The main point of this exercise is to use Green's Theorem to deduce a special case of the change of variable formula. Let U, V C R? be path connected open sets and let G:U → V be one-to-one and C2 such that the derivate DG(u) is invertible for all u E U. Let T CU be a regular region with piecewise smooth boundary, and let S = G(T). 2. (a) Prove that S is a regular region. [Hint: recall the proof that aS = G(@T)] %3D Show that the Jacobian JG : U (u, v) → det(DG(u, v)) E R continuous. (b) [Hint: Don't work hard. Use algebraic properties of continuous functions.] Deduce that Je is either everywhere positive or everywhere negative on U.
The main point of this exercise is to use Green's Theorem to deduce a special case of the change of variable formula. Let U, V C R? be path connected open sets and let G:U → V be one-to-one and C2 such that the derivate DG(u) is invertible for all u E U. Let T CU be a regular region with piecewise smooth boundary, and let S = G(T). 2. (a) Prove that S is a regular region. [Hint: recall the proof that aS = G(@T)] %3D Show that the Jacobian JG : U (u, v) → det(DG(u, v)) E R continuous. (b) [Hint: Don't work hard. Use algebraic properties of continuous functions.] Deduce that Je is either everywhere positive or everywhere negative on U.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....
Related questions
Question
The main point of this exercise is to use Green’s Theorem to deduce a special
case of the change of variable formula. Let U, V ⊆ R2 be path connected open sets and let
G : U → V be one-to-one and C2
such that the derivate DG(u) is invertible for all u ∈ U.
Let T ⊆ U be a regular region with piecewise smooth boundary, and let S = G(T).
Answer C
![The main point of this exercise is to use Green's Theorem to deduce a special
case of the change of variable formula. Let U, V CR? be path connected open sets and let
G :U → V be one-to-one and C2 such that the derivate DG(u) is invertible for all u E U.
Let T CU be a regular region with piecewise smooth boundary, and let S = G(T).
(a)
Prove that S is a regular region. [Hint: recall the proof that aS =
= G(ƏT)]
(b)
[Hint: Don't work hard. Use algebraic properties of continuous functions.]
Show that the Jacobian JG :U Ə (u, v) → det(DG(u, v)) ER continuous.
Deduce that JG is either everywhere positive or everywhere negative on U.
If JG(u, v) > 0 for all (u, v) E U, convert the formula Area(S) = - Sas ydx
(c)
(d)
into an integral over ÔT using a change of variable, and then apply Green's Theorem to
show that Area(S) = Sfr det(DG(u, v))dA.
(e)
Area(S) = - S, det(DG(u, v))dA. Where does the minus sign come from?
If JG(u, v) < 0 for all (u, v) E U, use a similar argument to show that
2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F717c41c9-7610-4369-a9d8-d14d52c86513%2F11b62810-8fe0-425b-8a14-3f73c89899d4%2Fllhcfb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The main point of this exercise is to use Green's Theorem to deduce a special
case of the change of variable formula. Let U, V CR? be path connected open sets and let
G :U → V be one-to-one and C2 such that the derivate DG(u) is invertible for all u E U.
Let T CU be a regular region with piecewise smooth boundary, and let S = G(T).
(a)
Prove that S is a regular region. [Hint: recall the proof that aS =
= G(ƏT)]
(b)
[Hint: Don't work hard. Use algebraic properties of continuous functions.]
Show that the Jacobian JG :U Ə (u, v) → det(DG(u, v)) ER continuous.
Deduce that JG is either everywhere positive or everywhere negative on U.
If JG(u, v) > 0 for all (u, v) E U, convert the formula Area(S) = - Sas ydx
(c)
(d)
into an integral over ÔT using a change of variable, and then apply Green's Theorem to
show that Area(S) = Sfr det(DG(u, v))dA.
(e)
Area(S) = - S, det(DG(u, v))dA. Where does the minus sign come from?
If JG(u, v) < 0 for all (u, v) E U, use a similar argument to show that
2.
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
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning