Extra Credit : Due December 14
You must show all work to receive full credit.
Let Bn denote the solid ball of radius one in Rn . This is the set of points
(x1 , x2 , . . . , xn ) Rn satisfying the condition 0 x2 + x2 + + x2 1.
n
1
2
1. Calculate the volum
Math 223 VECTOR CALCULUS
SOLUTION FOR QUIZ III (01/29)
February 3 (Wed), 2010
Instructor:
Yasuyuki Kachi
Line #: 67985
[I] (10pts)
68995.
Let 1 and 2 be two lines in R3 prescribed as
1 :
=
t, 2t, t ,
2 :
(1)
x, y, z
where t is the parameter,
x, y, z
=
1 +
Math 223 VECTOR CALCULUS
SOLUTION FOR QUIZ IV (02/10)
February 12 (Fri), 2010
Instructor:
Yasuyuki Kachi
Line #: 67985
[I] (15pts)
68995.
(1)
For
a1
A = b1
c1
we dene
a2
b2
c2
det A =
det A =
a
a3
b3 ,
c3
a1
b1
c1
and
a=
b=
a3
b3
c3
b1 , b2 , b3 ,
c=
a1 ,
Math 223 VECTOR CALCULUS
SOLUTION FOR QUIZ V (02/12)
February 17 (Wed), 2010
Instructor:
Yasuyuki Kachi
Line #: 67985
[I] (4pts)
system
(1)
r, , z
68995.
The conversion formulas between the cylindrical
and the Cartesian coordinate system
x, y, z
coordinat
Math 223 VECTOR CALCULUS
SOLUTION FOR QUIZ VI (03/12)
March 29 (Mon), 2010
Instructor:
Yasuyuki Kachi
Line #: 67985
[I] (12pts)
(1)
68995.
d 2x + 3y
(2)
d xy
(3)
d x2 + y 2
(4)
d y4
(5)
d
x+y
(6)
d
y
x
= 2 dx + 3 dy .
= y dx + x dy
the Leibniz rule
.
= 2x
Math 223 VECTOR CALCULUS
SOLUTION FOR QUIZ VII (03/24)
March 29 (Mon), 2010
Instructor:
Yasuyuki Kachi
Line #: 67985
[I] (50pts)
(1)
68995.
The graph of the curve prescribed by the equation
y=
is the upper half-plane
y0
x
1x
portion of the circle with rad
Math 223 VECTOR CALCULUS
SOLUTION FOR QUIZ VIII (03/31)
April 9 (Fri), 2010
Instructor:
Yasuyuki Kachi
Line #: 67985
[I] (26pts)
68995.
r R that ts in the box, if any. Here +
The largest
is the answer if there does exist r that ts in the box, and there i
Outline (Checklist) of Exam 2
Math 223 Vector Calculus
October 14, 2013
Extrema Problem.
(i) Local extrema and Second derivative test.
Section 4.2: # 6, 7, 29, 30.
(ii) Global extrema in a closed region.
Section 4.2: # 37.
(iii) Global extrema under co
Outline (Checklist) of Exam 1
Math 223 Vector Calculus
September 16, 2013
Vectors.
(i) Dot product: Denition, Theorem 3.3 on pg. 20, and projections.
(ii) Cross product: Algebraic denition and geometry meaning.
(iii) Line equation in R3 .
(iv) Plane equa
Name(Print):
Math 223 Vector Calculus
Fall 2013
Exam 3
11/22/13
Time Limit: 50 Minutes
This exam contains 6 pages (including the cover pages) and 6 problems. Check to see if any pages
are missing. PRINT your name on the top of this page, and put your init
Name(Print):
Math 223 Vector Calculus
Fall 2013
Exam 2
10/18/13
Time Limit: 50 Minutes
This exam contains 7 pages (including the cover pages) and 6 problems. Check to see if any pages
are missing. PRINT your name on the top of this page, and put your init
Math 223 Vector Calculus Exam 1
9/20/13
Please print your name below:
Name:
1
Math 223 Vector Calculus
Math 223 Vector Calculus
Fall 2013
Exam 1
9/20/13
Time Limit: 50 Minutes
Exam 1 - Page 2 of 9
9/20/13
Name (Print):
This exam contains 9 pages (includin
Math 223 VECTOR CALCULUS
SOLUTION FOR QUIZ I (01/25)
February 1 (Mon), 2010
Instructor:
Yasuyuki Kachi
Line #: 67985
68995.
Let a R be such that a > 0. In R3 with the coordinate system
x, y, z , the distance from the point P = 1, 2 a, a2
to
[I] (6pts)
(1)
Math 223 VECTOR CALCULUS
SOLUTION FOR MIDTERM EXAM I (IN-CLASS; 04/05)
April 7 (Wed), 2010
Instructor:
Yasuyuki Kachi
Line #: 67985
[I] (12pts)
(1b)
68995.
(1a)
x
d 3x + 8y
3x + 8y
(2a)
d xy
x
(2b)
d x2 + y 2
(3a)
(3b)
[II] (18pts)
xy
x
(1)
x2 + y 2
= 3 d
Name:
Instructor:
ID:
Math122 Final Exam, May 19, 2011
Instructions:
1. You will be given 150 minutes for this exam (4:30 - 7:00pm).
2. The exam has three parts: True-Flase, Multiple-Choice, and Essay Problems.
3. Fill in your name, Student ID, and your i
Math 223 : Exam 1
You must show all work to receive full credit.
1. Determine whether or not lim(x,y)(0,0) x+y
2
x +y 2
exists. (15pts)
2. Let f (x, y, z ) = x + 2y + 3z .
(a) Use the limit denition to show that f (x, y, z ) is dierentiable at
(1, 1, 1).
Math 223 : Exam 2
You must show all work to receive full credit.
1. Let F = (2xz + y 2 ) i + (2xy + z 2 ) j + (2yz + x2 ) k and X denote the
line segment from (0,0,0) to (2,3,4).
(a) Show that F is a conservative vector eld. (10 pts)
(b) Find the parametr
Math 223 : Exam 3
You must show all work to receive full credit.
1. Let B denote the solid ellipsoid in Euclidean 3-space given by the equation
Calculate
2. Calculate
(x 1)2 (y + 1)2 (z 2)2
+
+
1.
9
16
25
B 2z dV . (30 pts)
R3
e(x
3
2 +y 2 +z 2 ) 2
dV .
Math 223 : Sample Exam 1
This exam is practice for Exam 1, which will be similar in spirit, but not
identical to the sample exam.
1. Let f (x, y, z ) = 3x2 + 2yz . Find the directional derivative of f at
1
1
1
i
j
k
(1, 1, 1) in the direction of := 3 + 3
Math 223 : Sample Exam 2
This is a practice exam for Exam 2, which will be similar in spirit, but
not identical to this sample exam.
1. Consider the vector eld F = x3 i + y 3 j + z 3 k and the path in R3
given by X = cos(t) i + sin(t) j + t k , 0 t 2 .
(a
Math 223 : Sample Exam 3
This is a practice exam for Exam 3, which will be similar in spirit, but
not identical to this sample exam.
2
1. Find
B x + 3yz dV for B the solid parallelopiped spanned by the
vectors i + j + k , 2i 3j + k , and j + 2k .
2. Let W