a) Calculate the surface integral = ff, F.nds of the vector field: F(x, y, z) = x²yzi - xy²zj over the surface S of the unit cube defined by the intersection of the planes x = 0, x = 1, y = 0, y = 1, z = 0 and z 1, where n denotes the unit vector normal to the surface element dS pointing in the outward direction. b) Using Gauss's (divergence) theorem re-evaluate the surface integral in part a) above by means of a volume (triple) integral and comment on the result.
a) Calculate the surface integral = ff, F.nds of the vector field: F(x, y, z) = x²yzi - xy²zj over the surface S of the unit cube defined by the intersection of the planes x = 0, x = 1, y = 0, y = 1, z = 0 and z 1, where n denotes the unit vector normal to the surface element dS pointing in the outward direction. b) Using Gauss's (divergence) theorem re-evaluate the surface integral in part a) above by means of a volume (triple) integral and comment on the result.
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:a) Calculate the surface integral
= ff, F.nds of the vector field:
F(x, y, z) = x²yzi - xy²zj
over the surface S of the unit cube defined by the intersection of the planes x = 0,
x = 1, y = 0, y = 1, z = 0 and z = 1, where n denotes the unit vector normal to
the surface element dS pointing in the outward direction.
b) Using Gauss's (divergence) theorem re-evaluate the surface integral in part a)
above by means of a volume (triple) integral and comment on the result.
c) It can be shown that a vector field is conservative if and only if it can be written
as the gradient of a scalar field termed "potential function". By constructing such a
potential function, show that the vector field:
F(x, y, z) = 2xyl + (x² + 2yz)j + y²k
is conservative.
d) Calculate via direct integration the line integral
fF.dl
where F is defined in part c) and the path of integration is the straight line connecting
points (0,0,0) and (1,1,1). Verify your answer by using the potential function you have
constructed in part c).
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