The vector field F(r) is given by = (₁ F = X Y 1+x² + y² ¹ 1 + x² + y² ' 2z (a) Sketch the field F in the (x, y)-plane. (b) Showing your workings, explicitly evaluate the line integral F.dr C where C is a circle of radius 2 in the (x, y)-plane, traversed anti-clockwise. (c) Evaluate curl F and state why your answer is consistent with the result of part (b).
The vector field F(r) is given by = (₁ F = X Y 1+x² + y² ¹ 1 + x² + y² ' 2z (a) Sketch the field F in the (x, y)-plane. (b) Showing your workings, explicitly evaluate the line integral F.dr C where C is a circle of radius 2 in the (x, y)-plane, traversed anti-clockwise. (c) Evaluate curl F and state why your answer is consistent with the result of part (b).
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:2. The vector field F(r) is given by
= (₁
F =
X
Y
1 + x² + y²¹ 1 + x² + y²¹
(a) Sketch the field F in the (x, y)-plane.
(b) Showing your workings, explicitly evaluate the line integral
fo
C
d
dt
F.dr
where is a circle of radius 2 in the (x, y)-plane, traversed anti-clockwise.
(c) Evaluate curl F and state why your answer is consistent with the result of part (b).
(d) Find the scalar potential of F and sketch some contours of in the (x, y)-plane.
You may find it useful to recall that the derivative of the natural logarithm of a
function h(t) is given by
log h
J =
=
2z
1 dh
h dt
(e) Calculate the divergence of F. Hence, without explicitly evaluating a surface
integral, state whether J is positive, negative or zero, where J is defined as the
integral
$
S
F. dS
over the surface of a sphere of radius 3, centred on the point (1,1,1). Explain
your reasoning by referring to a relevant theorem.
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