Surface More formally, let S be a surface, and x be a point on the surface. Let \mathbf{v} be a vector at Then one can write uniquely \mathbf{v} as a sum \mathbf{v} = \mathbf{v}_\parallel + \mathbf{v}_\perp where the first vector in the sum is the tangential component and the second one is the normal component. It follows immediately that these two vectors are perpendicular to each other. To calculate the tangential and normal components, consider a
unit normal to the surface, that is, a
unit vector \hat\mathbf{n} perpendicular to S at Then, \mathbf{v}_\perp = \left(\mathbf{v} \cdot \hat\mathbf{n}\right) \hat\mathbf{n} and thus \mathbf{v}_\parallel = \mathbf{v} - \mathbf{v}_\perp where "\cdot" denotes the
dot product. Another formula for the tangential component is \mathbf{v}_\parallel = -\hat\mathbf{n} \times (\hat\mathbf{n}\times\mathbf{v}), where "\times" denotes the
cross product. These formulas do not depend on the particular unit normal \hat\mathbf{n} used (there exist two unit normals to any surface at a given point, pointing in opposite directions, so one of the unit normals is the negative of the other one).
Submanifold More generally, given a
submanifold N of a
manifold M and a point p \in N, we get a
short exact sequence involving the
tangent spaces: T_p N \to T_p M \to T_p M / T_p N The
quotient space T_p M / T_p N is a generalized space of normal vectors. If
M is a
Riemannian manifold, the above sequence
splits, and the tangent space of
M at
p decomposes as a
direct sum of the component tangent to
N and the component normal to
N: T_p M = T_p N \oplus N_p N := (T_p N)^\perp Thus every
tangent vector v \in T_p M splits as where v_\parallel \in T_p N and v_\perp \in N_p N := (T_p N)^\perp. ==Computations==