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). Solve all of them plz
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). Solve all of them plz
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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).
Solve all of them plz
![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%2Fad903e19-d65e-4e02-b839-aab08de2984a%2Fwvz7e87_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.
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