2. (a) Let ds² = v du² − du dv + 2 dv². (a) Find the largest subset U of R² for which this expression defines a Riemannian metric. (b) Let y: (1,2) → R², y(t) (t+1, t² + ¹). Calculate the length of y with respect to ds². (You may assume it is well-defined.) = (c) Let V = {(u, v) € R² : u > 0, u + v <3, u − v <-1}. Calculate the area of V with respect to ds². (You may assume it is well-defined.)
2. (a) Let ds² = v du² − du dv + 2 dv². (a) Find the largest subset U of R² for which this expression defines a Riemannian metric. (b) Let y: (1,2) → R², y(t) (t+1, t² + ¹). Calculate the length of y with respect to ds². (You may assume it is well-defined.) = (c) Let V = {(u, v) € R² : u > 0, u + v <3, u − v <-1}. Calculate the area of V with respect to ds². (You may assume it is well-defined.)
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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