Prove that a torus at R³ is equivalent to the Cartesian product of two circles at R² 3 that is, prove that a torus at R is homeomorphic to S¹ x S¹. Hint: Given a > b two positive reals, the torus at T = { [ (a + bcosø)cose, (a + bcosø)s ine, bsino ] E R Remember that the circle is defined as the set 2 S¹ = {(cosß, sinß) € R² 0≤ß≤2n} R R³ |0 ≤Ø≤ 2,0 ≤ 0 ≤ 2}. can be defined as the set:
Prove that a torus at R³ is equivalent to the Cartesian product of two circles at R² 3 that is, prove that a torus at R is homeomorphic to S¹ x S¹. Hint: Given a > b two positive reals, the torus at T = { [ (a + bcosø)cose, (a + bcosø)s ine, bsino ] E R Remember that the circle is defined as the set 2 S¹ = {(cosß, sinß) € R² 0≤ß≤2n} R R³ |0 ≤Ø≤ 2,0 ≤ 0 ≤ 2}. can be defined as the set:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![SOLVE STEP BY STEP IN DIGITAL FORMAT
۵۰۶۹
ツッシ
7
3 * * ! ! ?? !! ??! ¿¡ !?
√√√XXXXXOO
DO
-
3
Prove that a torus at R
that is, prove that a torus at R
3
Hint: Given a > b
is equivalent to the Cartesian product of two circles at R
is homeomorphic to S¹ x S¹.
two positive reals, the torus at
T = {[(a + bcosø)cosł, (a + bcosø)s in, bsinø ] €
ob WX
Remember that the circle is defined as the set
S¹ = {(cosß, sinß) € R |0 ≤ß≤2π}
J
R can be defined as the set:
3
R³ |0 ≤ø≤2¹, 0≤ 0 ≤ 2À }.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc95a7fa8-f21a-4709-81cf-64d281fbc8e2%2F02862c46-e4d6-48ec-8d31-d033c5825a64%2Fpfhx29n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:SOLVE STEP BY STEP IN DIGITAL FORMAT
۵۰۶۹
ツッシ
7
3 * * ! ! ?? !! ??! ¿¡ !?
√√√XXXXXOO
DO
-
3
Prove that a torus at R
that is, prove that a torus at R
3
Hint: Given a > b
is equivalent to the Cartesian product of two circles at R
is homeomorphic to S¹ x S¹.
two positive reals, the torus at
T = {[(a + bcosø)cosł, (a + bcosø)s in, bsinø ] €
ob WX
Remember that the circle is defined as the set
S¹ = {(cosß, sinß) € R |0 ≤ß≤2π}
J
R can be defined as the set:
3
R³ |0 ≤ø≤2¹, 0≤ 0 ≤ 2À }.
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