Surface area and surface integrals. (Sect. 16.5)
                          ◮ Review: Arc length and line integrals.
                          ◮ Review: Double integral of a scalar function.
                          ◮ Explicit, implicit, parametric equations of surfaces.
                          ◮ The area of a surface in space.
                                 ◮ The surface is given in parametric form.
                                 ◮ The surface is given in explicit form.
                 Review: Arc length and line integrals
                          ◮ The integral of a function f : [a,b] → R is
                              Z b                          n
                                   f (x)dx = lim Xf(x∗)∆x.
                                                  n→∞              i
                                a                        i=0
                          ◮ The arc length of a curve r : [t0,t1] → R3 in space is
                                        Z t
                                             1   ′
                              st ,t  =         |r (t)|dt.
                                1 0
                                          t
                                           0
                          ◮ The integral of a function f : R3 → R along a curve
                                                        Z             Z t
                                                                          1         
 ′
                              r : [t0, t1] → R3 is          f ds =          f  r(t) |r (t)|dt.
                                                          C             t
                                                                         0
                          ◮ The circulation of a function F : R3 → R3 along a curve
                                                        Z                  Z t
                                                                               1         
    ′
                              r : [t0, t1] → R3 is          F·uds =              F r(t) ·r (t)dt.
                                                          C                 t
                                                                             0
                          ◮ The flux of a function F : {z = 0}∩R3 → {z = 0}∩R3 along
                              a loop r : [t0,t1] → {z = 0} ∩ R3 is F = I F · nds.
                                                                                         C
            Surface area and surface integrals. (Sect. 16.5)
                   ◮ Review: Arc length and line integrals.
                   ◮ Review: Double integral of a scalar function.
                   ◮ Explicit, implicit, parametric equations of surfaces.
                   ◮ The area of a surface in space.
                        ◮ The surface is given in parametric form.
                        ◮ The surface is given in explicit form.
            Review: Double integral of a scalar function
                   ◮ The double integral of a function f : R ⊂ R2 → R on a region
                      R ⊂R2, which is the volume under the graph of f and above
                      the z = 0 plane, and is given by
                                ZZ                 n   n
                                     f dA = lim XXf(x∗,y∗)∆x∆y.
                                            n→∞              i  j
                                   R              i=0 j=0
                   ◮ The area of a flat surface R ⊂ R2 is the particular case f = 1,
                      that is, A(R) = ZZ dA.
                                         R
                 Wewill show how to compute:
                   ◮ The area of a non-flat surface in space. (Today.)
                   ◮ The integral of a scalar function f on a surface is space.
                   ◮ The flux of a vector-valued function F on a surface in space.
            Surface area and surface integrals. (Sect. 16.5)
                   ◮ Review: Arc length and line integrals.
                   ◮ Review: Double integral of a scalar function.
                   ◮ Explicit, implicit, parametric equations of surfaces.
                   ◮ The area of a surface in space.
                        ◮ The surface is given in parametric form.
                        ◮ The surface is given in explicit form.
            Explicit, implicit, parametric equations of surfaces
                 Review: Curves on R2 can be defined in:
                   ◮ Explicit form, y = f (x);
                   ◮ Implicit form, F(x,y) = 0;
                   ◮ Parametric form, r(t) = hx(t),y(t)i.
                      The vector r′(t) = hx′(t),y′(t)i is tangent to the curve.
                 Review: Surfaces in R3 can be defined in:
                   ◮ Explicit form, z = f (x,y);
                   ◮ Implicit form, F(x,y,z) = 0;
                   ◮ Parametric form, r(u,v) = hx(u,v),y(u,v),z(u,v)i.
                      Two vectors tangent to the surface are
                               ∂ r(u,v) = h∂ x(u,v),∂ y(u,v),∂ z(u,v)i,
                                u            u          u         u
                               ∂ r(u,v) = h∂ x(u,v),∂ y(u,v),∂ z(u,v)i.
                                v            v          v         v
            Explicit, implicit, parametric equations of surfaces
                Example                                       p
                Find a parametric expression for the cone z =    x2 +y2, and two
                tangent vectors.
                Solution: Use cylindrical coordinates: x = r cos(θ), y = r sin(θ),
                z = z. Parameters of the surface: u = r, v = θ. Then
                        x(r,θ) = r cos(θ),  y(r,θ) = r sin(θ),  z(r,θ) = r.
                Using vector notation, a parametric equation of the cone is
                                   r(r,θ) = hr cos(θ),r sin(θ),ri.
                Two tangent vectors to the cone are ∂ r and ∂ r,
                                                      r        θ
                      ∂ r = hcos(θ),sin(θ),1i,  ∂ r = h−r sin(θ),r cos(θ),0i.
                       r                          θ                             ⊳
            Explicit, implicit, parametric equations of surfaces
                Example
                Find a parametric expression for the sphere x2 + y2 + z2 = R2,
                and two tangent vectors.
                Solution: Use spherical coordinates:
                x = ρcos(θ)sin(φ), y = ρsin(θ)sin(φ), z = ρcos(φ).
                Parameters of the surface: u = θ, v = φ.
                    x = Rcos(θ)sin(φ),     y = Rsin(θ)sin(φ),    z = Rcos(φ).
                Using vector notation, a parametric equation of the cone is
                         r(θ,φ) = R hcos(θ)sin(φ), sin(θ)sin(φ), cos(φ)i.
                Two tangent vectors to the paraboloid are ∂ r and ∂ r,
                                                            θ       φ
                            ∂ r = R h−sin(θ)sin(φ), cos(θ)sin(φ), 0i,
                             θ
                         ∂φr = R hcos(θ)cos(φ), sin(θ)cos(φ), −sin(φ)i.         ⊳