Mock Final Exam – Multivariable Calculus
Math 53M, 2001. Instructor: E. Frenkel
1. For each statement below, determine whether it is true or not. Circle T if it is true,
or F if it is false.
(1) If
a
and
b
are two nonzero vectors in
R
3
and
a
·
b
= 0, then
a
×
b
=
0
.
(2) Let
f
be a function in two variables. If
f
x
(
x
0
, y
0
) =
f
y
(
x
0
, y
0
) = 0, then
f
has
a local maximum or a local minimum at the point (
x
0
, y
0
).
(3) There exists a vector field
F
, such that curl
F
=
x
3
i
+
y
3
j
+
z
3
k
.
(4) There exists a vector field
F
, such that div
F
=
x
3
+
y
3
+
z
3
.
(5) The line integral of the vector field
x
i
over any closed curve in
R
3
equals 0.
(6) Any surface in
R
3
, which belongs entirely to the
yz
–plane is orientable.
(7) The flux of any vector field across any closed surface in
R
3
without boundary
equals 0.
(8) The directional derivative
D
u
f
(
x
0
, y
0
) is 0 if
u
is perpendicular to
∇
f
(
x
0
, y
0
).
(9)
lim
(
x,y
)
→
(0
,
0)
x
2

y
2
x
2
+
y
2
exists.
(10) If
F
is a vector field in
R
3
, such that div
F
= 0, then the flux of
F
across any
surface in
R
3
equals 0.
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 Fall '08
 Lanzara
 Physics, Calculus, Derivative, Vector Calculus, Manifold, Stokes' theorem

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