Let ß be the vector-valued function 3u B(u, v) = { 32 ß: (-2, 2) × (0, 2π) → R³, 3u² + 4 COS(V) } u cos(v) VI + u²¹ sin(v), ) Sketch the image of ß (i.e. plot all values ß(u, v), for (u, v) in the domain of ß). ) On the sketch in part (a), indicate (i) the path obtained by holding v = varying u, and (ii) the path obtained by holding u = 0 and varying v. π/2 and

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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(5)
Let ß be the vector-valued function
3u
ß: (-2,2) × (0, 2π) → R³, B(U₁₂ v) = {
3u²
4
B (0,7), 0₁B (0,7), 0₂B (0,7)
u cos(v)
VI+ u², sin(v),
(a) Sketch the image of ß (i.e. plot all values ß(u, v), for (u, v) in the domain of ß).
(b) On the sketch in part (a), indicate (i) the path obtained by holding v = π/2 and
varying u, and (ii) the path obtained by holding u = O and varying v.
(c) Compute the following quantities:
(d) Draw the following tangent vectors on your sketch in part (a):
X₁ = 0₁B (0₂7) B(0)¹ X₂ = 0₂ß (0,7) p(0.4)*
'
cos(v)
√1+u²
+
Transcribed Image Text:(5) Let ß be the vector-valued function 3u ß: (-2,2) × (0, 2π) → R³, B(U₁₂ v) = { 3u² 4 B (0,7), 0₁B (0,7), 0₂B (0,7) u cos(v) VI+ u², sin(v), (a) Sketch the image of ß (i.e. plot all values ß(u, v), for (u, v) in the domain of ß). (b) On the sketch in part (a), indicate (i) the path obtained by holding v = π/2 and varying u, and (ii) the path obtained by holding u = O and varying v. (c) Compute the following quantities: (d) Draw the following tangent vectors on your sketch in part (a): X₁ = 0₁B (0₂7) B(0)¹ X₂ = 0₂ß (0,7) p(0.4)* ' cos(v) √1+u² +
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