-) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² + y? ' x² + y² not conservative by considering the line integral F· dr here C is the circle x2 + y = 1 oriented counterclockwise.
-) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² + y? ' x² + y² not conservative by considering the line integral F· dr here C is the circle x2 + y = 1 oriented counterclockwise.
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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Question
Calc 3 - Line
Transcribed Image Text:You may assume the following theorem:
Theorem (Test for Conservative Fields in the Plane). A vector field
F(x, y) = (M(x, y), N(x, y))
whose domain is an open region of the plane that is connected and simply connected is conservative if and only if
ON
ƏM
%3D
dx
ду
(a) Let
-y
M(x, y) =
x2 + y?
and
N(x, y) =
x2
+ y?
Show that
ON
ƏM
dx
ду
(b) Show that
)
-y
F(x, y) = (
x² + y² ’ x² + y²
is not conservative by considering the line integral
F· dr
C
where C is the circle x2 + y
= 1 oriented counterclockwise.
(c) Explain why parts (a) and (b) do not contradict the Test for Conservative Fields in the Plane.
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