Inmathematics, given avector at a point on acurve, that vector can be decomposed uniquely as a sum of two vectors, onetangent to the curve, called thetangential component of the vector, and another oneperpendicular to the curve, called thenormal component of the vector. Similarly, a vector at a point on asurface can be broken down the same way.

More generally, given asubmanifoldN of amanifoldM, and a vector in thetangent space toM at a point ofN, it can be decomposed into the component tangent toN and the component normal toN.

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 aunit normal to the surface, that is, aunit 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 thedot 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 thecross product.

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 asubmanifoldN of amanifoldM and a point${\displaystyle p\in N}$, we get ashort exact sequence involving thetangent spaces:$${\displaystyle T_{p}N\to T_{p}M\to T_{p}M/T_{p}N}$$Thequotient space${\displaystyle T_{p}M/T_{p}N}$ is a generalized space of normal vectors.

IfM is aRiemannian manifold, the above sequencesplits, and the tangent space ofM atp decomposes as adirect sum of the component tangent toN and the component normal toN:$${\displaystyle T_{p}M=T_{p}N\oplus N_{p}N:=(T_{p}N{)}^{\perp }}$$Thus everytangent vector${\displaystyle v\in T_{p}M}$ splits as${\displaystyle v=v_{\parallel }+v_{\perp }}$, where${\displaystyle v_{\parallel }\in T_{p}N}$ and${\displaystyle v_{\perp }\in N_{p}N:=(T_{p}N{)}^{\perp }}$.

SupposeN is given by non-degenerate equations.

IfN is given explicitly, viaparametric equations (such as aparametric curve), then the derivative gives a spanning set for the tangent bundle (it is abasis if and only if the parametrization is animmersion).

IfN is givenimplicitly (as in the above description of a surface, (or more generally as) ahypersurface) as alevel set or intersection oflevel surfaces for${\displaystyle g_i}$, then the gradients of${\displaystyle g_i}$ span the normal space.

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

• Lagrange multipliers: constrainedcritical points are where the tangential component of thetotal derivative vanish.
• Surface normal
• Frenet–Serret formulas
• Differential geometry of surfaces § Tangent vectors and normal vectors

• Rojansky, Vladimir (1979).Electromagnetic fields and waves. New York: Dover Publications.ISBN 0-486-63834-0.
• Crowell, Benjamin (2003).Light and Matter.

• Differential geometry