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Let M C R³ be the set obtained by rotating the circle {(x, 0, z) E R³ : (x – 2)² +2² = 1} about the z-axis. By explicitly constructing coordinate charts, show that M is a smooth surface in R³. This surface is an example of a torus.

Let M C R³ be the set obtained by rotating the circle {(x, 0, z) E R³ : (x – 2)² +2² = 1} about the z-axis. By explicitly constructing coordinate charts, show that M is a smooth surface in R³. This surface is an example of a torus.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Let M C R³ be the set obtained by rotating the circle {(x,0, 2) E R³ : (x – 2)² +z² = 1} about the z-axis. By explicitly constructing coordinate charts, show that M is a smooth surface in R3. This surface is an example of a torus.
Transcribed Image Text:Let M C R³ be the set obtained by rotating the circle {(x,0, 2) E R³ : (x – 2)² +z² = 1} about the z-axis. By explicitly constructing coordinate charts, show that M is a smooth surface in R3. This surface is an example of a torus.
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