Normalteil

Normalteil is a term used in vector calculus and physics to denote the normal component of a vector relative to a given surface or submanifold. For a surface with unit normal vector n at a point, the normalteil of a vector v is v_n = (v · n) n, and the scalar normal component is v · n. The remaining component is the tangential component v_t = v - v_n. This decomposition is unique when n is defined. The normal direction is perpendicular to the surface.

In differential geometry, the concept extends to vectors in the tangent space and the normal bundle; the projection of a vector field onto the normal bundle yields its normalteil.

Computation: if the surface normal is n, then compute the dot product to obtain the scalar projection s = v · n, and multiply by n to obtain the vector normalteil. If n is not unit, first normalize: n_hat = n/|n|, then v_n = (v · n_hat) n_hat.

Applications: evaluating fluxes across surfaces, stress analysis where forces act normal to a boundary, boundary conditions in fluid dynamics, and decomposition of velocity fields near interfaces.

Limitations: the concept requires a well-defined normal; at edges or corners or non-smooth surfaces the normal may be undefined or piecewise defined.

Etymology: German "Normal" meaning perpendicular, "Teil" meaning part or component.