Why are we minimizing the norm of the second derivative instead of just the second derivative, which is just the curvature?

Curvature is not the second derivative. Anyway I think the second derivative is a scalar quantity so we need to minimize its absolute value.

Why are we minimizing the norm of the second derivative instead of just the second derivative, which is just the curvature?

Curvature is not the second derivative. Anyway I think the second derivative is a scalar quantity so we need to minimize its absolute value.