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 formula_1 be a surface, and formula_2 be a point on the surface. Let formula_3 be a vector at formula_4 Then one can write uniquely formula_3 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 formula_7 perpendicular to formula_1 at formula_4 Then , and thus where "formula_12" denotes the dot product. Another formula for the tangential component is where "formula_14" denotes the cross product. Note that these formulas do not depend on the particular unit normal formula_7 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 formula_16, we get a short exact sequence involving the tangent spaces: The quotient space formula_18 is a generalized space of normal vectors . 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": Thus every tangent vector formula_20 splits as formula_21, where formula_22 and formula_23. 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 formula_24, then the gradients of formula_24 span the normal space. In both cases, we can again compute using the dot product; the cross product is special to 3 dimensions however.