Page 11, Nonlinear Dynamics and Chaos, by Strogatz, says the following:
This is the domain of classical applied mathematics and mathematical physics where the linear partial differential equations live. Here we find Maxwell's equations of electricity and magnetism, the heat equation, Schrödinger's wave equation in quantum mechanics, and so on. These partial differential equations involve an infinite "continuum" of variables because each point in space contributes additional degrees of freedom. Even though these systems are large, they are tractable, thanks to such linear techniques as Fourier analysis and transform methods.
I'm wondering what is meant by the following:
These partial differential equations involve an infinite "continuum" of variables because each point in space contributes additional degrees of freedom.
"Each point in space contributes additional degrees of freedom"? We have three spacial dimensions, but since we're talking about an "infinite" continuum, I'm assuming he's referring to the infinite possible coordinates for these dimensions? I still don't understand why this infinite number of coordinates means that there are infinite variables? In mathematics, the number of variables has nothing to do with the number of possible coordinate combinations (which would be uncountably infinite, since there are an uncountably infinite number of real numbers).
Am I misinterpreting something here?
I would greatly appreciate it if people could please take the time to clarify this.