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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}; 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++-π, 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}; 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++-π, is positive, zero or negative. No explanation is required.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section: Chapter Questions
Problem 10RE
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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}; 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++-π, 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}; 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++-π, is positive, zero or negative. No explanation is required.
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