Lecture XXII Surfaces Recall that an elementary region ˆ D contained in E 2 is called convex if for D , the line conecting them is contained in ˆ any points in ˆ D . Also recall that a R on D is called injective if distinct points in ˆ R . map ˆ D give diﬀerent values of Using these notions we will give a deFnition for surfaces. Defnition 1 (Parametric expression For a surFace) Let O be a fxed point in E 3 and let ˆ D be a convex elementary region in E 2 , where E 2 is the Cartesian R ( u, v ) be an injective continuous Function on ˆ plane oF coordinates u, v . Let D with values in E 3 . The surface S is the set oF all points that have R ( u, v ) as position vector For some ( u, v ) ∈ ˆ D . ±or example, in the uv plane we consider the rectangle 0 ≤ u ≤ π , 0 ≤ v ≤ 2 π , ˆ ˆ and the function R ( u, v ) = sin u cos v ˆ i +sin u sin vj +cos uk . Then the surface S that has parametric representation given by R is the sphere of radius 1 centered D to be the disk of radius 1 centered at O , and at O . Also, if we take
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