= Solution
A two-dimensional embedded submanifold is locally flattened by a continuously differentiable coordinate diffeomorphism of an ambient <open neighbourhood> onto a coordinate plane, or equivalently locally admits a regular two-parameter chart. A tangent vector at $x$ is the velocity $\gamma'(0)$ of a differentiable curve in $M$ through $x$. All such vectors form the two-dimensional <tangent space> $T_xM$, the image of the differential of a regular parametrization. Normal vectors form its <orthogonal> complement $(T_xM)^\perp$ in $\mathbb R^3$, which has \b[dimension one].
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