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Math 280
Homework Assignment 12
due date: Dec. 10, 2015
1. Let c be the top half of the unit circle, oriented from (−1, 0) to (1, 0), and F be the
vector field
F(x, y) = (ln(x + 5) + y 2 , 2xy − y 2 ).
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Check that Curl(F) =
Z 0 (the R curl), and use the flexibility theorem for curves to
compute the integral F · ds. Flexing the curve to the line joining (−1, 0) and (1, 0)
c
is a possibility.
2. Let F be the vector field
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F(x, y, z) = (yez − xexy , yexy + tan(z 2 + z + 1), x2 ).
Check that Div(F) = 0, and use the flexibility theorem for surfaces to compute
ZZ
F · dS
S
where S is the top half of the unit sphere oriented outwards. (“Flexing” it to the unit
disk seems like a good bet.)
3. Let c1 be the top half of the unit circle, oriented counterclockwise, c2 be the line
segment joining (1, 0) to (−1, 0), oriented from (1, 0) to (−1, 0), and let F be the vector
field
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F(x, y) = (x2 − y, xy − arcsin(y) + ey ).
Z
Z
F · ds.
F · ds and
(a) Use Green’s theorem to compute the difference between
(b) Use (a) to compute
Z
F · ds (by computing the easier
c1
Z
c1
c2
F · ds, of course. . . ).
c2
4. Questions about electricity and magnetism.
(a) Could E(x, y, z) = (x2 − y 2, ey − 2z, z 2 + 3 sin(y)) be an electric field (in a static
situation)?
(b) Could B(x, y, z) = (sin(x), sin(y), z 2 ) be a magnetic field (in a static situation)?
(c) Suppose that F and G are C 1 vector fields defined on all of R3 which have the
following property. For every oriented surface S with oriented boundary curve c,
we have the equality
Z
ZZ
F · ds =
G · dS.
S
c
What relationship must hold between F and G, and why?
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This homework assignment is due on or before Thursday, December 10th, at 4pm.
The assigment can be handed in to my office, 507 Jeffery Hall. (There is a mailbox with
my name on it to the right of my door. There is an even closer mailbox with someone
else’s name on it to the left of my door. Do not be tempted by the closer mailbox!)
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