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Let $\gamma : I \to \mathbb{R}^2$ be a smooth immersed curve, i.e. $\mathcal{C}^\infty$ and $\frac{d\gamma}{dt}\neq 0$ on $I$. I have found the following formula for the curvature $\kappa$ of $\gamma$, but I haven't seen a proof. Does anyone know how to prove:

$$\kappa(t) = \left|\frac{d\gamma}{dt}\right|^{-3} \left(-\frac{d^2\gamma_1}{dt^2}\frac{d\gamma_2}{dt} +\frac{d^2\gamma_2}{dt^2}\frac{d\gamma_1}{dt}\right)$$

Thanks in advance!

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MathWorld has a derivation. (If I could, I would vote to close as general-reference.) – Rahul Jan 27 '13 at 1:12

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