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Question 2. In this question we investigate the smooth surface S defined by z = r² - y². It's known as a hyperbolic paraboloid and it has an atlas consisting of a single regular chart σ : 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 σ. (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. (-2u, 2v, 1).

Question 2. In this question we investigate the smooth surface S defined by z = r² - y². It's known as a hyperbolic paraboloid and it has an atlas consisting of a single regular chart σ : 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 σ. (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. (-2u, 2v, 1).

Advanced Engineering Mathematics
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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Need help with (2) (c). 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 σ : 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 Ñ to S is given at each point p = o(u, v) by 1 √4u² + 4v² + 1 (c) Using №, find the second fundamental form of o. (d) Find the Weingarten map of S. (e) Show that the Gaussian curvature K and mean curvature H of S are given by K = Ñ = -4 " (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 = O₂ + O₂, and W = ou - ov. (a) Find a unit speed reparametrisation y(t) of (t) with y(0) = 3(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 4t² + 1 V₁V=0, V₂W = -W. (3) We now try to understand the geometric meaning of some of the above calculations! (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 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?
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 σ : 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 Ñ to S is given at each point p = o(u, v) by 1 √4u² + 4v² + 1 (c) Using №, find the second fundamental form of o. (d) Find the Weingarten map of S. (e) Show that the Gaussian curvature K and mean curvature H of S are given by K = Ñ = -4 " (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 = O₂ + O₂, and W = ou - ov. (a) Find a unit speed reparametrisation y(t) of (t) with y(0) = 3(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 4t² + 1 V₁V=0, V₂W = -W. (3) We now try to understand the geometric meaning of some of the above calculations! (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 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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