Imagine a particle is confinded onto a surface. The only rule that the partcle follows is always choosing the path that minimizes the lenght of travel.
$$ \boldsymbol{\mathrm r}(\tau) = u(\tau) \; \boldsymbol{\mathrm e_1} + v(\tau) \; \boldsymbol{\mathrm e_2} + w(\tau) \; \boldsymbol{\mathrm e_3} $$
If we wanna find the arclength. $$ \mathrm d s^2 = \mathrm d \boldsymbol{\mathrm r} \cdot\mathrm d \boldsymbol{\mathrm r} \\ S = \int_{\partial \mathcal M} \sqrt{\mathrm d \boldsymbol{\mathrm r} \cdot\mathrm d \boldsymbol{\mathrm r}} \\ S = \int_{\partial \mathcal M} {\sqrt{\frac{\mathrm d \boldsymbol{\mathrm r}}{\mathrm d \tau} \cdot \frac{\mathrm d \boldsymbol{\mathrm r}}{\mathrm d \tau}} \; \mathrm d \tau} = \int_{\partial \mathcal M} {\frac{\mathrm d \mathrm r } {\mathrm d \tau} \; \mathrm d \tau} = \int_{\partial \mathcal M} { \dot{\mathrm r} \; \mathrm d \tau} $$ And now vary and find the functional that minimizes the arclenght: $$ \delta S = \int_{\partial \mathcal M} { \delta \dot{\mathrm r} \; \mathrm d \tau} = 0\\ \delta \dot{\mathcal r} = \left | \frac{\mathrm d \dot{\boldsymbol {\mathrm r}}}{\mathrm d \tau} \right | \delta \tau \\ \delta S = \int_ {\partial \mathcal M} { \left | \frac{\mathrm d \dot{\boldsymbol {\mathrm r}}}{\mathrm d \tau} \right | \delta \tau \; \mathrm d \tau} = 0 \\ \therefore \left | \frac{\mathrm d \dot{\boldsymbol {\mathrm r}}}{\mathrm d \tau} \right | = 0 $$ This means that the $\boldsymbol{\mathrm r}$ is a straight line, thing that make sense. But what a about a particle that is confined to a curved surface. How would you find this function that minimizes arclenght? Do I have to take into account the basis vectors tangent to the surfaces, or the gradient?