April 17: Surface Area and Surface Integrals
                                          April 17: Surface Area and Surface Integrals
        Parametrized Surfaces
        How should we integrate a scalar function f(x,y,z) over a surface S?
        Step 1. Subdivide S into finitely many smaller surfaces Si of the same area
        ∆S. We are using ∆S for a small element of surface area.
        Step 2. Choose a point (x ,y ,z ) from the component S .
                                  i  i  i                        i
        Step 3. Multiply f(x ,y ,z ) by the size of each S to get f(x ,y ,z ) ∆S.
                             i i  i                       i          i  i  i
        Step 4. Add the products in Step 3 to get the Riemann sum : Σ f(x ,y ,z )∆S.
                                                                         i   i i  i
        Step 5. Take the limit of the Riemann sums as ∆S → 0, to get:
                                      Z Z f(x,y,z) dS,
                                           S
        the surface integral of f (x,y,) over S.
        Wewrite a double integral, since our domain of integration is two-dimensional.
                                                  April 17: Surface Area and Surface Integrals
        Parametrized Surfaces
        Following an analogy with curves, we should have:
         (i) A way to calculate surface area.
         (ii) A way to describe or parametrize a surface as a function of two variables.
        Definition. Given a surface S ⊆ R3, a parametrization of S will be a function
                               G(u,v) = (x(u,v),y(u,v),z(u,x)),
        such that S = G(D) for some domain D in the uv-plane.
        As usual, we assume that all first order partials exists and are continuous, at
        least on the interior of D
                                                   April 17: Surface Area and Surface Integrals
        Examples of parametrized surfaces
        Spherical and cylindrical coordinates tell us how to parametrize spheres and
        cylinders.
        Example. For the sphere of radius R centered at the origin we take:
                     G(φ,θ) = (R sin(φ)cos(θ),R sin(φ)sin(θ),R cos(φ)),
        with 0 ≤ φ ≤ π,0 ≤ θ < 2π.
                                                  April 17: Surface Area and Surface Integrals