The directional derivative is zero, so the change is zero to first order. The second order term can exist. (Consider what happens if you move along the tangent line to a circle. The distance from the circle goes ~quadratically, since the circle looks locally parabolic.) Hence ds2.
The directional derivative is zero, so the change is zero to first order. The second order term can exist. (Consider what happens if you move along the tangent line to a circle. The distance from the circle goes ~quadratically, since the circle looks locally parabolic.) Hence ds2.
I see, thanks!