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

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

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

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

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

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

Example 1 Evaluate the surface integral Surface Integrals7Dr. Erickson

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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