Two surfaces S and S^(-) with a common point p have contact order ≥ 2 at p if there exist parametrization x(u,v) and x^(-)(u,v) in p of S and S^(-) respectively such that xu = x^(-)u, xv = x^(-)v, xuu = x^(-)uu, xuv = x^(-)uv, xvv = x^(-)vv at p. Prove the following: a. Let S and S^(-) have contact order greater than or equal to 2 at p; x:U -> S and x^(-): U -> S^(-) be arbitrary parametrizations in p of S and S^(-) respectively and f: V c R^(3) -> R be a differentiable function in a neighborhood V of p in R^(3). Then the partial derivatives of order smaller than or equal to 2 of f o x^(-): U -> R are zero in x bar^(-1)(p) iff the partial derivatives of order smaller than or equal to 2 of f o x: U -> R are zero in x^(-1) (p). b. Let S and S^(-) have contact of order smaller than or equal to 2 at p. Let z = f(x, y), z = f^(-) (x, y) be the equations in a neighborhood of p, of S and S^(-) respectively where the xy plane is the common tangent plane at p = (0, 0). Then the function f(x, y) - f^(-) (x, y) has all partial derivatives of order smaller than or equal to 2 at (0, 0) equal to zero. c. Let p be a point in a surfaces S c R^(3). Let Oxyz be a Cartesian coordinate system for R^(3) such that O = p and the xy plane is the tangent plane of S at p. Show that the paraboloid z = 1/2 (x^(2) fxx + 2xy fxy + y^(2) fyy), obtained by neglecting third and higher order terms in the Taylor development around p = (0, 0) has contact of order greater than or equal to 2 at p with S. d. If a paraboloid has contact of order greater than or equal to 2 with a surface S at p, then it is the osculating paraboloid of S at p. e. If two surfaces have contact of order greater than or equal to 2 at p, then the osculating paraboloids of S and S^(-) at p coincide. Conclude that the Gaussian and mean curvatures of S and S^(-) at p are equal. f. The notion of contact of order greater than or equal to 2 is invariant by diffeomorphisms of R^(3) ; that is, if S and S^(-) have contact order greater than or equal to 2 at p and phi : R^(3) - R^(3) is a diffeomorphism then phi(S) and phi(S^(-)) have contact of order greater than or equal to 2 at phi(p). g. If S and S^(-) have contact of order greater than or equal to 2 at p then lim r -> 0 d/r^(2) = 0, where d is the length of the segment cut by the surfaces in a straight line normal to Tp(S) = Tp(S^(-)) which is a distance r from p.

Two surfaces S and S^(-) with a common point p have contact order ≥ 2 at p if there exist parametrization x(u,v) and x^(-)(u,v) in p of S and S^(-) respectively such that xu = x^(-)u, xv = x^(-)v, xuu = x^(-)uu, xuv = x^(-)uv, xvv = x^(-)vv at p. Prove the following: a. Let S and S^(-) have contact order greater than or equal to 2 at p; x:U -> S and x^(-): U -> S^(-) be arbitrary parametrizations in p of S and S^(-) respectively and f: V c R^(3) -> R be a differentiable function in a neighborhood V of p in R^(3). Then the partial derivatives of order smaller than or equal to 2 of f o x^(-): U -> R are zero in x bar^(-1)(p) iff the partial derivatives of order smaller than or equal to 2 of f o x: U -> R are zero in x^(-1) (p). b. Let S and S^(-) have contact of order smaller than or equal to 2 at p. Let z = f(x, y), z = f^(-) (x, y) be the equations in a neighborhood of p, of S and S^(-) respectively where the xy plane is the common tangent plane at p = (0, 0). Then the function f(x, y) - f^(-) (x, y) has all partial derivatives of order smaller than or equal to 2 at (0, 0) equal to zero. c. Let p be a point in a surfaces S c R^(3). Let Oxyz be a Cartesian coordinate system for R^(3) such that O = p and the xy plane is the tangent plane of S at p. Show that the paraboloid z = 1/2 (x^(2) fxx + 2xy fxy + y^(2) fyy), obtained by neglecting third and higher order terms in the Taylor development around p = (0, 0) has contact of order greater than or equal to 2 at p with S. d. If a paraboloid has contact of order greater than or equal to 2 with a surface S at p, then it is the osculating paraboloid of S at p. e. If two surfaces have contact of order greater than or equal to 2 at p, then the osculating paraboloids of S and S^(-) at p coincide. Conclude that the Gaussian and mean curvatures of S and S^(-) at p are equal. f. The notion of contact of order greater than or equal to 2 is invariant by diffeomorphisms of R^(3) ; that is, if S and S^(-) have contact order greater than or equal to 2 at p and phi : R^(3) - R^(3) is a diffeomorphism then phi(S) and phi(S^(-)) have contact of order greater than or equal to 2 at phi(p). g. If S and S^(-) have contact of order greater than or equal to 2 at p then lim r -> 0 d/r^(2) = 0, where d is the length of the segment cut by the surfaces in a straight line normal to Tp(S) = Tp(S^(-)) which is a distance r from p.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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