3. Suppose that each of G and H are functions of a single variable from R to R and are continuous with continuous first and second derivatives. Let F(x, y) = (H'(y), –G'(x)). (a) Show that V·F = 0. (b) Suppose that C is parameterized by r(t) = x(t)i + y(t)j, a
3. Suppose that each of G and H are functions of a single variable from R to R and are continuous with continuous first and second derivatives. Let F(x, y) = (H'(y), –G'(x)). (a) Show that V·F = 0. (b) Suppose that C is parameterized by r(t) = x(t)i + y(t)j, a
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.
Suppose that each of G and H are functions of a single variable from R to R and are continuous
with continuous first and second derivatives. Let F(x, y) = (H'(y), –G'(x)).
(a) Show that V·F = 0.
(b) Suppose that C is parameterized by r(t) = x(t)i + y(t)j, a <t< b. Show that the flux of F across C is
given by G(r(b)) – G(x(a)) + H(y(b)) – H(y(a)).
(c)
say about the total flux of F across this closed curve?
Suppose that C happened to be a simple closed curve. What does your result in (b)
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