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
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 perpendicular to at Then,
and thus
where "" denotes the dot product. Another formula for the tangential component is
where "" denotes the cross product.
Note that these formulas do not depend on the particular unit normal used.

Submanifold

More generally, given a submanifold N of a manifold M and
a point, we get a short exact sequence
involving the tangent spaces:
The Quotianifold, 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:
Thus every tangent vector splits as
where and.

Computations

Suppose N is given by non-degenerate equations.
If N is given explicitly, via parametric equations, then the derivative gives a spanning set for the tangent bundle.
If N is given implicitly 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 though.

Applications