Answered: (у, — 2х) F C: r(t) = (2 cos t, 4 sin…

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

Circulation Consider the following vector fields F and closed oriented curves C in the plane (see figure).
a. Based on the picture, make a conjecture about whether the circulation
of F on C is positive, negative, or zero.
b. Compute the circulation and interpret the result.

(у, — 2х)
F
C: r(t) = (2 cos t, 4 sin t), for 0 <is 2m
V4r? + y
(y. –2x)
F =
yA
V4x2 +
y²
C: r(t) = (2 cos t, 4 sin t)

Expert Solution

Step 1

Given:
Vector field: $F = \frac{<y, - 2 x>}{\sqrt{4 x^{2} + y^{2}}}$ ;
Curve: $C : r (t) = <2 \cos t, 4 \sin t>$ , for $0 \leq t \leq 2 π$ .

To Do:
(a) From the given picture we have to make a conjecture about whether the circulation of F on C is positive, negative, or zero.

(b) Compute the circulation and interpret the result.

Step 2

(a) We have,

Here we see that the vector field circulates in the clockwise direction, and curve C is oriented counterclockwise. Therefore the circulation will be negative.

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