University of Chicago
A Textbook for Advanced Calculus
John Boller and Paul J. Sally, Jr.
2
Chapter 0
Number Systems and Cardinality
Dans la prsente Note, on va essayer de prciser une
e
e
terminologie
Math 205
Spring 2014
Notes 4/23/2014
As said before, we would like to do calculus on curves (for example, integrate function on curves). As a
rst step, we want to dene the length of a curve. We will d
Math 205
Spring 2014
Homework Assignment 3
due Friday, May 2
Each problem is worth 10 points.
1. Exercise 5.10.6 in Boller-Sally. You may use the change of variables as stated in Boller-Sally
(where n
Math 205
Spring 2014
Homework Assignment 4
due Friday, May 9
Each problem is worth 10 points.
1.a. (5 points) Let 1 : [a, b] Rn and 2 : [c, d] Rn be two simple closed piecewise smooth
parameterized cu
Math 205
Spring 2014
Homework Assignment 6
due Friday, May 30
Each problem is worth 10 points.
1. For each k N, let Rk = 0, k [0, 1] and let k : Rk R3 be dened as k (x, y) =
2
(x, y, sin x). For each
Math 205
Spring 2014
Homework Solutions 6
1.a. Note that the 2-form dx dy measures the area of a parallelogram in R3 projected to the
xy-plane. Thus, we are looking at the area of this graph projected
Math 205
Spring 2014
Homework Solutions 4
1.a. As f = f , we may suppose that both 1 and 2 are clockwise parameterized. If
1 (a) = 2 (c), then there is nothing to prove as both 1 and 2 have the same s
Math 205
Spring 2014
Homework Solutions 3
1. We have that
cos 1 sin 2 r sin 1 sin 2 r cos 1 cos 2
D(r, 1 , 2 ) = sin 1 sin 2 r cos 1 sin 2 r sin 1 cos 2 .
cos 2
0
r sin 2
Using the formula for the det
Math 205
Spring 2014
Homework Solutions 1
1. Let R = [a1 , b1 ] . [an , bn ]. Each Ri gives rise to 2n hyperplanes of dimension n 1 as
determined by the faces of Ri . Note that each of these hyperplan
Math 205
Spring 2014
Homework Solutions 2
1. We will rst assume that R1 R2 . We will assume that R2 (the general case follows
easily from the two-dimensional case). Thus, R1 = [a, b] [c, d] and R2 = [
Math 205
Spring 2014
Homework Solutions 5
1. We have that
D1 ( )(x) = D(x)D1 (x)
We have that the surface area of : R1 Rn is
D1 ( )(x)
2
D2 ( )(x)
2
(D1 ( ) D2 ( )2
R1
2
D(x)D1 (x)
=
D(x)D2 (x)
2
(D
Important Info
Math 205, Section 53
Spring 2014
Sean Li
Oce: Ryerson 360J
E-mail: [email protected]
Oce hours: TBD
Text: A Textbook for Advanced Calculus, Boller and Sally, preprint.
General po
me Mela/u
WE. JMyIaou on 4?
a me
MI. ymym'ou cu Ru
5
2
i/"\/-\4Il
gt 2:1) I!
J :-._._-.1cfw_J4
6
x
I v R
LIIIIIII unmqnmtuun
Ruff. (A6631 Mm
6?foch = [91713, it
p
I
) )
% ._. vow mam
f
cfw_nwdl'm
)wa xmdanemfd 73mm f Mail.
cfw_990 7? .Lb [WWW [337-9 ag/x/(tplt a fray.
%, (9am) _
a? f: [M] MP be magma: an [5,67, $0 PM .
Moe Fm =/a";mar , dbxefqj. E x
.7720 72 4/5 MW cm [cw] (a :0m x a
mu F 4,
Math 205
Spring 2014
Homework Assignment 5
due Friday, May 23
Each problem is worth 10 points.
1. Let R1 and R2 be two compact rectangles in R2 and : R1 Rn and : R2 Rn . Let
: R1 R2 be a bijective ma
Math 205
Spring 2014
Homework Assignment 2
due Friday, April 18
Each problem is worth 10 points.
1. Let be a bounded domain and f : R be a bounded function. Suppose Df has
measure zero. Let R1 , R2 be
Math 205
Spring 2014
Homework Assignment 1
due Friday, April 11
Each problem is worth 10 points.
1. Let R Rn be a generalized rectangle and cfw_R1 , ., Rm be any nite collection of generalized
subrec
Math 205
Spring 2014
Notes 4/17/2014
Theorem 1. Let R Rn be a rectangle and U Rn be an open set containing R. If : U Rn is C 1 ,
1-1 on R, and D(x) is invertible for x R. then a bounded real-valued fu
Math 205
Spring 2014
Notes 4/23/2014
Theorem 1. Let : [a, b] Rn and : [c, d] Rn be simple, non-closed, piecewise smooth parameterized
curves with C = C . Then is equivalent to either or .
Proof. We ma
Math 205
Spring 2014
Notes 4/7/2014
Theorem 1 (Change of variables). Let V Rn be a bounded open set and U Rn be an open set such that
V U . Let : U Rn be C 1 on U and 1-1 on V , with D(x) invertible o
Math 205
Spring 2014
Notes 4/23/2014
Let (t) = (cos t, sin t) dened on [0, 2]. Then is the smooth closed parameterized curve of a circle
of radius one. One should expect before any computations that (
Math 205
Spring 2014
Notes 4/23/2014
We can now begin dening the line integral of vector valued functions, which is an orientation sensitive
line integral. We have the following setup. We let : [a, b]
Math 205
Spring 2014
Notes 4/23/2014
We now discuss Greens theorem in the plane, which relates line integrals to volume integrals. We will
rst prove the following case, which is a special case of a sp
Math 205
Spring 2014
Notes 4/23/2014
Theorem 1. Let U R2 be an open convex subset and let = P (x, y) dx + Q(x, y) dy be a C 1 1-form.
Then is exact if and only if P = Q .
y
x
Proof. We rst assume that
Math 205
Spring 2014
Notes 4/23/2014
We will now take what we know about parameterized curves and line integrals and develop a theory of
parameterized surfaces and surface integrals. We will be talkin
Math 205
Spring 2014
Notes 4/23/2014
a
Remember that when we dened integrals of real valued functions on intervals of R, we dened b f (t) dt =
b
a f (t) dt. This is because function f : [a, b] R can
Math 205
Spring 2014
Notes 4/23/2014
To generalize this to surfaces in Rn , we need to know how to compute the surface area of a parallelogram
given by two vectors u and v. Let be the angle between u
Math 205
Spring 2014
Notes 4/23/2014
I want to revise the notion of equivalence for parameterized surfaces, but rst we need to dene determinants for 2 2 matrices. Recall that we had dened | det T | as
Math 205
Spring 2014
Notes 4/23/2014
Here are some important properties of pullbacks.
Proposition 1. Let f : Rm be as above.
If , are k-forms on Rm and g, h : Rm R are functions (i.e. 0-forms), then
Denition 6.1.22 Let $1 : [the] 9 id." and e; :cfw_ng,bg] -+ R" be piecewieeamneth parametrizedenrvee
such that npng = tilli. The canentenatr'an til 2 1+h ale); and 955 is the piecewise smnnth paremetr
Denition $3.1 Let d : [aI b] i JR he a piecewise smooth parametrized curve. let U r; R" be an open
set containing Cm and let F : U r E. be a continuous function which is written in terms of its coordi