So I am doing an introductory course to PDEs and I have been introduced to the method of characteristics.
Now I have sort-of learnt the method used to solve the transport equation, where the transport is uniform. I also understand how it all plays out with the chain rule etc.
However, I am struggling to understand what exactly a characteristic is and the geometry behind this method. In a lot of the textbooks I read, there are often plots of characteristic lines, but what exactly are these lines?
I know my question may not be very clear, but that is because I don't really understand much at the moment. If it helps, I am studying out of 'Introduction to Partial Differential Equations' by 'Peter J. Olver'