For a scalar function f over a surface parameterized by u and v, the surface integral is given by Φ | = | integral_S f d a | = | integral_S f(u, v) left bracketing bar T_uxT_v right bracketing bar d u d v, where T_u and T_v are tangent vectors and axb is the cross product. For a vector function over a surface, the surface integral is given by Φ | = | integral_S F·d a | = | integral_S(F·n^^) d a | = | integral_S f_x d y d z + f_y d z d x + f_z d x d y, where a·b is a dot product and n^^ is a unit normal vector.

integral | path integral | surface area | surface parameterization | volume integral