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Parameterized Surfaces Definition A parameterized surface x U R2 R3 is a differentiable map x from an open set U R2 into R3 The set x U R3 is called the trace of x x is regular if the differential dxq R2 R3 is one to one for all q U i e the vectors x u x v are linearly independent for all q U A point p U where dxp is not one toone is called a singular point of x Proposition Let x U R2 R3 be a regular parameterized surface and let q U Then there exists a neighborhood V of q in R2 such that x V R3 is a regular surface Tangent Plane Definition 1 By a tangent vector to a regular surface S at a point p S we mean the tangent vector 0 0 of a differentiable parameterized curve S with 0 p Proposition 1 Let x U R2 S be a parameterization of a regular surface S and let q U The vector subspace of dimension 2 dxq R2 R3 coincides with the set of tangent vectors to S at x q Definition 2 By Proposition 1 the plane dxq R2 which passes through x q p does not depend on the parameterization x This plane is called the tangent plane to S at p and will be denoted by Tp S The choice of the parameterization x determines 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 the basis associated to a parameterization x are determined as follows w is the velocity vector 0 0 of a curve x where U is given by t u t v t with 0 q x 1 p Thus d d x 0 x u t v t 0 dt dt 0 xu q u 0 xv q v 0 0 0 0 w Thus in the basis xu q xv q w has coordinates u0 0 v 0 0 where u t v t is the expression of a curve whose velocity vector at t 0 is w Let S1 and S2 be two regular surfaces and let V S1 S2 be a differentiable mapping of an open set V of S1 into S2 If p V then every tangent vector w Tp S1 is the velocity vector 0 0 of a differentiable parameterized curve V with 0 p The curve is such that 0 p and therefore 0 0 is a vector of T p S2 Proposition 2 In the discussion above given w the vector 0 0 does not depend on the choice of The map d p Tp S1 T p S2 defined by d p w 0