Tangential and normal components

More formally, let ${\displaystyle S}$ be a surface, and ${\displaystyle x}$ be a point on the surface. Let ${\displaystyle \mathbf {v} }$ be a vector at ${\displaystyle x}$. Then one can write uniquely ${\displaystyle \mathbf {v} }$ as a sum

$${\displaystyle \mathbf {v} =\mathbf {v} _{\parallel }+\mathbf {v} _{\perp }}$$

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 ${\displaystyle {\hat {\mathbf {n} }}}$ perpendicular to ${\displaystyle S}$ at ${\displaystyle x}$. Then,

$${\displaystyle \mathbf {v} _{\perp }=\left(\mathbf {v} \cdot {\hat {\mathbf {n} }}\right){\hat {\mathbf {n} }}}$$

and thus

$${\displaystyle \mathbf {v} _{\parallel }=\mathbf {v} -\mathbf {v} _{\perp }}$$

where "${\displaystyle \cdot }$" denotes the dot product. Another formula for the tangential component is

$${\displaystyle \mathbf {v} _{\parallel }=-{\hat {\mathbf {n} }}\times ({\hat {\mathbf {n} }}\times \mathbf {v} ),}$$

where "${\displaystyle \times }$" denotes the cross product.

Note that these formulas do not depend on the particular unit normal ${\displaystyle {\hat {\mathbf {n} }}}$ 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).

More generally, given a submanifold N of a manifold M and a point ${\displaystyle p\in N}$, we get a short exact sequence involving the tangent spaces:

$${\displaystyle T_{p}N\to T_{p}M\to T_{p}M/T_{p}N}$$

The quotient space ${\displaystyle T_{p}M/T_{p}N}$ 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:

$${\displaystyle T_{p}M=T_{p}N\oplus N_{p}N:=(T_{p}N)^{\perp }}$$

Thus every tangent vector ${\displaystyle v\in T_{p}M}$ splits as ${\displaystyle v=v_{\parallel }+v_{\perp }}$, where ${\displaystyle v_{\parallel }\in T_{p}N}$ and ${\displaystyle v_{\perp }\in N_{p}N:=(T_{p}N)^{\perp }}$.