Question 3. In this problem show that, if we take two different regular surface patches for the same surface, the matrices of their metrics are closely related. Let o(u, v): U →→ R³ be a regular surface patch for a surface S. Let õ(u, v): Ũ R³ be a regular reparametrisation of o, and let : U Ũ be a transition map, so that (Þ(u, v)) = o(u, v). Let the metrics of o and ỡ be g = E du² + 2F du dv + G dv² and 9 = Ẽ du² + 2F dũ dĩ +Ğ dv² respectively. Show that the first fundamental forms g and 9 are related by [E(u, v) F(u, v)] [Ẽ(Þ(u, v)) Ẽ(Þ(u, v))] D (4,0) B. F(u, v) G(u, v) (D(u,v) Þ)T =
Question 3. In this problem show that, if we take two different regular surface patches for the same surface, the matrices of their metrics are closely related. Let o(u, v): U →→ R³ be a regular surface patch for a surface S. Let õ(u, v): Ũ R³ be a regular reparametrisation of o, and let : U Ũ be a transition map, so that (Þ(u, v)) = o(u, v). Let the metrics of o and ỡ be g = E du² + 2F du dv + G dv² and 9 = Ẽ du² + 2F dũ dĩ +Ğ dv² respectively. Show that the first fundamental forms g and 9 are related by [E(u, v) F(u, v)] [Ẽ(Þ(u, v)) Ẽ(Þ(u, v))] D (4,0) B. F(u, v) G(u, v) (D(u,v) Þ)T =
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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![Question 3. In this problem show that, if we take two different regular surface
patches for the same surface, the matrices of their metrics are closely related.
Let o (u, v): U →→→→ R³ be a regular surface patch for a surface S. Let õ(ũ, v): Ū →
R³ be a regular reparametrisation of o, and let Þ: U →Ũ be a transition map, so
that õ(Þ(u, v)) = o(u, v). Let the metrics of σ and ỡ be
g=E dữ +2F du du +G dữ and g=Ẽ dữ +2F du dữ +ữ để
respectively. Show that the first fundamental forms g and ğ are related by
F(Þ(u,v))]
[F(u, v) G(u, v)] - (Dub) [E(+(u, v)) F(x(u, v))] D.) 4.
D(u, v) Þ.
U₂
(Hint for one possible approach: the matrix for the metric is (Do)¹ Do.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fe1fc35-672a-49fd-831b-9642c77888ed%2Feb7c6902-ed97-4a08-8f76-7a694208a3d3%2Fcs63mop_processed.png&w=3840&q=75)
Transcribed Image Text:Question 3. In this problem show that, if we take two different regular surface
patches for the same surface, the matrices of their metrics are closely related.
Let o (u, v): U →→→→ R³ be a regular surface patch for a surface S. Let õ(ũ, v): Ū →
R³ be a regular reparametrisation of o, and let Þ: U →Ũ be a transition map, so
that õ(Þ(u, v)) = o(u, v). Let the metrics of σ and ỡ be
g=E dữ +2F du du +G dữ and g=Ẽ dữ +2F du dữ +ữ để
respectively. Show that the first fundamental forms g and ğ are related by
F(Þ(u,v))]
[F(u, v) G(u, v)] - (Dub) [E(+(u, v)) F(x(u, v))] D.) 4.
D(u, v) Þ.
U₂
(Hint for one possible approach: the matrix for the metric is (Do)¹ Do.)
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