1. Parametric Surfaces
We can use parametric equations to describe a curve.
Because a curve is one dimensional, we only need one parameter.
If we want to describe a surface (two dimensions) using parametric equations, we will need two parameters.
 r(u,v) where the domain refers to values in the uv-plane.

2. Ex. Identify the surface with vector equation r(u,v) = 2cos ui + vj + 2sin uk.

3. Ex. Identify the surface with vector equation r(u,v) = 5ui + (2u + v)j + v2k.

4. Ex. Find the rectangular equation of the surface with vector equation r(u,v) = 2ucos vi + ½u2 j + 2usin vk.

5. Ex. Find a parametric representation of the sphere x2 + y2 + z2 = a2.

6. Ex. Find a parametric representation of the elliptic paraboloid z = 2x2 + y2.

7. For surfaces created by rotating the function y = f (x) about the x-axis, the parametric equations would be
x = u y = f (u) cos v z = f (u) sin v
 These can be adapted for rotation around the y- or z-axis.

8. Ex. Find the parametric equation of the surface generated by rotating z = ln y about the y-axis.

9. Let r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k
Each of these are tangent to the surface

10. Ex. Find the equation of the plane tangent to the surface r(u,v) = u2i + v2j + (u + 2v)k at the point (1,1,1).

11. Thm. If a surface S is defined by the vector function r(u,v), defined on the region D in the uv-plane, then the surface area of S is

12. Ex. Find the surface area of the sphere of radius a.

13. Ex. Find the area of the surface defined by z = f (x,y).

14. Ex. Find the area of the part of the paraboloid z = x2 + y2 that lies under the plane z = 9.

15. Surface Integrals
Line integrals added the values of a function at every point on a curve.
Surface integrals add the value of a 3-D function at every point on a surface.

16. Let S be a surface with equation z = g(x,y), and let R be the projection of S onto the xy-plane.

17. Ex. Evaluate , where S is the surface z = x + y2,

18. Ex. Evaluate , where S is the first octant
portion of 2x + y + 2z = 6.

19. If the surface can not be written as z = g(x,y), then we need to parameterize it like last time.
Thm. If S can be represented parametrically by
, then
where D is the domain in the uv-plane.

20. Ex. Evaluate , where S is given by

21. Ex. Evaluate , where S is the unit sphere.

22. Ex. Evaluate , where S is the surface of the region
bounded by x2 + y2 = 1, z = 0, and z = 1 + x.

24. Surface Integrals of Vector Fields
Let S be an oriented surface with unit normal vector n. The surface integral of F over S is also called the flux integral of F over S.
 If F is a force causing energy to flow through our surface, the flux integral gives the rate of flow through S.

25. For surface defined by z = g(x,y), the unit normal vector is

26. Ex. Evaluate , where F = yi + xj + zk and S is
the boundary of the solid enclosed by z = 1 – x2 – y2 and z = 0.

27. For a surface that is defined parametrically, the unit normal vector is

28. Ex. Find the flux of the vector field F = zi + yj + xk across the unit sphere.