Parametric Surfaces and their Area Part II

Parametric Surfaces – Tangent Plane The line u = u 0 is mapped to the gridline C 2 =r(u 0,v) Consider the parametric surface defined over some domain D. The line v = v 0 is mapped to the gridline C 1 =r(u,v 0 )

Parametric surfaces – Tangent Plane We can use this fact to find the equation of the tangent plane to a parametric surface at a given point. Thus the normal vector to the tangent plane is Cartesian coordinates of the point corresponding to u = 2, Example 6: Find the equation of the tangent plane to Equation of tangent plane: 0(x−0) − 2(y − 2)+2(z − 2)=0 Note that the surface is a cone. Can you see that from the parametric equations?

Parametric Surfaces – Surface Area Consider the parametric surface S described by the equation defined over some domain D.

Area of the surface S is Parametric Surfaces and their area

Parametric Surfaces and their area – Example 7 Find the surface area of the helicoid

Parametric Surfaces and their area – Example 8 Find the surface area of a sphere of radius a The sphere can be parameterized by We then have: =1

Parametric Surfaces and their area Surface Area:

Parametric Surfaces and their area – Example 9 Since z is positive, the surface can be written in explicit form as Find the curve C of intersection:

Parametric Surfaces and their area – Example 9 continued Changing to polar coordinates:

on the domain Parametric Surfaces and their area – Example 10