VECTORS NOTES
Vectors can be introduced as displacements in space, called position vectors. To describe a position vector,
5
we need to specify its direction and its length or magnitude (to say how far we go in the given direction).
This is a geometric de
MATH 116: HOMEWORK 8
DUE BY 5PM ON WEDNESDAY, NOVEMBER 9
1) From Apostols chapter 12, sect. 12.11, p.460-462, #1,5,12, 17(a,b),
18, 20 (In #20, we say that the n-space together with this distance
function is a metric space.)
2) From Apostols chapter 12, s
MATH 116: HOMEWORK 9
DUE BY 5PM ON FRIDAY, NOVEMBER 18
1) From Apostols chapter 13, sect. 13.5 p.477 #2,3,8,9.
2) From Apostols chapter 13, sect. 13.8 p.482-3 #2,12.
3) From Apostols chapter 13, sect. 13.11, p.487-8 # 1(a,e),2(b),3(c),8(a).
4) From Aposto
MATH 116: HOMEWORK 11
DUE BY 5PM ON WEDNESDAY, DECEMBER 7
1) From Apostols chapter 15, sect. 15.12, p.566-8 # 1(b,e),4,8,13(a,b,c,d).
2) From Apostols chapter 15, sect. 15.16 p.576 #1(a),2(b),4.
3) From Apostols chapter 16, sect. 16.4, p.582-3 # 1,4,7,25.
Math 114: Final Exam
May 4, 2012
Instructions:
1. Please sign your name and indicate the name of your instructor and your teaching
assistant:
A. Your Name:
B. Your Instructor:
C. Your Teaching Assistant:
2. This exam is 2 hours long and there are 16 quest
Chris X: Why Do Americans Spend So Much Money At Christmas?
Americans spend an exorbitant amount of money, time, and effort during the major gift
purchasing season between Thanksgiving and Christmas. This paper explores American gift
giving through four d
52
THEFREEMAN
IDEAS ON Ll0ERTV
The Constraints
on Helping
There are laws
that explain
why social
programs not
only do not but
cannot produce
the intended
effects.
by Charles Murray
.
Charles Murray is a senior
research fellow at the Manhattan
Institute fo
Full Bibliography
Bourdieu, Pierre. 1992. The Logic of Practice. Stanford: Stanford University Press
Davis, Kingsley, and Wilbert E. Moore. 1945. "Some Principles of Stratification." American Sociological Review 10: 242-49.
Duncan, Cynthia. 2015. Worlds A
POVERTY IN AMERICA: A HANDBOOK
Introduction
Poverty will always be with This is not a new idea. From the Gospel of John to today
many have despaired that poverty is an enduring feature of society, even as they search
for ways to alleviate it. But is this
MATH 116: HOMEWORK 7
DUE BY 5PM ON WEDNESDAY, NOVEMBER 2
1) Let f be a function of class C 1 on all of R. Recall this means the
function is dierentiable, and its derivative is continuous on R. Let
[a, b] be given, then there exists a constant C > 0 such t
MATH 116: HOMEWORK 6
DUE BY 5PM ON WEDNESDAY, OCTOBER 26
1) From Apostols chapter 3, sect. 3.20, p.155 # 6,7,8.
2) From Apostols chapter 4, sect. 4.6, p.167-168 # 12,38.
3) From Apostols chapter 4, sect. 4.9, p.173 # 1,8,11.
4) From Apostols chapter 4, se
MATH 116: HOMEWORK 5
DUE BY 5PM ON WEDNESDAY, OCTOBER 19
1) From Apostols chapter 3, sect. 3.6, p.138-9 # 1,3,5,11,27,28,32,33
2) From Apostols chapter 3, sect. 3.11, p.145 # 1, 2(a),3, 5
3) For each of the following functions f dened on [0, 1], determine
Vector functions of one or more variables
(See Thomas 13.1)
In many physical contexts one is interested in vectors that vary with position or time. For example, the
position of a point can be described by a vector r. Thus, if we consider a moving particle
Vector Fields
(See Thomas 16.2)
Henceforward we shall be concerned mostly with vectors (and scalars) which depend on position in
threedimensional
space, i.e. which are functions of three variables x, y, z. Sometimes they may depend also on a
fourth variab
1.2 Trigonometric functions
1.2.1 Values
(See Thomas 1.6)
We can quickly estimate the value of a trigonometric function for any argument in [0,2 ] by remembering a
few things. First we have the table
0 30 =
6 radians 45 =
4 rad. 60 =
3 rad. 90 =
2 rad
The Curl Operator
(See Thomas 16.7)
The curl of a vector field F is defined to be
F =
_
Fz
y
Fy
z
_
i+
_
Fx
z
Fz
x
_
j+
_
Fy
x
Fx
y
_
k.
Note that curlF is another vector field.
We can write F as curlF again the two notations are interchangeable
Scalar and vector products
We have defined vector addition and subtraction, but not multiplication of vectors. This is more complicated
because to obtain another vector we need to define both a magnitude and a direction (and in general division
cannot be
Ln, or loge, exp, and hyperbolic functions
(See Thomas section 7.2)
The natural logarithm ln x can be defined as
lnx =
Zx
1
dt
t
.
This implies ln1 = 0. Note that this is not a good definition if x < 0, but it is easy to show R that for negative u,
du/u =
Gradients and directional derivatives
In Calculus II you met functions of more than one variable. Those that were discussed there were scalar
functions, i.e. their values at a particular point were numbers. Such a scalar quantity (magnitude but no
directi
Double and triple integrals
(See Thomas 15.1 and 15.4)
First let us revise the idea of 2-D integration.
Example 1.7. Integrate the function f (x,y) = x2y2 over the triangular area R: 0 x 1, 0 y x.
We can write this integral as Z
R
f (x,y)dA,
where dA is a
1.1
Curves and surfaces
We shall need a number of geometrical shapes for use in examples, so we need the equations for them. The
main ones are so-called conic sections in two dimensions, and related three-dimensional surfaces. First we
discuss curves in t
MATH 116: HOMEWORK 1
DUE BY 5PM ON WEDNESDAY, SEPTEMBER 14
1) From Apostols introduction, sect. I 2.5, p. 16 # 13, 15, 19.
In #19, if B is viewed as the universal set, then B A is the complement
of A in B, often denoted Ac or A. Then the two statements in
MATH 116: HOMEWORK 2
DUE BY 5PM ON WEDNESDAY, SEPTEMBER 21
1) From Apostols introduction, sect. 3.3, p. 19 # 2,4,8
2) From Apostols introduction, sect. 3.5, p. 21 # 2,9
3) From Apostols introduction, sect. 3.12, p. 28 # 2,3,12.
For #12 remember that to sh
MATH 116: HOMEWORK 3
DUE BY 5PM ON WEDNESDAY, SEPTEMBER 28
1) Prove no rational number is a solution to x2 = 3.
2) Find all complex numbers that are solutions to z 8 = 16, and write
all of them in the form a + ib.
3) From Apostols chapter 9, sect. 9.6, p.
MATH 116: HOMEWORK 4
DUE BY 5PM ON WEDNESDAY, OCTOBER 5
1) From Apostols chapter 1, sect. 1.26, pp.83-84 # 22 (a), 23, 25.
Verify the conclusion in #25 explicitly for x4 and x5 .
2) Using just the denition of limit, prove that limx0 (7x + 4) = 4.
3) Dene