SurfaceElement - crowlogic/arb4j GitHub Wiki

A surface element in a Euclidean space is a differential quantity used to measure a small portion of a surface embedded in the space. It is often used in the context of surface integrals, which are integral calculations performed over a surface.

In a 3-dimensional Euclidean space, a surface element is typically denoted by dS or dA, representing the infinitesimal area of a small patch of the surface. To compute a surface integral, you sum up the contributions of all these tiny surface elements across the entire surface. The concept can be generalized to higher-dimensional Euclidean spaces as well, where you'd be considering hypersurfaces.

To work with surface elements mathematically, you often need to express them in terms of local coordinates (e.g., using parametric or implicit representations of the surface). In the case of parametric representation, a surface is described by a vector function R(u, v), where u and v are local coordinates (parameters). The surface element can be computed as the cross product of the partial derivatives of R with respect to u and v:

dS = ||dR/du × dR/dv|| dudv

Here, dR/du and dR/dv are the tangent vectors to the surface in the u and v directions, respectively, and || · || denotes the Euclidean norm (magnitude) of a vector. The cross product of these tangent vectors gives a normal vector whose magnitude is equal to the area of the parallelogram spanned by the tangent vectors, representing the infinitesimal area dS.