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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 15EQ
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
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
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
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,
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
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