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Chapter 16 – Vector Calculus

Chapter 16 – Vector Calculus. 16.7 Surface Integrals. Objectives: Understand integration of different types of surfaces. Surface Integrals. The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length.

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Chapter 16 – Vector Calculus

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  1. Chapter 16 – Vector Calculus 16.7 Surface Integrals • Objectives: • Understand integration of different types of surfaces 16.7 Surface Integrals

  2. Surface Integrals • The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length. 16.7 Surface Integrals

  3. Surface Integrals • Suppose a surface S has a vector equation r(u, v) = x(u, v) i+ y(u, v) j + z(u, v) k (u, v) D 16.7 Surface Integrals

  4. Surface Integrals • In our discussion of surface area in Section 16.6, we made the approximation ∆Sij ≈ |rux rv| ∆u ∆v where: are the tangent vectors at a corner 16.7 Surface Integrals

  5. Surface Integrals - Equation 2 • If the components are continuous and ruand rv are nonzero and nonparallel in the interior of D, it can be shown that: 16.7 Surface Integrals

  6. Surface Integrals • Formula 2 allows us to compute a surface integral by converting it into a double integral over the parameter domain D. • When using this formula, remember that f(r(u, v) is evaluated by writing x =x(u, v), y =y(u, v), z =z(u, v) in the formula for f(x, y, z) 16.7 Surface Integrals

  7. Example 1 • Evaluate the surface integral. 16.7 Surface Integrals

  8. Graphs • Any surface S with equation z =g(x, y) can be regarded as a parametric surface with parametric equations x =x y =y z =g(x, y) • So, we have: 16.7 Surface Integrals

  9. Graphs • Therefore, Equation 2 becomes: 16.7 Surface Integrals

  10. Graphs • Similar formulas apply when it is more convenient to project Sonto the yz-plane or xy-plane. • For instance, if Sis a surface with equation y =h(x, z) and D is its projection on the xz-plane, then 16.7 Surface Integrals

  11. Example 2 – pg. 1145 # 9 • Evaluate the surface integral. 16.7 Surface Integrals

  12. Oriented Surface • If it is possible to choose a unit normal vector n at every such point (x, y, z) so that n varies continuously over S, then • S is called an oriented surface. • The given choice of n provides S with an orientation. 16.7 Surface Integrals

  13. Possible Orientations • There are two possible orientations for any orientable surface. 16.7 Surface Integrals

  14. Positive Orientation • Observe that n points in the same direction as the position vector—that is, outward from the sphere. 16.7 Surface Integrals

  15. Negative Orientation • The opposite (inward) orientation would have been obtained if we had reversed the order of the parameters because rθ x rΦ = –rΦ x rθ 16.7 Surface Integrals

  16. Closed Surfaces • For a closed surface—a surface that is the boundary of a solid region E—the convention is that: • The positive orientation is the one for which the normal vectors point outward from E. • Inward-pointing normals give the negative orientation. 16.7 Surface Integrals

  17. Flux Integral (Def. 8) • If F is a continuous vector field defined on an oriented surface S with unit normal vector n, then the surface integral of F over Sis: • This integral is also called the flux of Facross S. 16.7 Surface Integrals

  18. Flux Integral • In words, Definition 8 says that: • The surface integral of a vector field over Sis equal to the surface integral of its normal component over S (as previously defined). 16.7 Surface Integrals

  19. Flux Integral • If Sis given by a vector function r(u, v), then n is • We can rewrite Definition 8 as equation 9: 16.7 Surface Integrals

  20. Example 3 – pg. 1145 # 26 • Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 16.7 Surface Integrals

  21. Vector Fields • In the case of a surface Sgiven by a graph z =g(x, y), we can think of x and y as parameters and write: • From this, formula 9 becomes formula 10: 16.7 Surface Integrals

  22. Vector Fields • This formula assumes the upward orientation of S. • Fora downward orientation, we multiply by –1. 16.7 Surface Integrals

  23. Example 4 • Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 16.7 Surface Integrals

  24. Other Examples • In groups, please work on the following problems on page 1145: #’s 12, 14, and 28. 16.7 Surface Integrals

  25. Example 5 – pg. 1145 # 12 • Evaluate the surface integral. 16.7 Surface Integrals

  26. Example 6 – pg. 1145 # 14 • Evaluate the surface integral. 16.7 Surface Integrals

  27. Example 7 – pg. 1145 # 28 • Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 16.7 Surface Integrals

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