# What Arnold Sommerfeld is doing here?

" Let P be a point of the volume element under consideration and $$\displaystyle x, ~,y, ~z$$ its coordinate in a rectangular system the origin of which, O, lies within the element. In a general motion of the body the points P and O will both experience change of position, which we denote by $$\displaystyle \xi,~\eta,~\zeta$$ and $$\displaystyle \xi_0,~\eta_0,~\zeta_0$$ respectively, referring to the chosen space-fixed coordinate system. Taylor's formula then gives for the displacement of P $$\xi = \xi_0 + \frac{\partial \xi}{\partial x} x + \frac{\partial \xi}{\partial y} y + \frac{\partial \xi}{\partial z}z+ .... \\ \eta = \eta_0 + \frac{\partial \eta}{\partial x}x + \frac{\partial \eta}{\partial y}y+ \frac{\partial \eta}{\partial z}z +... \\ \zeta = \zeta_0 + \frac{\partial \zeta }{\partial x}x + \frac{\partial \zeta }{\partial y}y + \frac{\partial \zeta}{\partial z}z + ... \label{1}$$ "
Now, I must say that I'm shocked! First of all the notation are very very unusual, is $$\displaystyle \xi$$ the same thing as $$\displaystyle \Delta x$$ and similarly $$\displaystyle \eta$$ is $$\displaystyle \Delta y$$ and $$\displaystyle \zeta$$ is $$\displaystyle \Delta z$$ ? And second question how he got those equations and what do they mean? I know something like this $$\Delta f \left(x,y,z\right) = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y}\Delta y + \frac{\partial f}{\partial z}\Delta z$$ which is quite understandable as $$\displaystyle \frac{\partial f}{\partial x} \Delta x = \Delta f~\textrm{in x direction}$$ and similarly for other terms and hence adding all of them give us net change in $$\displaystyle f$$. But what does those equations mean? What is the idea behind them?