(a) nsider the change of variable from cartesian coordinates (x, y, z) to coordi- nates (r, y, 0) in R². X = exp(r) cos y cos 0, y = exp(r) sin 0, We will assume that y = [0, 2π) and 0 € [-π/2, π/2]. (i) Show that exp(r) = x² + y² + z²2. (ii) Compute the Jacobian of this transformation. I.e. find the determinant of the matrix z = exp(r) sin cos 0. ах ах ах др до до ду ду ду ar ap 20 əz əz əz or οφ οθ (iii) For € (0, π) and 0 € (-π/2, π/2), express r, y and in terms of x, y and z.
(a) nsider the change of variable from cartesian coordinates (x, y, z) to coordi- nates (r, y, 0) in R². X = exp(r) cos y cos 0, y = exp(r) sin 0, We will assume that y = [0, 2π) and 0 € [-π/2, π/2]. (i) Show that exp(r) = x² + y² + z²2. (ii) Compute the Jacobian of this transformation. I.e. find the determinant of the matrix z = exp(r) sin cos 0. ах ах ах др до до ду ду ду ar ap 20 əz əz əz or οφ οθ (iii) For € (0, π) and 0 € (-π/2, π/2), express r, y and in terms of x, y and z.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![(a) Consider the change of variable from cartesian coordinates (x, y, z) to coordi-
nates (r, y, 0) in R².
x = exp(r) cos y cos 0,
y = exp(r) sin 0,
We will assume that y = [0, 2π) and 0 € [-π/2, π/2].
(i) Show that exp(r) = √x² + y² + z².
(ii) Compute the Jacobian of this transformation. I.e. find the determinant of
the matrix
'дх
др
ах ах
до
до
ду
z = exp(r) sin cos 0.
до до
əz əz əz
др
др до до
(iii) For y € (0, π) and 0 € (−π/2, π/2), express r, 4 and 0 in terms of x, y
and z.
(b) Consider a box with square base and no lid which has width x > 0 and height
h> 0. It has area a(x, h) = x² + 4hx and volume v(x, h) = hx².
(i) What is the maximum volume of such a box when it has fixed area A?
(ii) Is it also possible to minimise the volume of the box, keeping the area fixed?
Explain your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea7bbfe4-f0d9-4343-839c-f9e5c129de8e%2Fe8daac8b-3cf3-4681-b5b5-f9d97504e6a4%2Fosnljb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Consider the change of variable from cartesian coordinates (x, y, z) to coordi-
nates (r, y, 0) in R².
x = exp(r) cos y cos 0,
y = exp(r) sin 0,
We will assume that y = [0, 2π) and 0 € [-π/2, π/2].
(i) Show that exp(r) = √x² + y² + z².
(ii) Compute the Jacobian of this transformation. I.e. find the determinant of
the matrix
'дх
др
ах ах
до
до
ду
z = exp(r) sin cos 0.
до до
əz əz əz
др
др до до
(iii) For y € (0, π) and 0 € (−π/2, π/2), express r, 4 and 0 in terms of x, y
and z.
(b) Consider a box with square base and no lid which has width x > 0 and height
h> 0. It has area a(x, h) = x² + 4hx and volume v(x, h) = hx².
(i) What is the maximum volume of such a box when it has fixed area A?
(ii) Is it also possible to minimise the volume of the box, keeping the area fixed?
Explain your answer.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning