Tangential and normal components

In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector. Similarly a vector at a point on a surface can be broken down the same way.

More generally, given a submanifold N of a manifold M, and a vector in the tangent space to M at a point of N, it can be decomposed into the component tangent to N and the component normal to N.

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

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