# 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*.

## Formal definition

### Surface

More formally, let be a surface, and be a point on the surface. Let be a vector at . Then one can write uniquely as a sum

To calculate the tangential and normal components, consider a unit normal to the surface, that is, a unit vector perpendicular to at . Then,

where "" denotes the cross product.

These formulas do not depend on the particular unit normal 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 , we get a short exact sequence involving the tangent spaces:

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*:

## Computations

Suppose *N* is given by non-degenerate equations.

If *N* is given explicitly, via parametric equations (such as a parametric curve), then the derivative gives a spanning set for the tangent bundle (it is a basis if and only if the parametrization is an immersion).

If *N* is given implicitly (as in the above description of a surface, (or more generally as) a hypersurface) as a level set or intersection of level surfaces for , then the gradients of span the normal space.

In both cases, we can again compute using the dot product; the cross product is special to 3 dimensions however.

## Applications

- Lagrange multipliers: constrained critical points are where the tangential component of the total derivative vanish.
- Surface normal
- Frenet–Serret formulas
- Differential geometry of surfaces § Tangent vectors and normal vectors