To compute the area, we need to evaluate the double integral over the unit disk u+u in the parametric domain D A 1+42+42dud To perform the integration, let us change variables u= rcos(), v=rsin(0), and du do= r dr de √1+4r2rdrd x(5√5-1) 3.4 Tangent plane Tangent plane at a point r(uo, vo) is the union of tangent vectors of all curves on the surface pass through r(uo, vo), as shown in Figure 3.5. Since the tangent vector of a curve on a parametric surface is given by d =r, d+r d, the tangent plane lies on the plane of the vectors Tu and ru. The equation of the tangent plane is Tp(u, v)=r(u, v)+ Aru(u, v)+ur(u, u) where a and u are real variables parameterizing the plane 式+ Figure 3.5: The tangent plane at a point on a surface 3.5 Normal vector The surface normal is the vector at point r(uo, vo) perpendicular to the tangent plane, see Figure 3.6. And therefore (3.10) Note that ru and ru are not necessarily perpendicular.To compute the area, we need to evaluate the double integral over the unit disk u 2 +v 2 ≤ 1 in the parametric domain D; A = Z Z u2+v 2≤1 p 1 + 4u 2 + 4v 2 du dv. To perform the integration, let us change variables. u = r cos(θ), v = r sin(θ), and du dv = r dr dθ A = Z Z r≤1 p 1 + 4r 2 r dr dθ = Z 2π 0 Z 1 0 p 1 + 4r 2 r dr dθ = π 6 (5√ 5 − 1) 3.4 Tangent plane Tangent plane at a point r(uo, vo) is the union of tangent vectors of all curves on the surface pass through r(uo, vo), as shown in Figure 3.5. Since the tangent vector of a curve on a parametric surface is given by dr dt = r u du dt + rv dv dt , the tangent plane lies on the plane of the vectors ru and rv. The equation of the tangent plane is Tp(u, v) = r(u, v) + λru(u, v) + µrv(u, v) (3.9) where λ and µ are real variables parameterizing the plane. x y z r=ruu+rvv r(u0,v0) Tp Figure 3.5: The tangent plane at a point on a surface. 3.5 Normal vector The surface normal is the vector at point r(uo, vo) perpendicular to the tangent plane, see Figure 3.6. And therefore N = ru × rv |ru × rv| (3.10) Note that ru and rv are not necessarily perpendicular. 6