1. ~et 5 be the part of the surface z
In
= 1 + 3.y2 + 3y2
between the planes z = 4 and z
=7
the first octant.
(a) Compute the surface area of S. (10 points)
A ~ JJJ\+ (t)\r+j)1-oIA
?
p
!IJ I.(,x)'-r("f]"'
:0
JA
o
-; J J
f
I l- 1. (.l':'- <- :lb~"\. .,./
A
Math 39200 Quizz #3
Group A
NAME:
Points (maximum = 12):
Problem 1 (8 points). Let S be the surface dened by x = y 2 + z 2 and x 9. The
boundary curve C = S is oriented counterclockwise seen from the positive x-axis
(seen from points (x, 0, 0, ) with x >
MATH 392 TEST 3A
May 1, 2013
Name: SMITF.l
Note that both sides of each sheet has printed material
Instructions:
1. Read the instructions.
2. Dont panic!
3. Complete all problems!
4. Show ALL your work to receive full credit. You will get 0 credit for sim
MATH 392 QUIZ 2 - Version B Answers
1. Complete the following rules. Let r(t) =< x(t), y(t), z(t) > where appropriate:
b
(a) Let C be a smooth curve, then
f (x(t), y(t) (x (t)2 + (y (t)2 dt
f (x, y) ds =
a
C
(b) Given f (x, y, z), we have f = < fx (x, y,
MATH 392 TEST 33
May 1, 2013
Name: V O N M i TH
Note that both sides of each sheet has printed material
Instructions:
1. Read the instructions.
2. Dont panic!
3. Complete all problems!
4. Show ALL your work to receive full credit. You will get 0 credit fo
MATH 392 QUIZ 2 - Version A Answers
1. Complete the following rules. Let r(t) =< x(t), y(t), z(t) > where appropriate:
(a) r (t) = < x (t), y (t), z (t) >
b
(b)
f (x(t), y(t)
f (x, y) ds =
(x (t)2 + (y (t)2 dt , where C is a smooth curve.
a
C
(c) Given f
MATH 392 TEST 1 REVIEW
October 5, 2011
From Spring 2005 Final:
1. Compute the vector which describes the direction of greatest increase for the function
f (x, y) = x2 y 3 at the point with coordinates (2, 1).
2. Find the equation of the tangent plane for
MATH 392 TEST 213
April 4, 2013
Name: , )HGVON ngTPl
Note that both sides of each sheet has printed material
Instructions:
1. Read the instructions.
2. Complete all problems!
3. Show ALL your work to receive full credit. You will get 0 credit for simply w
Math 39200 Quizz #2
Group B
NAME:
Points (maximum = 10):
Problem 1 (5 points). Compute divergence and curl of the vector field
2
y + 6xz, 2xy, 3x2 .
Is it irrotational? Is it conservative? Is it incompressible (source-free)?
Problem 2 (5 points). Let the
Math 39200 Quizz #3
Group B
NAME:
Points (maximum = 12):
Problem 1 (4 points). Let T be the tetrahedron which is contained in the first
octant below the plane 3x + 2y + z = 6 with outward pointing normal vector ~n and
let F be the vector field F = hx, y +
Math 39200 Quizz #2
Group A
NAME:
Points (maximum = 10):
Problem 1 (5 points). Compute divergence and curl of the vector eld
3y 2 , z 2 + 6xy, 2zy .
Is it irrotational? Is it conservative? Is it incompressible (source-free)?
Problem 2 (5 points). Let the
Math 39200 Quizz #1
Group B
NAME:
Points (maximum = 10):
Let R be the region in the x, y-plane contained between the curves x = y 2 + 2y + 2
and x + y = 2. Let C be the boundary of R, oriented counterclockwise. Evaluate
the integral
2dx + 3xdy
C
a) direct
Marices (basic denitions, multiplication)
An m n matrix (m by n or m time n) is a table of numbers
usually real numbers) consisting of m rows and n columns. Notation:
a1,1 a1,2 a1,n
a2,1 a2,2
a2,n
A = (ai,j ) = (ai,j )i,j = (ai,j )1im = .
.
.
.
.
1jn
.
Math 39200 Quizz #1
Group A
NAME:
Points (maximum = 10):
Let R be the region in the x, y-plane contained between the curves x = y 2 4y 1
and x + y = 1. Let C be the boundary of R, oriented counterclockwise. Evaluate
the integral
2dx + 3xdy
C
a) directly a
Math 39200 Exam #1
Group A
1.
26 March 2009
2.
Name:
3.
4.
Use simplied integer fractions, avoid decimals. You are not allowed to use books,
notes or calculators.
Problem 1 (8 points). Let R be the quarter-disk x2 + y 2 4, x 0, y 0, in
the xy-plane. Let C
Math 39200 Exam #1
Group B
1.
26 March 2009
2.
Name:
3.
4.
Use simplied integer fractions, avoid decimals. You are not allowed to use books,
notes or calculators.
Problem 1 (6 points). Compute divergence and curl of the vector eld
F = xy, cos y x, z 2 .
I
Math 39200 Exam #2
1.
Group A
5 May 2009
2.
Name:
3.
4.
Use simplied integer fractions, avoid decimals. You are not allowed to use books, notes
or calculators.
Problem 1 (6 points). Let T be the part of the solid ball x2 + y 2 + z 2 16 where y 2.
Let S be
Math 39200 Exam #2
1.
Group B
5 May 2009
2.
Name:
3.
4.
Use simplied integer fractions, avoid decimals. You are not allowed to use books, notes
or calculators.
Problem 1 (8 points). Compute the inverse matrix A1 by
(a) the adjoint matrix (Gaussian method)
MATH 392 TEST 2A
April 4, 2013
Name: Ahie/Cw S MITH
Note that both sides of each sheet has printed material
Instructions:
1. Read the instructions.
2. Complete all problems!
3. Show ALL your work to receive full credit. You will get 0 credit for simply wr
Math 392 1XC Course Information Sheet
Instructor: Prof. Paolillo
Office: NAC 6/273
Email: apaolillo@ccny.cuny.edu
Times and Place: T, W, Th. 1:20 3:00 in NAC 6/113
Text 1: Thomas Calculus, 13th edition, by Thomas, Weir, Hass and Heil
isbn: 0 321 87896 5 .
Name
Math 3921XC
Test#2
So kf,-~.s
-~-~-
Switch off and put away all electronic devices, including cell-phones and calculators. Show your work for full credit.
1. A surface of revolution
S is parametrized by r( u, B) = (u 2 + 1, u cos( B), u sin( B) ,0 :s
Math 3921XC
Midterm 1
Switch off and put away all electronic devices, including cell phones and calculators. You have the entire class period
to complete this exam. Show all work for full credit.
1 3
1. a) let A = 2 4 1:1
(
1 2
J.Compute A-' by doing elem
M392:
Linear systems of ordinary differential equations
Definitions: is an eigenvalue of an m by m matrix A provided
the eigenvector equation Ax = x with x = the m by 1 matrix
x1 x2 , , , , xm T has nontrivial solutions x . The trivial
solution is x = 0 =