Tangent Planes and Normal Vectors
(Page: 1
 2 ) Text If both partial derivatives for each coordinate function of the parametrized surface (x(u,v), y(u,v), z(u,v)) exist at a point (u_{0},v_{0}), then the tangent plane of the surface at (u_{0},v_{0}) is the plane determined by the vectors
(x_{u}(u_{0},v_{0}), y_{u}(u_{0},v_{0}), z_{u}(u_{0},v_{0})) and
(x_{v}(u_{0},v_{0}), y_{v}(u_{0},v_{0}), z_{v}(u_{0},v_{0})).
The unit normal to the point (x(u_{0},v_{0}), y(u_{0},v_{0}), z(u_{0},v_{0})) is the cross product of the two vectors above divided by this cross product's magnitude.
Demos
Tangent Plane to a Parametrized Surface
 
For this demonstration, choose a point (u_{0},v_{0}) in the domain. You will see a graph of the parametrized surface (x(u,v), y(u,v), z(u,v)) and the u and vslice curves at the point (x(u_{0},v_{0}), y(u_{0},v_{0}), z(u_{0},v_{0})). The tangent lines to each of the slice curves at (x(u_{0},v_{0}), y(u_{0},v_{0}), z(u_{0},v_{0})) are also drawn, and it is clear that together they determine a plane. This plane, which is shown in yellow, is tangent to the graph at (x(u_{0},v_{0}), y(u_{0},v_{0}), z(u_{0},v_{0})).

Normal Vector to a Parametrized Surface
 
This demo shows the unit normal vector to the tangent plane for a given point (u_{0},v_{0}) in the domain.

