Math 291
Workshop 1
Please work in groups of two or three. These exercises are not to be turned in.
Problem 1.1. Sketch the following sets. Compute their dimensions.
1. cfw_x R | x = 1.
2. cfw_x R | x2 = 1.
3. cfw_x R | x2 4.
4. cfw_x R | x2 1 and x2 4.
5
Second Practice Final Exam, Math 291 Fall 2012
Eric A. Carlen1 Rutgers University December 16, 2012
1: (a) Let a = (2, 1, 2) and b = (1, 2, -2). The set of points x R3 satsifying ax=b is a line. (a) Find a parameterization of this line. (b) Find the dista
Practice Test I, Math 291 Fall 2012
October 11, 2012
1: (a) Find a right handed orthonormal basis cfw_u1 , u2 , u3 such that u1 is a positive multiple of
(2, 2, 1) and u2 is orthogonal to (1, 1, 0).
In the rest of this problem, cfw_u1 , u2 , u3 refers t
Practice Test IB, Math 291 Fall 2012
October 16, 2012
1: Let
a = (1, 4, 8)
b = (9, 0, 9)
and c = (11, 11, 1) .
(a) Find a right handed orthonormal basis cfw_u1 , u2 , u3 such that u1 is a multiple of a, and u3 is
orthogonal to b.
(b) Find numbers y1 , y2
Practice Test IIA, Math 291 Fall 2012
Eric A. Carlen1
Rutgers University
November 25, 2012
1: Let f (x, y ) = 3xy 3 + 2x + 1 x4 + 9 y 2 .
4
2
(a) Find all of the critical points of f . Evaluate the Hessian matrix of f at each of these critical
points, and
MULTIVARIABLE CALCULUS, LINEAR
ALGEBRA AND DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
September, 2011
2
Contents
1 GEOMETRY, ALGEBRA AND THE ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES
1.1
1
n
Algebra and geometry in
MULTIVARIABLE CALCULUS, LINEAR
ALGEBRA AND DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
September, 2012
2
Contents
1 GEOMETRY, ALGEBRA AND Analysis IN SEVERAL VARIABLES
1.1
n
1
Algebra and geometry in R . . . . . . . .
Challenge Exercise Set I, Math 291 Fall 2012
Eric A. Carlen1
Rutgers University
October 7, 2012
1
Introduction
This Challenge Exercise Set will give you an introduction to doing proofs. The proofs will be based
on computations, and the key to each proof i
Challenge Exercise Set 2, Math 291 Fall 2012
Eric A. Carlen1
Rutgers University
October 9, 2012
1
Introduction
This Challenge Exercise Set concerns reections and rotations. We will use the notation form the
previous Challenge Exercise Set.
Recall that in
Challenge Exercise Set 3, Math 291 Fall 2012
Eric A. Carlen1
Rutgers University
October 10, 2012
1
Introduction
This Challenge Exercise Set concerns geodesics shortest paths on the 2-dimensional sphere in
R3 , and also the n-dimensional sphere in Rn+1 for
Challenge Problem Set 4, Math 291 Fall 2012
Eric A. Carlen1
Rutgers University
October 23, 2012
1
Linear functions and their matrices
This challenge problem set concerns some basic facts about linear functions and matrices. We focus
on the important case
Challenge Problem Set 5, Math 291 Fall 2012
Eric A. Carlen1
Rutgers University
November 7, 2012
This challenge problem set illustrates realistic use of Lagrange Multiplies and Newtons
Method to solve an interesting optimization problem. It can be done wit
Challenge Problem Set 6, Math 291 Fall 2012
Eric A. Carlen1
Rutgers University
November 20, 2012
This challenge problem set concerns planetary motion, and completes the analysis of Newtons
derivation of Keplers Laws from his Universal Theory of Graviation
Calculus III
Preface
Here are my online notes for my Calculus III course that I teach here at Lamar University.
Despite the fact that these are my class notes they should be accessible to anyone wanting to
learn Calculus III or needing a refresher in some
Calculus III
Preface
Here are my online notes for my Calculus III course that I teach here at Lamar University.
Despite the fact that these are my class notes they should be accessible to anyone wanting to
learn Calculus III or needing a refresher in some
Calculus III
Preface
Here are my online notes for my Calculus III course that I teach here at Lamar University.
Despite the fact that these are my class notes they should be accessible to anyone wanting to
learn Calculus III or needing a refresher in some
Calculus III
Preface
Here are my online notes for my Calculus III course that I teach here at Lamar University.
Despite the fact that these are my class notes, they should be accessible to anyone wanting to
learn Calculus III or needing a refresher in som
Calculus III
Preface
Here are my online notes for my Calculus III course that I teach here at Lamar University.
Despite the fact that these are my class notes, they should be accessible to anyone wanting to
learn Calculus III or needing a refresher in som
Homework Problems for Math 291, Fall 2012, Due Mon., Nov 26
Eric A. Carlen1 Rutgers University November 18, 2012
1 3. Do problems 6.4, 6.6, and 6.8 from the text. 4. Let f (x, y) = x3 + y 3 - 3xy. (a) Find all critical points of f , and then compute the H
MATH 291 PROBLEM SOLVING This is a comprehensive guide on how to answer every type of question you'll face in MATH 291. It is divided into chapters, and there are NO proofs. Chapter 1 Given: Three vectors in R^2 (vectors a, b, c) Find: Express each of the
Math 291
Workshop 2
Please work in groups of two or three.
Problem 2.1. A few warm-up problems: Let i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1).
These are the unit coordinate vectors.
1. Compute all of the pairwise dot products: i i, i j, and so on. T
Math 291
Workshop 3
Please work in groups of two or three.
Problem 3.1 (Warm-up). Let H be the set of points (x, y, z ) in R3 satisfying the
equation x2 + y 2 = 1 + z 2 . Sketch at least ve representative z traces (ie cross sections
of H by planes of the
Math 291
Workshop 4
Please work in groups of two or three. Please explain all answers carefully.
Suppose p(t) is a space curve. The following formulas may be useful (all derivatives
are taken with respect to t):
T=
=
|T |
,
|p |
p
,
|p |
N=
T
,
|T |
B
= N
Math 291
Workshop 5
Please work in groups of two or three. Please explain all answers carefully.
Problem 5.1 (Warm-up). For each of the following functions, plot enough of the
gradient vectors to get a feel for the gradient vector eld: the function
(x, y
Math 291
Workshop 6
Please work in groups of two or three. Please explain all answers carefully.
Be sure to acknowledge any and all help you receive with your write-up. As usual,
you can think with other people but you should not write with anybody. Pleas
Math 291
Workshop 7
The point of todays workshop is to become familiar with computing with matrices
and the total derivative. Well discuss the geometric meaning of these objects next class.
Problem 7.1 (Matrices). An nm matrix is a rectangular array, with
Math 291
Workshop 8
Todays workshop focuses on computing integrals using the change of variables formula. Remember the determinant of the total derivative (the Jacobian) measures the
local change in length (or area, or volume) and so it shows up in the co
Math 291
Workshop 9
Todays workshop is designed to help you practice computations with the derivatives
involved in Greens Theorem and Stokes Theorem.
We will use for the del operator, f for scalar functions, and F for vector elds.
(Be sure to pronounce th
Math 291
Workshop 10
Many exercises on vector elds.
Problem 10.1. (Found on a exam review of Prof. Greenelds.) Evaluate the line
integral C F dr where F(x, y ) = x2 y 3 , y x and r(t) = (t2 , t3 ) for 0 t 1.
Problem 10.2 (Review). Show that the vector eld
Math 291
Workshop 11
Please write in complete English sentences. Helpful gures are always welcome.
Stokes Theorem states that:
=
D
d.
D
It follows that there are two trivial ways for an integral to vanish. That is, if asked
to integrate the left hand side