# Shouldn't these two definitions for curvature agree?

In $\mathbb R^n$ the defintion of curvature of a smooth regular curve $\gamma : \mathbb R \to \mathbb R^n$ is $$\kappa (t) = \|\gamma''(t)\| / \|\gamma '(t)\|$$

In $\mathbb R^2$ the definition for the curvature of an arbitrary smooth regular curve $\gamma : \mathbb R \to \mathbb R^2$ given as $\gamma (t) = (x(t),y(t))$ is

$$\kappa (t) = {x'(t) y''(t) - x'' (t) y' (t) \over (x^{'2} + y^{'2})^{3/2}}$$

I assumed these should be equal because I thought the second formula was simply for convenience and not in fact different. But calculating a simple example says otherwise:

If $\gamma (t) = (2 \cos t, \sin t)$ is an ellipse then

$$\|\gamma ''\|/\|\gamma'\| = {\sqrt{4 \cos^2 t + \sin^2 t} \over \sqrt{4 \sin^2 t + \cos^2 t}}$$

whereas

$$\kappa(t) = {2\over (4 \sin^2 t + \cos^2 t)^{3/2}}$$

Similarly, a different curvature for $\gamma (t) = (t,t^2)$.

Why is the curvature in $\mathbb R^2$ defined differently than for $\mathbb R^n$? It seems bizarre to me that they are not equal.

Your first formula is wrong. Or, rather, it is only correct when $||\gamma'(t)||$ is constant. Otherwise, its value depends of parametrization: for example, that formula gives non-zero curvature even for a straight line parametrized as $(t^3+t,0)$.
The (not oriented) curvature of $\gamma$ is the length of the derivative of its unit speed vector in respect of arclength, i.e., the length of $$\frac{1}{\|\gamma'\|}\left(\frac{\gamma'}{\|\gamma'\|}\right)'.$$ It’s easy to calculate this length as $$\frac{1}{\|\gamma'\|^3}\sqrt{\|\gamma'\|^2\|\gamma''\|^2-\langle \gamma',\gamma''\rangle^2},$$ which equals $$\frac{1}{\|\gamma'\|^2}\sqrt{\|\gamma''\|^2-\langle\frac{\gamma'}{\|\gamma'\|},\gamma''\rangle}.$$ As the radicand is the Gram-matrix of $\gamma''$ and $\gamma'/\|\gamma'\|$ the curvature admits the following interpretation: It is the area of the parallelogram spanned by the unit speed vector and the acceleration, divided by the squared velocity.
The main difference regarding $R^2$ vs. $R^n$ for $n>2$ is that we can't naturally establish an orientated curvature.
Edit: In case of $R^2$ we easily achieve that $$\sqrt{\|\gamma'\|^2\|\gamma''\|^2-\langle \gamma',\gamma''\rangle^2}=\sqrt{(x'y''-x''y')^2}=|x'y''-x''y'|.$$