Derivatives and quantities with units

The units of quantities are conveniently ignored in the definition of a derivative, but they should not be. A derivative should be defined as a function of two quantities, both with their own units:

$\mathbf{r'}(\mathbf{r},s)=\lim_{\Delta&space;s\rightarrow&space;0}\frac{\mathbf{r}(s+\Delta&space;s)-\mathbf{r}(s)}{\Delta&space;s}$

where r’ is a vector function of two quantites and r is a vector functon of one quantity. The second derivative is then a function of three quantities, each with their own units:

$\mathbf{r''}(\mathbf{r'},\mathbf{r},s)&space;=\lim_{\Delta&space;s\rightarrow&space;0}\frac{\mathbf{r'}(\mathbf{r},s+\Delta&space;s)-\mathbf{r'}(\mathbf{r},s)}{\Delta&space;s}$

Other derivatives should be defined similarly.