VecdA

VecdA, usually written as dA_vec or dA with an arrow, denotes the differential vector area element used in surface integrals in vector calculus. It encodes both the small patch of surface area and its orientation, given by the surface normal.

For a parametric surface r(u,v), the differential area vector is dA_vec = (∂r/∂u × ∂r/∂v) du dv. Equivalently, it can be expressed as dA_vec = n_hat dA, where n_hat is the unit normal to the surface and dA is the scalar area element. The sign and direction of dA_vec depend on the chosen orientation, often specified by the order of parameters (right-hand rule).

In applications, dA_vec is used to compute fluxes and surface integrals. A common form is the flux of a vector field F through a surface S: ∬_S F · dA_vec. This notation also appears in the divergence theorem and Stokes’ theorem, linking surface integrals to volume or line integrals.

Examples aid intuition. For a plane z = 0 with outward normal k, dA_vec = k dx dy. For a sphere of radius R, dA_vec = n_hat R^2 sinθ dθ dφ, where n_hat is the radial unit vector and (θ, φ) are the usual spherical angles.

See also: area vector, surface integral, flux, divergence theorem, Stokes’ theorem, normal vector. VecdA is a standard tool in physics and engineering for expressing how fields interact with curved surfaces.