. . Name: £le 0")
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Math 290-3: Midterm 1 Practice
1. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
(a) If f is a continuous function and 1]; f(x, y) (111 = 0 fo
True. )Vcse go\u\:5¥\$
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Northwestern University
Name:
Student ID:
Math 290-3: Midterm 2 Practice
1. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
(a) Let F be a vector eld on Rn and let C be a curve in Rn parametrized by x(t),
a t b. I
Math 290-3: Final Exam Practice
Page 8 of 13
!
3. Let F = (y + sin cos xz)i + (x9 cos ey z)j + (yx y)k. Compute S curl F dS where S
is the piece of the paraboloid x = 5 y2 z2 with x 1, oriented with outward pointing
normal vectors.
ANSWER: The boundary of
Practice Problems for Midterm 2
1
True/False
Determine whether each of the following is true or false. Briey justify.
1. If T is a transformation whose Jacobean is equal to one, then T is a rotation (for our
purposes, the identity is a rotation by zero de
Math 290-3: Midterm 2 Practice Corrections
No Math 290 Practice Midterm would be complete without mistakes in the posted solutions.
So, here are some corrections. (Our track record is taking a beating!)
True/False 1. Not a big deal, but this technically d
Name:
Student ID:
Northwestern University
Math 290-3: Midterm 1 Practice
1. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
(a) If f is a continuous function and D f (x, y) dA = 0 for any region D in the xyplane,
REVIEW OF TECHNIQUES OF INTEGRATION
1. Integrals to memorize
xn dx
=
xn+1
+ C (n = 1)
n+1
dx
x
=
ln |x| + C
sin(x)dx
= cos(x) + C
cos(x)dx
=
sin(x) + C
tan(x)dx
=
ln |sec(x)| + C
sec(x)dx
=
ln |tan(x) + sec(x)| + C
ex dx
dx
1 x2
dx
1 + x2
= ex + C
=
sin1
How to Compute Line Integrals
Northwestern, Spring 2013
Computing line integrals can be a tricky business. First, theres two types of line integrals:
scalar line integrals and vector line integrals. Although these are related, we compute them in
dierent w
Math 290-3: Homework 7
Northwestern University, Spring 2014
Due: June 2, 2014
Colley, Section 7.3. 1, 8, 11, 12, 13, 18, 20 (Youll probably need to look up some trig identities
to nish 11, or use Wolfram Alpha)
TF. There exists a vector eld F such that
zx
Math 290-3: Homework 6
Northwestern University, Spring 2014
Due: May 28, 2014
Colley, Section 7.1. 3, 13, 14, 27
Colley, Section 7.2. 1, 2, 3, 7, 8, 15, 18, 24
TF. Let S denote the unit sphere, oriented with inward pointing normal vectors. Then the value
Notes on Greens Theorem
Northwestern, Spring 2013
The purpose of these notes is to outline some interesting uses of Greens Theorem in situations
where it doesnt seem like Greens Theorem should be applicable. Such applications arent really
mentioned in our
Math 290-3: Homework 4
Northwestern University, Spring 2014
Due: May 9, 2014
Colley, Section 3.3. 4, 9, 25 (To be clear in 25, we didnt talk about ow lines in class but
the idea is as follows. A ow line of a vector eld F is a curve x(t) such that at each
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Math 2908 Midterm 2 Spring Quarter 2013 Page 2 of 9
l. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
. (30 L11 LEO!de = I: OZSinm)rdrd6
TRUE
(b) Let X 1 [0, 7r] » R2 be the path KO?) 2 (2 cos(2t),3 sin(2f
Math 290-3: Lecture Notes, Week 1
Northwestern University, Spring 2014
March 31, 2014: Double Integrals
Final quarter of the year! Today I gave a brief overview of the course, which is all about integration.
We then started with material on double integra
Math 290-3: Lecture Notes, Week 9
Northwestern University, Spring 2014
May 28, 2014: More on Stokes Theorem
Today we continued talking about Stokes Theorem, and used it to show that surface integrals
of curls have a certain surface-independence property,
Math 290-3: Lecture Notes, Week 10
Northwestern University, Spring 2014
June 2, 2014: More on Gausss Theorem
Today, unfortunately, was the last class of the entire year. Kudos to you all for making it through
Math 290! We looked at a few more examples, an
Math 290-3: Lecture Notes, Week 8
Northwestern University, Spring 2014
May 19, 2014: Vector Surface Integrals
Today we started talking about vector surface integrals, which arise when we integrate vector
elds over surfaces. Just as with vector line integr
Math 290-3: Linear Algebra & Multivariable Calculus
Northwestern University, Spring 2014
Course Information
Instructor: Santiago Caez
n
Email: [email protected]
Website: http:/math.northwestern.edu/~scanez/courses/math290/spring14/
Oce Hours: W 1:30
Math 290-3: Lecture Notes, Week 7
Northwestern University, Spring 2014
May 12, 2014: More on Conservative Fields
Today we spoke more about conservative vector elds, moving beyond the denition and looking
at their important properties. The end result is th
Math 290 Schedule
Monday
Wednesday
Friday
3/31-4/4
Areas and Volumes
Double Integrals
More on Double Integrals
4/7-4/11
Changing Order of Integration
Triple Integrals
More on Triple Integrals
4/14-4/18
Yet More on Triple Integrals
Change of Variables
More
Math 290-3: Lecture Notes, Week 6
Northwestern University, Spring 2014
May 5, 2014: Vector Line Integrals
Today we spoke about the second type of line integral we care about, which arises when integrating
a vector eld along a curve. Such line integrals ha
Math 290-3: Lecture Notes, Week 4
Northwestern University, Spring 2014
April 21, 2014: Cylindrical and Spherical Integrals
Today we nished talking about the concept of making a change of variables in an integral, focusing
on rewriting triple integrals in
Math 290-3: Lecture Notes, Week 5
Northwestern University, Spring 2014
April 28, 2014: Divergence and Curl
Today we spoke about the notions of the divergence and curl of a vector eld. These will play crucial
roles in some theorems we will soon look at, an
Math 290-3: Lecture Notes, Week 3
Northwestern University, Spring 2014
April 14, 2014: Yet More on Triple Integrals
Today we continued with more examples of setting up triple integrals, and in particular looked at
some instances where we have to split our
Math 290-3: Lecture Notes, Week 2
Northwestern University, Spring 2014
April 7, 2014: Changing Order of Integration
Today we spoke about changing the order of integration in a double integral, which can be useful in
situations where integrating with respe
Name:
Student ID:
Math 290-3: Midterm 2 Practice
1. Determine whether each of the following statements is TRUE or FALSE. Justify your
answer.
(a) Let F be a vector eld on R" and let C be a curve in R parametrized by 31(1),
a s r s b. If F(x(r) is perpen
Math 290-3: Homework 3
Northwestern University, Spring 2014
Due: April 25, 2014
Additional Triple Integral Problem. Rewrite the following with respect to the orders dz dx dy
and dy dz dx.
1
0
1z
0
1+y
1
1z
1y
f (x, y, z) dx dy dz
f (x, y, z) dx dy dz +
0
Math 290-3: Homework 5
Northwestern University, Spring 2014
Due: May 16, 2014
Colley, Section 6.2. 2, 8, 13, 17, 15b (do 17 before 15b), 22, 27
Additional Greens Theorem Problem. Let F = (xy x cos x)i + (x2 yey )j and let C be
the path which starts at (1,