(2.3) Prove that each of the following systems a) Parabolic cylindrical coordinates (u, v, z) defined by * = uv, y = (u? – u²], z = z b) Paraboloidal coordinate system (u, v, ) defined by I = uv cos ø, · y = uv sin ø, z = [u? – v²] is orthogonal, and then find the following for each of them: The scale factors, the base unit vectors, the vector de, the element de? and the element of volume dr. [hint: to show that the system is orthogonal prove that h12 h23 = 0] h13 %3D %3D
(2.3) Prove that each of the following systems a) Parabolic cylindrical coordinates (u, v, z) defined by * = uv, y = (u? – u²], z = z b) Paraboloidal coordinate system (u, v, ) defined by I = uv cos ø, · y = uv sin ø, z = [u? – v²] is orthogonal, and then find the following for each of them: The scale factors, the base unit vectors, the vector de, the element de? and the element of volume dr. [hint: to show that the system is orthogonal prove that h12 h23 = 0] h13 %3D %3D
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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CHAPTER 2. CURVILINEAR COORDINATES
(2.3) Prove that each of the following systems
a) Parabolic cylindrical coordinates (u, v, z) defined by
x = u v, y = [u? – u²], z = z
b) Paraboloidal coordinate system (u, v, 4) defined by
I = UV COS¢, y = uv sin o, z =
(u² – v]
%3D
is orthogonal, and then find the following for each of them:
The scale factors, the base unit vectors, the vector de, the element
de? and the element of volume dr.
[hint: to show that the system is orthogonal prove that h12
h23 = 0]
h13
Gradient, Divergence, Laplacian and Curl](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1325eda-7fed-487d-ae84-b47193687fa6%2F1c4c8c81-a613-4d5b-a5c4-336c06999af9%2F7yvsgs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:154
CHAPTER 2. CURVILINEAR COORDINATES
(2.3) Prove that each of the following systems
a) Parabolic cylindrical coordinates (u, v, z) defined by
x = u v, y = [u? – u²], z = z
b) Paraboloidal coordinate system (u, v, 4) defined by
I = UV COS¢, y = uv sin o, z =
(u² – v]
%3D
is orthogonal, and then find the following for each of them:
The scale factors, the base unit vectors, the vector de, the element
de? and the element of volume dr.
[hint: to show that the system is orthogonal prove that h12
h23 = 0]
h13
Gradient, Divergence, Laplacian and Curl
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