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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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