PX275 - G2 - functions of a multiple variables

in physics, a common pairing is a spatially varying quantity, with a spatial dimension, $x$ , and time, $t$
eg: density of particles:
- $c (x, t)$ is the concentration of particles
- $Q (x, t)$ is the heat energy density
- $u (x, t)$ is the displacement of a string
partial derivatives are defined as:

\frac{\partial u (x, t)}{\partial x} and \frac{\partial u (x, t)}{\partial t}

this leads to partial differential equations (PDES)
it can be extended to an arbitrary combination of independent variables; eg: multiple spatial dimensions: $f (\vec{r}, t)$ , where $\vec{r} = (x, y, z)$ or $\vec{r} = (r, θ, φ)$ , or, ant set of independent variables: $f (x_{1}, x_{2}, \dots, x_{N}) ⟹ \frac{\partial f}{\partial x_{i}}$ , where, $i \in [1, N]$
the wave equation is an example of a PDE:

\frac{\partial^{2} u (x, t)}{\partial t^{2}} = c^{2} \frac{\partial^{2} u (x, t)}{\partial x^{2}}

note: the notation may differ:
- $\dot{u}$ is often associated with the time derivative, $\frac{\partial}{\partial t}$
- $u^{'}$ is often associated with the $\frac{\partial}{\partial x}$
- or subscripts: $u_{t} = \frac{\partial u}{\partial t}$ and $u_{x} = \frac{\partial u}{\partial x}$
the wave equation can also be written as:

\begin{aligned} u_{t t} & = c^{2} u_{x x} \\ \ddot{u} & = c^{2} u^{″} \end{aligned}