y_0) \in \mathbb{R}^2$,
This yields many trajectory lines, each with tangents parallel to
$$\begin{pmatrix}b(x,y)\\a(x,y)\end{pmatrix}=\frac{dy}{dx}$$
It is then proven that u is constant along the solutions of the PDE. Then it says
As $u(x(t),y(t))$ is constant along (x(t),y(t)), a solution to the PDE, the curves $(x(t),y(t))$ describe contour lines of u. Different values of u are taken, depending on $(x_0,y_0)$.
I have problems with understanding the above. I understand the given PDE, and understand that it can be solved for any initial point $(x_0,y_0)$. I don't understand why considering solutions of $(*)$ solves the given PDE. I don't understand what "This yields many trajectory lines, each with tangents parallel to..." means. I mean to say that I don't really understand what that vector represents, what trajectory lines are, or why the tangents to the trajectory lines should be parallel to it. Perhaps I should emphasize that this is one of the main issues.
The proof is given of u being constant along the solutions to the PDE. I think I understand that, so have not included it.
As far as I understand what comes next, I understand that there is a solution $(x(t),y(t))$ (a parametrized curve in $\mathbb{R}^2$), and I can visualize how, if u is constant along this curve, this somehow looks like a contour line on a map. Things break down now, because I don't really understand why I'm visualizing some sort of "hilly" region in $\mathbb{R}^3$, even though I am doing that and that seems to make sense in the context of the lecture notes. Is that hilly region just what $u(x,y)$ "looks like"?
Any help is appreciated.
Matt
I think that what is described here is the method of characteristics. Try looking up the geometric interpretation of this method, it makes a lot of sense. Take this as an example...