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:

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

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:

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

Other derivatives should be defined similarly.

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