Let u = (U1, U2, U3): R³ → R³ be an arbitrary (smooth) vector field in R³ = {x = (x₁, x2, x3)} (as in vector calculus). (We don't care about the difference between subscripts and superscripts here.) (a) Define a 1-form on R3 corresponding to u as follows: u= u₁(x) dx₁+U₂(x) dx2 + u3(x) dx3. Find the 2-form dub. What vector field in R3 corresponds to it? (b) Suppose that a vector field u satisfies u = df with f:R³ R. What is the relationship between u and f in the vector calculus terms? (c) The Poincaré Lemma says any closed 1-form in R³ is exact. What is the corresponding result in vector calculus?
Let u = (U1, U2, U3): R³ → R³ be an arbitrary (smooth) vector field in R³ = {x = (x₁, x2, x3)} (as in vector calculus). (We don't care about the difference between subscripts and superscripts here.) (a) Define a 1-form on R3 corresponding to u as follows: u= u₁(x) dx₁+U₂(x) dx2 + u3(x) dx3. Find the 2-form dub. What vector field in R3 corresponds to it? (b) Suppose that a vector field u satisfies u = df with f:R³ R. What is the relationship between u and f in the vector calculus terms? (c) The Poincaré Lemma says any closed 1-form in R³ is exact. What is the corresponding result in vector calculus?
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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Transcribed Image Text:3 Let u = (u1, u2, U3): R³ → R³ be an arbitrary (smooth) vector field in R3 = {x = (x1,x2, x3)} (as in vector
calculus). (We don't care about the difference between subscripts and superscripts here.)
(а)
Define a 1-form on R3 corresponding to u as follows:
u' = u1(x) dx1 + u2(x) dx2 + u3(x) dx3.
Find the 2-form du'. What vector field in R3 corresponds to it?
(b
Suppose that a vector field u satisfies u' = df with f:R3 → R. What is the relationship between u
and f in the vector calculus terms?
(c)
The Poincaré Lemma says any closed 1-form in R³ is exact. What is the corresponding result in
vector calculus?
(d)
| Define a 2-form on R³ corresponding to u as follows:
u* = u1(x) dx2 a dx3 + u2(x) dx3 A dx1 + u3(x) dx1 A dx2.
Calculate the 3-form du*.
(e)
The Poincaré Lemma says any closed 2-form in R' is exact. What is the corresponding result in
vector calculus?
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