Illustration of tangential and normal components of a vector to a surface.
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 submanifoldN of a manifoldM, 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.
Note that 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).
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 though.