Homework Assignment 2, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
January 30, 2014
1. Let v(x) = sin(x). For all 0 x , Fnd all solutions of
x (t) = v(x(t) ,
x(0) = x0 .
For which values of t is each solution dened?
Hint: It will probably hel
M.Naumova
Math 251
Chapter 14 (14.1 only) - Key-points
Functions of Two or More Variables.
Definitions
The domain D of a function f(x1, . . . , xn) of n variables is the set of all n-tuples
(a1 , . . . , an ) in Rn for which f (a1 , . . . , an ) is defin
01:640:251 Calculus III,
Review Problems for the Final Exam
Fall 2016
Department of Mathematics, Rutgers University
INSTRUCTOR: M.Naumova
Some answers are given in small type.
PART 1
1. Write the vector u = h2, 3, 1i in the form v + w, where v is a vector
Blitz Review Problems - Final Exam Math 251, M.Naumova
1. There is a vector field F(x, y) which has the property curl(F) = div(F) = C, where C
is a some constant.
2. Assume a vector field F(x, y) is a gradient field, then
center (0, 0, 1), radius 1, locat
DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
January, 2014
2
Contents
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1
1
What dierential equations are, and what it means to solve them . . . . . . . . . . . .
1
The descript
DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
January, 2014
2
Contents
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1
What dierential equations are, and what it means to solve them . . . . . . . . . . . .
1.1.1
n
1
1
Some
MULTIVARIABLE CALCULUS, LINEAR
ALGEBRA AND DIFFERENTIAL EQUATIONS,
PART II
Eric A. Carlen
Professor of Mathematics
Rutgers University
January, 2012
2
Contents
1 WHAT DIFFERENTIAL EQUATIONS ARE
1.1
1
Dierential Equations . . . . . . . . . . . . . . . . . .
Prep Problems for Quiz 5 01:220:320:04 Spring 2014
You have 24 hours to allocate between leisure () and work ( = 24 ). Every hour you work you
receive a wage $. (If you work a fraction of an hour, you get the corresponding fraction of .) You buy
a consump
M.Naumova
Math 251
17. Fundamental Theorems of Vector Analysis
17.1. Greens Theorem.
Key Points
Orientation of a simple closed curve and the orientation of the boundary D of a
region bounded by a finite union of simple closed curves.
Scalar curl of a co
M.Naumova
Math 251
Chapter 14 - Key-points
14.1. Functions of Two or More Variables
Key Points
The domain D of a function f (x1 , . . . , xn ) of n variables is the set of all n-tuples
(a1 , . . . , an ) in Rn for which f (a1 , . . . , an ) is defined. T
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M.Naumova
Math 251
Chapter 12 - Key-points
Vector Geometry
12.1 Vectors in the Plane
A vector v = P Q is determined by a basepoint or tail P and a terminal point or
head Q. The length of v = P Q, denoted kvk, is the distance from P to Q.
The vector v =
M.Naumova
Math 251
16. Line and Surface Integrals
16.1. Vector Fields
Key Points
Definition of a vector field in R2 and R3 .
The divergence of a vector field F =< F1 , F2 , F3 > is defined by
F1 F2 F3
div(F) =
+
+
x
y
z
The (vector) curl of a continuou
M.Naumova
Math 251
Chapter 15 - Key-points
15.1. Integration in Two Variables
Key Points
The double integral of f (x, y) over a rectangle R is defined as
Z Z
N X
M
X
f (x, y) dA = lim
f (Pij )xy,
M,N
R
i=1 j=i
where Pij is any point in the subrectangle
M.Naumova
Math 251
Chapter 13 - Key-points
Calculus of Vector-Valued Functions
13.1. Vector-Valued Functions.
A curve in R3 can be parametrized using a three dimensional vector valued function
r(t) = x(t)i + y(t)j + z(t)k = hx(t), y(t), z(t)i
The projec
Practice Test for Test 2, Math 292, April 25, 2013
Eric A. Carlen1
Rutgers University
April 24, 2014
1. The dierential equation
t2 x (t) 3tx (t) + 4x(t) = 0
has polynomial coecients.
(a) Find one polynomial solution to this equation.
(b) Find the general
Practice Final Exam, Math 292, 2014
Eric A. Carlen1
Rutgers University
May 6, 2014
1. Find the general solution of
t3 x (t) + t2 x(t) x2 (t) = 2t4 .
2. Cinside the two equations
I (y )2 + y 2 = 1
and
II (y )2 y 2 = 1 .
One has a unique solution with y(t)
Solutions for the Exercises from Chapter 1
Eric A. Carlen1
Rutgers University
February 10, 2014
1.1 Find the general solution of the dierential equation
tx = 3x + t4
for t > 0. Find the corresponding ow transformation, and the particular solution with x(1
Homework Assignment 4, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
February 22, 2014
1. (10 points) Let A be the matrix A =
0 1
0
.
(a) Compute A2 , A3 and A4 Observe the patterns, and deduce a formula for Ak for all positive
integers k. (Yo
Solutions for Homework Assignment 2, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
February 11, 2014
1. Let v(x) = sin(x). For all 0 x , Find all solutions of
x (t) = v(x(t) ,
x(0) = x0 .
For which values of t is each solution dened?
Hint: It w
Challenge Problem Set 1, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
February 11, 2014
This challenge problem set concerns the matrix exponential function. For any n n matrix A
this is dened by
k
t k
A
etA =
k!
k=0
where A0 = I by denition. W
Challenge Problem Set 2, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
February 26, 2014
Let K =
3 2
2 6
. The object of this challenge problem set is to nd and study the solution of
x (t) = Kx(t) + f (t)
with x(0) = (1, 2)
and
and x (0) = (1, 1
Challenge Problem Set 4, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
April 23, 2014
This challenge problem set concerns the construction of Greens functions for Lu = f subject
to boundary conditions other than u(a) = u(b) = 0. As always, we ta
Note on the Brachistochrone Problem
Eric A. Carlen1
Rutgers University
April 21, 2013
1
introduction
Consider two points p and q in the x, y plane.
We suppose that the height of q is lower than that of p, so that the straight line segment from
p to q runs
Challenge Problem Set 5, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
April 30, 2014
This challenge problem set concerns nding particular solutions of higher order third order in
this case solutions of inhomogeneous linear equations.
Consider t
Homework Assignment 5, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
March 14, 2014
1. Find the exact solution of x (t) = v(x(t), t) and x(0) = 0 for
v(x, t) = 2t(1 + x) .
Stating from X0 = 0, compute the next 4 terms in the Picard iteration, n