Physically/conceptually, I understand why the unit normal vector will always point toward the side of concavity on a curve. It's because the unit normal's direction is the derivative of the unit tangent, and the unit tangent's change in direction is always toward the concave side. But how is this represented mathematically? I tried to prove it, but I'm stuck. I've looked, but I couldn't find any proofs about this. Does anyone know?