1. State Stokestheorem.
Let S be an oriented piecewise-smooth surface that is bounded by a
simple, closed, piecewise-smooth boundary curve C with positive orientation. Let F be a vector eld whose components have continuous
partial derivatives on an open r
1. Evaluate
ZZ
the origin.
ex
2 +y 2
dA where D is the disk of radius 2 with center at
D
R2 R2
R 2 h1
r2
r2
ir=2
In polar coordinates this is 0 0 e r dr d = 0 2 e
d =
r=0
R
2
1
(e4 1) d = (e4 1).
2 0
ZZ
2. Evaluate
sin y 2 dA where D is the triangle with
1. Let F (x; y) be a function whose partial derivatives take on the following
values:
(0; 3) ( 1; 2) (0; 1) (3; 0)
Fx
1
5
3
2
Fy
2
3
4
1
Let g (t) = F (t2
t). Compute g 0 (0) and g 0 (1).
1; 2
We have dF = Fx dx + Fy dy = Fx 2t dt
g 0 (t) =
so g 0 (0) =
2
1. State Greens theorem.
Let D be the region bounded by a positively oriented, picewise-smooth,
simple closed curve C in the plane. If P and Q have continuous partial
derivatives on an open region containing D, then
ZZ
Z
P dx + Q dy =
(Qx Py ) dA
C
D
2. L
1a. Convert (4; 7 =4; =6) from spherical to rectangular coordinates.
7
7
4 sin cos ; 4 sin sin ; 4 cos
6
4
6
4
6
=
=
p
p ! p !
4 2 4
2
4 3
;
;
2 2 2
2
2
p
p p
2;
2; 2 3
1b. Convert ( 1; 2; 3) from rectangular to cylindrical coordinates.
q
p
r = ( 1)2 + 22
1. Find the directional derivative of the function
(a) ln (x2 + y 2 ) at the point (1; 2) in the direction of the vector h 1; 3i.
2x
2y
which is equal to h2=5; 4=5i
r ln (x2 + y 2 ) =
; 2
2
2
x + y x + y2
at the point (1; 2).
p
The unit vector in the dire
1. Find the unit tangent vector to the curve
The derivative is
1
p ;
2 t
p
1 3t2
;
t2 1 + t3
t; 1=t; ln (1 + t3 ) at t = 1.
.
At t = 1, this is h1=2; 1; 3=2i. The length of this vector is
p
1p
1=4 + 1 + 9=4 =
14
2
so the unit tangent vector is
h1=2; 1; 3=
1. What is a conservative vector eld? Give an example of a vector eld
that is not conservative.
A vector eld F is conservative if F = rf for some scalar function f .
See page 1061 of the text. Dont write down everything youve ever
heard about conservative
1. Find the local maximum and minimum values, and the saddle points,
of the function f (x; y) = xy 2 x2 2y 2 .
rf = hy 2 2x; 2xy 4yi so the critical points are obtained by solving
the equations y 2 2x = 0 and 2xy 4y = 0. From the second equation
we get 2y
1. Dene the dot product of two vectors.
The dot product of two vectors ha1 ; a2 ; a3 i and hb1 ; b2 ; b3 i is the scalar
a1 b 1 + a2 b 2 + a3 b 3 .
2. Dene what the cross product of two vectors is.
The cross product of two vectors ha1 ; a2 ; a3 i and hb1
1. Let F (x; y) =
Z
y2
arctan t dt. Find Fx (2; 3) and Fy (2; 3).
x
Use the fundamental theorem of calculus to compute Fx (x; y) = arctan x
and Fy (x; y) = 2y arctan y 2 . So Fx (2; 3) = arctan 2 and Fy (2; 3) =
6 arctan 9.
2. Suppose x2 2y 3 + 3z 4 = 6 d