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Let \( \displaystyle c : I \subset \mathbb{R} \longrightarrow S \subset \mathbb{R}^3 \) be a differentiable curve on a regular oriented surface \( \displaystyle S \), parametrized by arc length. If \( \kappa \) is the curvature of \( \displaystyle c \) as a curve in \( \displaystyle \mathbb{R}^3 \), \( \kappa_{g} \) is the geodesic curvature of \( \displaystyle c \) and \( \kappa_{n} \) is the normal curvature of \( \displaystyle c \), then show that \[ \displaystyle \kappa^2 = \kappa_{g}^2 + \kappa_{n}^2 \]