Parameterized SurfacesDefinition:A parameterized surface x : U ⊂ R2→ R3is adifferentiable map x from an open set U ⊂ R2into R3. The set x(U) ⊂ R3is called the traceof x.x is regular if the differential dxq: R2→ R3is one-to-one for all q ∈ U (i.e., the vectors∂x/∂u, ∂x/∂v are linearly independent for allq ∈ U). A point p ∈ U where dxpis not one-to-one is called a singular point of x.Proposition:Let x : U ⊂ R2→ R3be a regular parameter-ized surface and let q ∈ U. Then there exists aneighborhood V of q in R2such that x(V ) ⊂ R3is a regular surface.Tangent PlaneDefinition 1:By a tangent vector to a regular surface S ata point p ∈ S, we mean the tangent vectorα0(0) of a differentiable parameterized curveα : (−², ²) → S with α(0) = p.Proposition 1:Let x : U ⊂ R2→ S be a parameterization ofa regular surface S and let q ∈ U. The vectorsubspace of dimension 2,dxq(R2) ⊂ R3coincides with the set of tangent vectors to Sat x(q).Definition 2:By Proposition 1, the plane dxq(R2), whichpasses through x(q) = p, does not depend onthe parameterization x. This plane is called thetangent plane to S at p and will be denoted byTp(S).The choice of the parameterization x deter-mines a basis {(∂x/∂u)(q), (∂x/∂v)(q)} of Tp(S),called the basis associated to x.The coordinates of a vector w ∈ Tp(S) in thebasis associated to a parameterization x aredetermined as follows:w is the velocity vector α0(0) of a curve α =x ◦ β, where β : (−², ²) → U is given by β(t) =(u(t), v(t)), with β(0) = q = x−1(p). Thus,α0(0) =ddt(x ◦ β)(0) =ddtx(u(t), v(t))(0)= xu(q)u0(0) + xv(q)v0(0)= wThus, in the basis {xu(q), xv(q)}, w has coor-dinates (u0(0), v0(0)), where (u(t), v(t)) is theexpression of a curve whose velocity vector att = 0 is w.Let S1and S2be two regular surfaces and letϕ : V ⊂ S1→ S2be a differentiable mappingof an open set V of S1into S2. If p ∈ V , thenevery tangent vector w ∈ Tp(S1) is the velocityvector α0(0) of a differentiable parameterizedcurve α : (−², ²) → V with α(0) = p. Thecurve β = ϕ ◦ α is such that β(0) = ϕ(p), andtherefore β0(0) is a vector of Tϕ(p)(S2).Proposition 2:In the discussion above, given w, the vectorβ0(0) does not depend on the choice of α.The map dϕp: Tp(S1) → Tϕ(p)(S2) defined bydϕp(w) = β0(0) is linear.This proposition shows that β0(0) depends onlyon the map ϕ and the coordinates (u0(0), v0(0))of w in the basis {xu, xv}.The linear map dϕpis called the differential ofϕ at p ∈ S1. In a similar way, we can define thedifferential of a differentiable function f : U ⊂S → R at p ∈ U as a linear map dfp: Tp(S) → R.Proposition 3:If S1and S2are regular surfaces and ϕ : U ⊂S1→ S2is a differentiable mapping of an openset U ⊂ S1such that the differential dϕpof ϕat p ∈ U is an isomorphism, then ϕ is a localdiffeomorphism at p.The First Fundamental FormDefinition 1:The quadratic form Ip(w) = < w, w >p= |w|2≥0 on Tp(S) is called the first fundamental formof the regular surface S ⊂ R3at p ∈ S.The first fundamental form is merely the ex-pression of how the surface S inherits the nat-ural inner product of R3. And by knowing Ip,we can treat metric questions on a regular sur-face without further references to the ambientspace R3.In the basis of {xu, xv} associated to a param-eterization x(u, v) at p, since a tangent vectorw ∈ Tp(S) is the tangent vector to a param-eterized curve α(t) = x(u(t), v(t)), t ∈ (−², ²),with p = α(0) = x(u0, v0), we haveIp(α0(0)) = < α0(0), α0(0) >p= < xuu0+ xvv0, xuu0+ xvv0>p= < xu, xu>p(u0)2+ 2< xu, xv>pu0v0+ < xv, xv>p(v0)2= E(u0)2+ 2F u0v0+ G(v0)2where the values of the functions involved arecomputed for t = 0, andE(u0, v0) = < xu, xu>pF (u0, v0) = < xu, xv>pG(u0, v0) = < xv, xv>pare the coefficients.Definition 2:Let R ⊂ S be a bounded region of a regularsurface contained in the coordinate neighbor-hood of the parameterization x : U ⊂ R2→ S.The positive numberA =Z Z|xu× xv| dudv=Z Zq(EG − F2) dudvis called the area of R.Gauss MapIn the study of regular curve, the rate of changeof the tangent line to a curve C leads to an im-portant geometry entity, the curvature.Here, we will try to measure how rapidly a sur-face S pulls away from the tangent plane Tp(S)in a neighborhood of a point p ∈ S. This isequivalent to measuring the rate of change atp of a unit normal vector field N on a neigh-borhood of p, which is given by a linear mapon Tp(S).Definition 1:Given a parameterization x : U ⊂ R2→ S ofa regular surface S at a point p ∈ S, a unitnormal vector can be chosen at each point ofx(U) by the ruleN(q) =xu× xv|xu× xv|(q)This way, we have a differentiable map N :x(U) → R3that associates to each q ∈ x(U) aunit normal vector N(q).More generally, if V ⊂ S is an open set in Sand N : V → R3is a differentiable map whichassociates to each q ∈ V a unit normal vectorat q, we say that N is a differentiable field ofunit normal vectors on V.Definition 2:A regular surface is orientable if it admits a dif-ferentiable field of unit normal vectors definedon the whole surface, and the choice of sucha field N is called an orientation of S.An orientation N on S induces an orientationon each tangent plane Tp(S), p ∈ S, as follows.Define a basis {v, w ∈ Tp(S)} to be positive if< v × w, N > is positive.While every surface is locally orientable, notall surfaces admit a differentiable field of unitnormal vectors defined on the whole surface(i.e., the Mobius strip).Definition 3:Let S ⊂ R3be a surface with an orientation N.The map N : S → R3takes its values in theunit sphereS2= {(x, y, z) ∈ R3; x2+ y2+ z2= 1}The map N : S → S2, thus defined, is calledthe Gauss map of S.The linear map dNp: Tp(S) → Tp(S) operatesas follows. For each parameterized curve α(t)in S with α(0) = p, we consider the param-eterized curve N ◦ α(t) = N(t) in the sphereS2, this amounts to restricting the normal vec-tor N to the curve α(t). The tangent vectorN0(0) = dNP(α0(0)) is a vector in Tp(S). Itmeasures the rate of change of the normal vec-tor N, restricted to the curve α(t), at t = 0.Thus, dNpmeasures how N pulls away fromN(p) in a neighborhood of p.Definition 4:A linear map A : V → V is self-adjoint if <Av, w >=< v, Aw > for all v, w ∈ V .Proposition 1:The differential dNp: Tp(S) → Tp(S) of theGauss map is a self-adjoint linear map.This proposition allows us to associate to dNpa quadratic form Q in Tp(S), given by Q(v)