Parametric Surfaces.

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Presentation transcript:

Parametric Surfaces

1. Graph the vector-valued function (Similar to p.1109 #1-6)

Definition of Parametric Surface Let x, y, and z be functions of u and v that are continuous on a domain D in the uv-plane. The set of points (x, y, z) given by r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k Is called a parametric surface. The equations x = x(u, v), y = y(u, v), and z = z(u, v) Are the parametric equations for the surface

2. Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph (Similar to p.1109 #7-10)

3. Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. (Similar to p.1109 #7-10)

4. Find a vector-valued function whose graph is the indicated surface 4. Find a vector-valued function whose graph is the indicated surface. (Similar to p.1109 #21-30)

5. Find a vector-valued function whose graph is the indicated surface 5. Find a vector-valued function whose graph is the indicated surface. (Similar to p.1109 #21-30)

6. Find a vector-valued function whose graph is the indicated surface 6. Find a vector-valued function whose graph is the indicated surface. (Similar to p.1109 #21-30)

7. Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. (Similar to p.1110 #31-34)

8. Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. (Similar to p.1110 #31-34)

Finding a Tangent Plane to a Parametric Surface Find what u and v are equal to given the point (xo, yo, zo) Find the partials ru and rv Find the cross product ru x rv Plug in your u and v values into the cross product which gives the normal vector ai + bj + ck Tangent plane is: a(x – xo) + b(y – yo) + c(z – zo) = 0

9. Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. (Similar to p.1110 #35-38)

Area of a Parametric Surface Let S be a smooth parametric surface r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k Defined over an open region D in the uv-plane. If each point on the surface S corresponds to exactly one point in the domain D, then the surface area of S is given by

10. Find the area of the surface over the given region. (Similar to p

11. Find the area of the surface over the given region. (Similar to p