parametrized surface

A surface in 3-space can be parameterized by two variables (or coordinates) u and v such that x | = | x(u, v) y | = | y(u, v) z | = | z(u, v). If a surface is parameterized as above, then the tangent vectors T_u = (dx)/(du) x^^ + (dy)/(du) y^^ + (dz)/(du) z^^ T_v = (dx)/(dv) x^^ + (dy)/(dv) y^^ + (dz)/(dv) z^^ are useful in computing the surface area and surface integral.

parameter | parameterization | smooth surface | surface area | surface integral