Surface Parameterization

A surface in 3-Space can be parameterized by two variables (or coordinates) $u$ and $v$ such that

$$x = x(u,v)$$ (1)

$$y = y(u,v)$$ (2)

$$z = z(u, v).$$ (3)

If a surface is parameterized as above, then the tangent Vectors

$${\bf T}_u = {\partial x\over \partial u}\hat {\bf x} + {\pa... ...tial u}\hat {\bf y} + {\partial z\over \partial u}\hat {\bf z}$$ (4)

$${\bf T}_v = {\partial x\over \partial v}\hat {\bf x} + {\pa... ...tial v}\hat {\bf y} + {\partial z\over \partial v}\hat {\bf z}$$ (5)

are useful in computing the Surface Area and Surface Integral.

See also Smooth Surface, Surface Area, Surface Integral