Chapter 15
Section 15.5: Parametric Surfaces

• Parametric lines: We often say that we have parametric lines
  In a plane: `bbr(t) = x(t)bbi + y(t)bbj `
  In space:   `bbr(t) = x(t)bbi + y(t)bbj + z(t)bbk `
  but what we really mean in that the line is the set of endpoints of these vectors, in standard position.
• Parametric surfaces: if we have
  `bbr(u,v) =x(u,v)bbi +y(u,v)bbj + z(u,v)bbk`
  for `u` and `v` in some domain, much as we had `a <= t <= b`   in the case of a line.
  See http://www.flashandmath.com/, but note that they use "s,t" instead of "u,v".
• convert from `z=f(x,y)`  use `x(u,v)=x, \quad y(u,v)=y \quad z(u,v)=f(u,v) ` to get `bbr(u,t)=ubbi + ubbj + f(u,t)bbk`, which we could rewrite as `bbr(x,y)=xbbi + ybbj + f(x,y)bbk`.
• For rotation about an axis
  1. about the x-axis
     Tanslate `y=f(x)` or `z=f(x)` for `a<=x<=b` use `x(u,v)=u, \quad y(u,v)=cos(v)*f(u) \quad z(u,v)=sin(v)*f(u)` for `a<=u<=b` and `0<=v<=2pi` or `bbr(u,v)=ubbi + cos(v)*f(u)bbj + sin(v)*f(u)bbk` for `a<=u<=b` and `0<=v<=2pi `
  2. about the y-axis
     Tanslate `x=f(y)` or `z=f(y)` for `a<=y<=b` use `x(u,v)=cos(v)*f(u), \quad y(u,v)=u \quad z(u,v)=sin(v)*f(u)` for `a<=u<=b` and `0<=v<=2pi` or `bbr(u,v)=cos(v)*f(u)bbi + ubbj + sin(v)*f(u)bbk` for `a<=u<=b` and `0<=v<=2pi `
  3. about the z-axis
     Tanslate `x=f(z)` or `y=f(z)` for `a<=z<=b` use `x(u,v)=cos(v)*f(u), \quad y(u,v)=sin(v)*f(u) \quad z(u,v)=u` for `a<=u<=b` and `0<=v<=2pi` or `bbr(u,v)=cos(v)*f(u)bbi + sin(v)*f(u)bbj + ubbk` for `a<=u<=b` and `0<=v<=2pi `
• Within the parametric equation for a surface, `bbr(u,v) =x(u,v)bbi +y(u,v)bbj + z(u,v)bbk`, we can take partial derivatives with respect to the variables `u` and `v` (not the unit vector `bbu`).
• `bbr_u(u,v) =(del x)/(del u)(u,v)bbi +(del y)/(del u)(u,v)bbj +(del z)/(del u)(u,v)bbk` and
  `bbr_v(u,v) =(del x)/(del v)(u,v)bbi +(del y)/(del v)(u,v)bbj +(del z)/(del v)(u,v)bbk`

  each is a tangent vector at a point and they are tangent to the point in the direction of the line (curve) generated by holding `v` or `u` constant, respectively. Thus, `bbr_u(u_0,v_0)` is tangent to the surface at `(u_0,v_0)` in the direction of the curve `bbr(u,v_0)`.

• Since we have the two tangent vectors, `bbr_u " and" bbr_v`, we can get the normal vector, `bbN`, and therefore, the coefficients of the tangent plane, by looking at `bbN = bbr_u(u_0,v_0) xx bbr_v(u_0,v_0) = |[bbi,bbj,bbk],[(del x)/(del u),(del y)/(del u),(delz)/(del u)],[(del x)/(del v),(del y)/(del v),(delz)/(del v)]|`
• Area of a Parametric Surface:
  Let `S` be a smooth parametric surface `bbr(u,v) =x(u,v)bbi +y(u,v)bbj + z(u,v)bbk` defined over an open region `D` in the `uv`-plane. If each point on the surface `S` corresponds to exactly one point in the domain `D`, then the surface area of `S` is given by
  `"Surface area" = int_Sint dS = int_Dint ||bbr_u xx bbr_v|| dA`
  where `bbr_u(u,v) =(del x)/(del u)(u,v)bbi +(del y)/(del u)(u,v)bbj +(del z)/(del u)(u,v)bbk` and
  `bbr_v(u,v) =(del x)/(del v)(u,v)bbi +(del y)/(del v)(u,v)bbj +(del z)/(del v)(u,v)bbk`