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Consider the sets A, B, C, D and E defined below. Briefly justify, by reference to a previous exercise or the Regular Level Set theorem, that each set is a surface. A= {(x, y, z) = R³: : xy + z = = 0}; B = {(x, y, z) = R³ : x² + y² + z² = 1}; C={(x, y, z) = R³: z = 0}; D= {(x, y, z) = R³ : y² + z² = 1}; E = {(x, y, z) = R³ : x² + y² - z = 0}. A triangle, T, on one of these surfaces, is constructed in the natural way by connect- ing 3 distinct points (the vertices of the triangle) with the shortest length curves between them. (To avoid degenerate cases, we insist that the region bounded by these curves has positive area. Thus, our three vertices are not "collinear".) Sketch pictures of the surfaces above and of an arbitrary triangle on the surface in each case. Denoting by a, ẞ and y, the angles of such a triangle decide whether the angle difference, Angle Difference = a++ is positive, zero or negative. No explanation is required.

Consider the sets A, B, C, D and E defined below. Briefly justify, by reference to a previous exercise or the Regular Level Set theorem, that each set is a surface. A= {(x, y, z) = R³: : xy + z = = 0}; B = {(x, y, z) = R³ : x² + y² + z² = 1}; C={(x, y, z) = R³: z = 0}; D= {(x, y, z) = R³ : y² + z² = 1}; E = {(x, y, z) = R³ : x² + y² - z = 0}. A triangle, T, on one of these surfaces, is constructed in the natural way by connect- ing 3 distinct points (the vertices of the triangle) with the shortest length curves between them. (To avoid degenerate cases, we insist that the region bounded by these curves has positive area. Thus, our three vertices are not "collinear".) Sketch pictures of the surfaces above and of an arbitrary triangle on the surface in each case. Denoting by a, ẞ and y, the angles of such a triangle decide whether the angle difference, Angle Difference = a++ is positive, zero or negative. No explanation is required.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter10: Analytic Geometry
Section10.6: The Three-dimensional Coordinate System
Problem 41E: Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?
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Consider the sets A, B, C, D and E defined below. Briefly justify, by reference to a previous exercise or the Regular Level Set theorem, that each set is a surface. A= {(x, y, z) = R³: : xy + z = = 0}; B = {(x, y, z) = R³ : x² + y² + z² = 1}; C={(x, y, z) = R³: z = 0}; D= {(x, y, z) = R³ : y² + z² = 1}; E = {(x, y, z) = R³ : x² + y² - z = 0}. A triangle, T, on one of these surfaces, is constructed in the natural way by connect- ing 3 distinct points (the vertices of the triangle) with the shortest length curves between them. (To avoid degenerate cases, we insist that the region bounded by these curves has positive area. Thus, our three vertices are not "collinear".) Sketch pictures of the surfaces above and of an arbitrary triangle on the surface in each case. Denoting by a, ẞ and y, the angles of such a triangle decide whether the angle difference, Angle Difference = a++ is positive, zero or negative. No explanation is required.
Transcribed Image Text:Consider the sets A, B, C, D and E defined below. Briefly justify, by reference to a previous exercise or the Regular Level Set theorem, that each set is a surface. A= {(x, y, z) = R³: : xy + z = = 0}; B = {(x, y, z) = R³ : x² + y² + z² = 1}; C={(x, y, z) = R³: z = 0}; D= {(x, y, z) = R³ : y² + z² = 1}; E = {(x, y, z) = R³ : x² + y² - z = 0}. A triangle, T, on one of these surfaces, is constructed in the natural way by connect- ing 3 distinct points (the vertices of the triangle) with the shortest length curves between them. (To avoid degenerate cases, we insist that the region bounded by these curves has positive area. Thus, our three vertices are not "collinear".) Sketch pictures of the surfaces above and of an arbitrary triangle on the surface in each case. Denoting by a, ẞ and y, the angles of such a triangle decide whether the angle difference, Angle Difference = a++ is positive, zero or negative. No explanation is required.
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