(d) When you calculated the principal curvatures K₁, K₂ at p= (1, 1,0), you could have known in advance that K₁ = -K₂. Why?
(d) When you calculated the principal curvatures K₁, K₂ at p= (1, 1,0), you could have known in advance that K₁ = -K₂. Why?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 18T
Related questions
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Need help with part (3) (d). Please explain each step and neatly type up. Thank you :)
![Question 2. In this question we investigate the smooth surface S defined by z =
x² - y². It's known as a hyperbolic paraboloid and it has an atlas consisting of a
single regular chart o : R² → R³, o(u, v) = (u, v, u² − v²).
(1) First, let's compute some standard differential-geometric quantities for S.
(a) Calculate the Riemannian metric g of o.
(b) Show that a unit normal vector field N to S is given at each point
p = o(u, v) by
Ñ
1
√4u² + 4v² + 1
(c) Using №, find the second fundamental form of σ.
(d) Find the Weingarten map of S.
(e) Show that the Gaussian curvature K and mean curvature H of S are
given by
K =
-4
25
(4u²+ 4v² + 1)²
(-2u, 2v, 1).
H =
4 (v² — u²)
(4u² +4v² +1) ³/2
(f) At the point p =
(1, 1,0), find the two principal curvatures and principal
directions of S. Express the principal directions as vectors in R³ and
verify they are orthogonal.
(2) We now consider a curve along S and some vector fields along it. Let 3 be a
curve defined by (t, t, 0), and let V, W be tangent vector fields on S defined
by V = Ou + O₂ and W = σu - ov.
(a) Find a unit speed reparametrisation y(t) of (t) with y(0) = B(0).
(b) Writing V(t) and W(t) for the vector fields V, W at y(t), calculate V (t)
and W(t) as a function of t.
(c) Calculate the covariant derivatives of V, W along y and show that
4t
V₂V=0, V₂W
(3) We now try to understand the geometric meaning of some of the above
calculations!
=
-W.
4t² + 1
(a) For which points on S is K positive, negative, and zero?
(b) Show that the image of the Gauss map G of S is precisely one hemisphere
of the unit sphere S². In fact, the four subsurfaces of S obtained by
cutting along the planes x = 0 and y = 0 map to four octants of S²
how?
(c) Is the Gauss map orientation preserving or reversing? Why?
(d) When you calculated the principal curvatures K₁, K₂ at p = (1, 1,0), you
could have known in advance that K₁ = -K₂. Why?
(e) Without doing any further calculations, explain why y is a geodesic of
S, and in fact Kg = kn = 0.
(f) Bonus: Calculate the total curvature of S and show that J K dA
-2. This answer could have been predicted in advance why?
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cb672f7-47ed-4ee3-be4e-71db737c6150%2Fdf718feb-c2b4-47c0-8039-4e1f7b91c0d0%2Fkq20tx5_processed.png&w=3840&q=75)
Transcribed Image Text:Question 2. In this question we investigate the smooth surface S defined by z =
x² - y². It's known as a hyperbolic paraboloid and it has an atlas consisting of a
single regular chart o : R² → R³, o(u, v) = (u, v, u² − v²).
(1) First, let's compute some standard differential-geometric quantities for S.
(a) Calculate the Riemannian metric g of o.
(b) Show that a unit normal vector field N to S is given at each point
p = o(u, v) by
Ñ
1
√4u² + 4v² + 1
(c) Using №, find the second fundamental form of σ.
(d) Find the Weingarten map of S.
(e) Show that the Gaussian curvature K and mean curvature H of S are
given by
K =
-4
25
(4u²+ 4v² + 1)²
(-2u, 2v, 1).
H =
4 (v² — u²)
(4u² +4v² +1) ³/2
(f) At the point p =
(1, 1,0), find the two principal curvatures and principal
directions of S. Express the principal directions as vectors in R³ and
verify they are orthogonal.
(2) We now consider a curve along S and some vector fields along it. Let 3 be a
curve defined by (t, t, 0), and let V, W be tangent vector fields on S defined
by V = Ou + O₂ and W = σu - ov.
(a) Find a unit speed reparametrisation y(t) of (t) with y(0) = B(0).
(b) Writing V(t) and W(t) for the vector fields V, W at y(t), calculate V (t)
and W(t) as a function of t.
(c) Calculate the covariant derivatives of V, W along y and show that
4t
V₂V=0, V₂W
(3) We now try to understand the geometric meaning of some of the above
calculations!
=
-W.
4t² + 1
(a) For which points on S is K positive, negative, and zero?
(b) Show that the image of the Gauss map G of S is precisely one hemisphere
of the unit sphere S². In fact, the four subsurfaces of S obtained by
cutting along the planes x = 0 and y = 0 map to four octants of S²
how?
(c) Is the Gauss map orientation preserving or reversing? Why?
(d) When you calculated the principal curvatures K₁, K₂ at p = (1, 1,0), you
could have known in advance that K₁ = -K₂. Why?
(e) Without doing any further calculations, explain why y is a geodesic of
S, and in fact Kg = kn = 0.
(f) Bonus: Calculate the total curvature of S and show that J K dA
-2. This answer could have been predicted in advance why?
=
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