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

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Surface Integrals The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length Surface Integrals2Dr. Erickson

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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 Integrals3Dr. Erickson

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Surface Integrals In our discussion of surface area in Section 16.6, we made the approximation ∆S ij ≈ |r u x r v | ∆u ∆v where: are the tangent vectors at a corner 16.7 Surface Integrals4Dr. Erickson

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Surface Integrals - Equation 2 If the components are continuous and r u and r v are nonzero and nonparallel in the interior of D, it can be shown that: 16.7 Surface Integrals5Dr. Erickson

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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 Integrals6Dr. Erickson

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Example 1 Evaluate the surface integral Surface Integrals7Dr. Erickson

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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 Integrals8Dr. Erickson

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Graphs Therefore, Equation 2 becomes: 16.7 Surface Integrals9Dr. Erickson

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Graphs Similar formulas apply when it is more convenient to project S onto the yz-plane or xy-plane. For instance, if S is a surface with equation y = h(x, z) and D is its projection on the xz-plane, then 16.7 Surface Integrals10Dr. Erickson

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Example 2 – pg # 9 Evaluate the surface integral Surface Integrals11Dr. Erickson

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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 Surface Integrals12Dr. Erickson

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Possible Orientations There are two possible orientations for any orientable surface Surface Integrals13Dr. Erickson

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Positive Orientation Observe that n points in the same direction as the position vector—that is, outward from the sphere Surface Integrals14Dr. Erickson

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Negative Orientation The opposite (inward) orientation would have been obtained if we had reversed the order of the parameters because r θ x r Φ = –r Φ x r θ 16.7 Surface Integrals15Dr. Erickson

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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 Surface Integrals16Dr. Erickson

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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 S is: ◦ This integral is also called the flux of F across S Surface Integrals17Dr. Erickson

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Flux Integral In words, Definition 8 says that: ◦ The surface integral of a vector field over S is equal to the surface integral of its normal component over S (as previously defined) Surface Integrals18Dr. Erickson

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Flux Integral If S is given by a vector function r(u, v), then n is We can rewrite Definition 8 as equation 9: 16.7 Surface Integrals19Dr. Erickson

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Example 3 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 Surface Integrals20Dr. Erickson

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Vector Fields In the case of a surface S given 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 Integrals21Dr. Erickson

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Vector Fields ◦ This formula assumes the upward orientation of S. ◦ For a downward orientation, we multiply by – Surface Integrals22Dr. Erickson

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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 Surface Integrals23Dr. Erickson

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Other Examples In groups, please work on the following problems on page 1145: #’s 12, 14, and Surface Integrals24Dr. Erickson

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Example 5 – pg # 12 Evaluate the surface integral Surface Integrals25Dr. Erickson

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Example 6 – pg # 14 Evaluate the surface integral Surface Integrals26Dr. Erickson

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Example 7 – pg # 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 Surface Integrals27Dr. Erickson

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