2- 3
16.9
11
10
09
7.
Let F .
(a) Compute the outward flux of F through the sphere .
[6pts]
(b) Compute the outward flux of F through the ellipsoid .
[6pts]
8.
Let F and be the boundary surface of the solid region
enclosed by the paraboloid and the p
2- 3
16.8
11
10
09
6. Use Stokes' theorem to evaluate
F where
F
and is the triangle with vertices and , oriented counter
clockwise as viewed from above. [7pts]
2- 3
16.7
10
09
5. Let be the surface obtained by rotating the curve , about
the -axis, where and is continuous.
(a) Find the vector equation of the surface . [3pts]
(b) Show that the area of is
.
[6pts]
08
7. [10pts] Let be the surface obtained by rot
2- 3
16.5
11
09
4.
(a) Show that di v , where is a scalar function defined on
R .
Assume that has continuous second order partial derivatives on .
[5pts]
(b) Use the result of (a) to show that if is harmonic (that is, ) on , and
if on the boundary curve
2- 3
16.4
11
09
2. (a) Let be the region bounded by a positively oriented, piecewise-smooth,
simple closed curve in the plane.
given by
Prove that the area of the region is
.
[6pts]
(b) Use the formula in (a) to find the area under one arch of the cyc
2- 3
16.3
11
10
09
1.
(a)
Let F .
Determine whether
or
not
the vector
field
conservative, find a function such that F .
(b) Find
F where :
[6pts]
sin
F
is conservative.
If
[6pts]
s i n s i n cos cos
it
is
2- 2
15.8
11
10
09
8. (a) Evaluate
.
[5pts]
(b) Evaluate
.
[5pts]
08
7.[8pts] Use spherical coordinates to find the volume of the solid that lies under
the plane and above the cone .
8. [8pts] The volume of the closed ball
in
is , by evaluat
2- 2
15.6
11
09
7. Evaluate the following integrals.
(a)
cos
cos
ar ctan
(b)
[6pts]
[6pts]
(c)
, where and denotes the greatest
integer which is not larger than . [6pts]
(d)
[6pts]
10.
Determine whether the statement is true or false
2- 2
15.3
11
10
09
7. Evaluate the following integrals.
(a)
cos
cos
ar ctan
(b)
[6pts]
[6pts]
(c)
, where and denotes the greatest
integer which is not larger than . [6pts]
(d)
10.
[6pts]
Determine whether the statement is true or fa
2
14.2
11
10
09
8. Find the limit, if it exists, and prove your statement using
definition of limit, or show that the limit does not exist.
(a)
(b)
08
lim
[7pts]
lim
[7pts]
8. Find the limit, if it exists, and prove your statement using
definition
2
13.3~13.4
08
3.[10pts] Show that is perpendicular to the binormal vector and the
unit tangent vector , and deduce from them that
for some number (called the torsion of curve). (Here is the
principal unit normal vector.)
4. [7pts] Reparametrize the curv
2
13.2~13.3
13.2
10
13.3
11
10
09
4. Let be the unit tangent, unit normal, and binormal
vector, respectively on
a smooth space curve and be a
point of the curve.
(a) Show that the vector at lies in the plane determined by
and ,
constants.
that is,
where
2- 1
12.5
11
10
09
2. Find the volume of the tetrahedron with sides
, , and .
[5pts]
3. Find parametric equations for the line through the point that
is parallel to the plane and perpendicular to the line
, , . [5pts]
08
1.[7pts] Let and be the lines wi
2- 1
12.4
11
10
09
1.(a) Show that ,
where and are noncoplanar vectors, and
(Hint: , )
[7pts]
(b) Show that for any 3-dimensional nonzero vectors and , is
orthogonal to .
[5pts]
11.1 Sequences
(Limit Laws for sequences)
If
1.
and
are convergent sequences and
is a constant, then
(Definition 1)
lim lim lim
A sequence
has the limit ,
lim lim
lim
lim
lim lim lim
if we can make the terms as close to as
we like by taking s
10.1 Curve Defined by Parametric
.
Equations.
()
.
1. Parametric Equations
3 (parameter; )
(parametric equation; )
.
: initial point
: terminal point
,
,
(Ex 2)
(parametric curve; ) .
What curve is represented by the following
parametr
8.1 Arc Length
(The Arc Length Formula)
If
is continuous on ( is called
1. Arc Length
smooth),
, is
then
the
.
(Proof)
,
, .
(Definition)
The length of the curve with ,
, as the limit of the lengths of
these inscribed polygons
lim
corve
5.2
( 3)
If is continuous on , or if has only
1.
a finite number of removable or jump
( )
discontinuities, then is integrable on
If is a function defined for , we
; that is, the definite integral
divide the interval into subintervals
exists.
of equal
2011 (2) 3
1. [4pts each]
Let si n si n cos .
(a) Show that the vector field is conservative.
[ Use an appropriate theorem not the definition of conservativeness. ]
(b) Find a function such that .
(c) Evaluate the line integral
where consists
,
of and
1. [3pts each] The surface is implicitly defined by arcsin .
[ Note that sin . ]
(a) Find the direction in which changes fastest at and find the maximum rate of change of
at that point.
(b) Find the equation of the tangent plane at .
(c) Find the equat