Chapter 16 – Vector Calculus

1. Chapter 16 – Vector Calculus 16.6 Parametric Surfaces and their Areas • Objectives: • Understand the various types of parametric surfaces. • Compute the area using vector functions. 16.6 Parametric Surfaces and their Areas

2. Vector Calculus • So far, we have considered special types of surfaces: • Cylinders • Quadric surfaces • Graphs of functions of two variables • Level surfaces of functions of three variables 16.6 Parametric Surfaces and their Areas

3. Vector Calculus • Here, we use vector functions to describe more general surfaces, called parametric surfaces, and compute their areas. • Then, we take the general surface area formula and see how it applies to special surfaces. 16.6 Parametric Surfaces and their Areas

4. Introduction • We describe a space curve by a vector function r(t) of a single parameter t. • Similarly,we can describe a surface by a vector function r(u, v) of two parameters u and v. 16.6 Parametric Surfaces and their Areas

5. Equation 1 • We suppose that r(u, v) = x(u, v) i + y(u, v) j + z (u, v) kis a vector-valued function defined on a region D in the uv-plane. 16.6 Parametric Surfaces and their Areas

6. Equation 2 • The set of all points (x, y, z) in 3 such that x = x(u, v) y = y(u, v) z = z(u, v)and (u, v) varies throughoutD, is called a parametric surface S. • Equations 2 are called parametric equationsof S. 16.6 Parametric Surfaces and their Areas

7. Example 1 – pg. 1132 #2 • Determine whether the points P and Q lie on the given surface. 16.6 Parametric Surfaces and their Areas

8. Parametric Surfaces • Each choice of uand v gives a point on S. • By making all choices, we get all of S. • In other words, the surface S is traced out by the tip of the position vector r(u, v) as (u, v) moves throughout the region D. 16.6 Parametric Surfaces and their Areas

9. Example 2 – pg. 1132 # 5 • Indentify the surface with the given vector equation. 16.6 Parametric Surfaces and their Areas

10. Example 3 – pg. 1132 • Match the equations with the graphs labeled I – VI and give reasons for your answers. 16.6 Parametric Surfaces and their Areas

11. Parametric Representation • In Example 1 we were given a vector equation and asked to graph the corresponding parametric surface. • In the following examples, however, we are given the more challenging problem of finding a vector function to represent a given surface. • In the rest of the chapter, we will often need to do exactly that. 16.6 Parametric Surfaces and their Areas

12. Example 4 • Find a parametric representation of the spherex2 + y2 + z2 = a2 16.6 Parametric Surfaces and their Areas

13. Applications – Computer Graphics • One of the uses of parametric surfaces is in computer graphics. • The figure shows the result of trying to graph the sphere x2 + y2 + z2 = 1 by: • Solving the equation for z. • Graphing the top and bottom hemispheres separately. 16.6 Parametric Surfaces and their Areas

14. Computer Graphics • Part of the sphere appears to be missing because of the rectangular grid system used by the computer. 16.6 Parametric Surfaces and their Areas

15. Computer Graphics • The much better picture here was produced by a computer using the parametric equations found in the example 2. 16.6 Parametric Surfaces and their Areas

16. Parameters • In general, a surface given as the graph of a function of x and y—an equation of the form z = f(x, y)—can always be regarded as a parametric surface by: • Taking x and y as parameters. • Writing the parametric equations as x = xy = yz = f(x, y) 16.6 Parametric Surfaces and their Areas

17. Example 5 • Find a parametric representation for the surface. 16.6 Parametric Surfaces and their Areas

18. Tangent Planes • We now find the tangent plane to a parametric surface S traced out by a vector function r(u, v) = x(u, v) i + y(u, v) j + z(u, v) kat a point P0with position vector r(u0, v0). 16.6 Parametric Surfaces and their Areas

19. Tangent Planes • Keeping u constant by putting u = u0, r(u0, v) becomes a vector function of the single parameter vand defines a grid curve C1 lying on S. 16.6 Parametric Surfaces and their Areas

20. Tangent Planes – Equation 4 • The tangent vector to C1 at P0 is obtained by taking the partial derivative of r with respect to v: 16.6 Parametric Surfaces and their Areas

21. Tangent Planes • Similarly, keeping v constant by putting v = v0, we get a grid curve C2 given by r(u, v0) that lies on S. • Its tangent vector at P0 is: 16.6 Parametric Surfaces and their Areas

22. Smooth Surface • If ru x rvis not 0, then the surface is called smooth(it has no “corners”). • For a smooth surface, the tangent plane is the plane that contains the tangent vectors ru and rv, and the vector ru x rvis a normal vector to the tangent plane. 16.6 Parametric Surfaces and their Areas

23. Example 6 • Find an equation of the tangent plane to the given parametric surface at the specified point.. 16.6 Parametric Surfaces and their Areas

24. Definition 6 – Surface Area • Suppose a smooth parametric surface S is: • Given by the equation : r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k (u, v)D • Covered just once as (u, v) ranges throughout the parameter domain D. 16.6 Parametric Surfaces and their Areas

25. Definition 6 continued • Then, the surface areaof S is where: 16.6 Parametric Surfaces and their Areas

26. Surface Area of the Graph of a Function • Now, consider the special case of a surface S with equation z = f(x, y), where (x, y) lies in D and f has continuous partial derivatives. • Here, we take x and y as parameters. • The parametric equations are:x = xy = yz = f(x, y) 16.6 Parametric Surfaces and their Areas

27. Surface Area of the Graph of a Function • Then, the surface area formula in Definition 6 becomes: • (this is formula 9) 16.6 Parametric Surfaces and their Areas

28. Example 7 • Find the area of the surface. 16.6 Parametric Surfaces and their Areas

29. Example 8 – pg. 1133 • Find the area of the surface. 16.6 Parametric Surfaces and their Areas