Introduction to Analysis II¶

Notes for Introduction to Analysis II. The screenshots are from the lecture notes of Prof. MC Li.

Introduction to Analysis II

week 1-1 (3/3/2020)¶

Reimann Intergral¶

refinement = add points to partition
definition of Reimann sum (\(U_f(P), L_f(P), R_f(P)\))
- show \(L_f(P) \le L_f(Q)\) and \(U_f(P) \ge U_f(Q)\)
- hint : \(Q = P \cup (Q-P)\)
Reimann intergral
Reimann intergrability :::info Proof of Prop. 31.3

31.3.1 since f is \(bounded\), by the \(completeness\) of \(R\), \(m_i \le f(x_i) \le M_i\) are all well defined.

31.3.2

:::

Week 1-2¶

Reimann Intergrability¶

http://home.iitk.ac.in/~psraj/mth101/lecture_notes/lecture15-16.pdf
inf and suf and reimann intergral definition

Note: proof of equivalence part use the property of inf and sup

boundedness and monotonicity => integrability¶

P = \(\{a=x_0<x_1<...<x_n=b\}\) where \(x_i = a+i\frac{b-a}{n}\)
\(U_f(P) - L_f(P) = \frac{b-a}{n}[f(b)-f(a)]\)
Q.E.D
end of class

Week 1-3(3/5/2020)¶

part 1 (pdf 32)¶

Theorem 32.4 : continuity on a closed interval => integrability¶

closed and bounded on \(R\) \(\implies\) compact
teacher hint : continuity and compact relation
review: compact - f(cont.) -> compact
- not used here
Theorem 11.3 says f is continuous on a compact space then it is uniformly continuous
\(f\) is \(uniformly\ continuous\) on \([a, b]\)
the left is trivial
let P be a partition with \(|x_{i+1}-x_i|\lt\delta\)
then \(M_i-m_i \le \epsilon\ \forall i\) by uniform continuouity
(actually) \(\forall \epsilon\ \exists \delta\)
then \(U_f(P) - L_f(P) = \sum (M_i-m_i)*(x_i-x_{i-1})\)
\(\le \epsilon \sum (x_i-x_{i-1})\)
\(= \epsilon (b-a)\)
then for more formal proof replace \(\epsilon\) with \(\frac{\epsilon}{3*(b-a)}\)
\(Q.E.D.\)

Theorem 32.5 (the Intergrability Theorem)¶

\(If\ f\ :\ [a, b]\rightarrow\mathbb{R}\ be\ continous\ at\ all\ except\ finite\ many\ points\ in\ [a,b]\)
\(then\ f\ is\ Reimann\ integrable\ on\ [a,b]\)
proof:
key idea : remove not continous points
let M, m be global \(suf\) \(inf\)
reduce to \(((M-m)+(b-a))*\epsilon\)

To review, when will \(Min, Max\) exist
- compact in \(\mathbb{R}\), closed and bounded
- idea : proof \(I\) bounded + \(sup/inf \in I\)?

Definition 32.6 (content zero and measure zero)¶

see pdf definition
show \(\{\frac{1}{n}\mid n\in\mathbb{N}\}\) is content zero
- only finite elements out of \(epsilon\) range
show Q is measure zero
- show that length of \({\bigcup^{\infty}_{0}}I_i\) is finite(by \(\epsilon(1+\frac{1}{2}+\frac{1}{4}+...)\) trick

Theorem 32.7 (contious except content zero implies integrability)¶

similar method with 32.5
sketch of proof
try to remove the points like we did in Theorem 32.5
use definition of content zero
show \([a,b]-{\bigcup^{k}_{0}}I_i\) is compact by removing boundary of \(I_i\)

Theorem 32.8 (the Riemann-Lebesgue theorem in one variable: Lebesgue’s criterion for Riemann-integrability)¶

mentioned, will cover if have time
Theorem 32.9 : my guess by substraction and 32.5
Theorem 33.1 (the fundamental theorem of calculus)

Week 2-1(03/10/2020)¶

Theorem 33.1 (the fundamental theorem of calculus)¶

part1
proof sketch:
- Riemann integrable \(\implies\) bounded on \([a,b]\)
- \(\exists M > 0 \ni |f(t)| \le M \forall t \in (a,b)\)
- then uniform continuity on \([a, b]\) is trivial
  - \(\forall x,y \in [a,b] \ni |x-y|<\delta\)
  - \(|\int_{a}^{x}f(t)dt - \int_{a}^{y}f(t)dt| \le M*\delta\)
  - \(\epsilon - \delta\) trick, M is constant
- def of f continuous
- \(for\ x \in [a,b] \ni x > x_0\)
- express \(f(x_0)\) as \(f(x_0)\frac{\int_{x0}^{x}1dt}{x-x_0} = \frac{\int_{x0}^{x}f(x_0)dt}{x-x_0}\)
- we have \(\frac{\int_{a}^{x}f(t)dt - \int_{a}^{x0}f(t)dt}{x-x_0} - f(x_0) = \frac{\int_{a}^{x}f(t)dt - \int_{a}^{x0}f(t)dt}{x-x_0} - \frac{\int_{x0}^{x}f(x_0)dt}{x-x_0}\)
- thus if \(x \in [a,b] \ni x_0<x<x_0+\delta\)
- then \(|\frac{\int_{a}^{x}f(t)dt - \int_{a}^{x0}f(t)dt}{x-x_0} - \frac{\int_{x0}^{x}f(x_0)dt}{x-x_0}| = |\frac{\int_{x_0}^{x}f(t)-f(x_0)dt}{x-x_0}| \le \frac{\int_{x_0}^{x}|f(t)-f(x_0)|dt}{|x-x_0|}\)
- since f is continuous
- \(\frac{\int_{x_0}^{x}|f(t)-f(x_0)|dt}{|x-x_0|} < \frac{\int_{x_0}^{x} \epsilon dt}{x-x_0} = \epsilon\)
- \(x < x_0\) case can be done similarly
- above proves \(\(\frac{d}{dt}(\int_{a}^{x}f(t)dt)\mid_{x=x_0} = f(x_0)\ \forall x_0\ at\ which\ f\ is\ continous\)\)
part2 : \(\int_{a}^{b}f(t)dt = F(b)-F(a)\)
proof sketch:
- by def \(\forall \epsilon \exists\ partition\ P \ni U_f(P)-L_f(P) < \epsilon\)
- make a refinement Q of P to exlude finite indifferentiable points
- apply MVT(mean value theorem) to intervals
  - \(F(b)-F(a) = \sum F(x_i)-F(x_{i-1})\)
  - = \(\sum F^{'}(t_i)(x_i-x_{i-1})\)
  - = \(\sum f(t_i)(x_i-x_{i-1})\)
- \(L_f(Q) \le F(b)-F(a) \le U_f(Q)\)
- \(L_f(P) \le \int_{a}^{b}f(t)dt \le U_f(P)\)
- then \(|\int_{a}^{b}f(t)dt-(F(b)-F(a))| < \epsilon\)

Week 2-2¶

pass

the Generalized Mean Value Theorem for Integrals¶

Integration by parts¶

advanced practice problem¶

Week 2-3(2020/3/12)¶

Several variables¶

rectangle
- def is similar with 2d rectangle partition
for general bounded region
- for such S, use rectangle "disk" \(D\supseteq S\) + characteristic function \(\chi_S(x, y)\)
- def characteristic function \(\chi_S(x, y) : \mathbb{R}^2 \rightarrow \{0,1\}\)
- then can define \(\iint_SfdA = \iint_D(f_{\chi _S})dA\)
fact : \(S1 \cup S2\) is Reimann intergrable doesnt \(\implies\) \(S1\ and\ S2\) are
- consider s1 + s2 = 0

Jordan Measurable sets and Riemann Intergrability¶

content zero and Jordan measurable sets¶

content zero¶

bounded and \(\subset \mathbb{R}^k\)
if for any \(\epsilon\) > 0 there exists a finite collection of k-dimensional rectangular boxes \(D_1, D_2, ..., D_n \ni Z \subseteq \cup_{i=1}^{n}D_i\) and \(\sum volume(D_i)<\epsilon\)

! Jordan measurable¶

A bounded set \(S \subset \mathbb{R}\)
\(\mathbb{\delta} S\) is content zero

Case study¶

consider \(\mathbb{Q}\) / Cantor Set / Fat Cantor Set / set bounded by Sierpinski Curve all of them are not Jordan measurable

continuity except content zero implies Riemann integrability on a rectangle¶

sufficient conditions for content zero¶

set relationship
- subset => ok
- finite union => ok
- countable union => content X, measure V
let \(f : (a,b)\rightarrow \mathbb{R}^2\ be\ C^1\ function\)
\(f([c,d])\) is content zero when \([c,d] \in (a,b)\) ::: info \(C^1 \implies M = max\{|f'(x)| \mid x \in [c,d]\}\) exists let \(\epsilon>0\), Take \(n \in \mathbb{N} \ni \frac{?}{n} \lt \epsilon\) partition [c, d] to n equal-length parts let \(f = (f_1,f_2)\), by MVT to \(f_1\) and \(f_2\) \(|f_k(x)-f_k(x_i)| \le M*(x_{i}-x) \forall x \in [x_{i-1}, x]\) \(|f_k(x)-f_k(x_i)| \le M*\frac{d-c}{n}\) Hence \(f([x_{i-1},x_i])\) is contained in the rectangle \(D_i\) centered at \(\vec{f(x_i)}\) with side length \(\frac{2M(d-c)}{n}\) then volume of cover is \((\frac{2M(d-c)}{n})^2*n \lt \epsilon\) :::
case for \(C^0\) not holds
- Peano Curve : \(C^0(\mathbb{R}, \mathbb{R}^2)\)
- Cantor function restrictedto the Cantor set of measure zero

sufficient conditions for Riemann integrability on a bounded region¶

zero Riemann integral and its corollaries¶

Inter and Outer Areas¶

HW1¶

Reference¶

Intergrability Critiriens

My part - Theorem 32.7 - Continuity except content zero -> intergrability¶

:::info proof: by the definition of content zero proof by content zero intergral = 0 not the cutting technique \(\int_{[a,b]}f = \int_{S}f + \int_{S^{'}}f\) where \(S^{'}\) is the content zero part proof each part of S' is intergrable and sum of them abs <= 0 , hint : use \(max(|M|, |m|)\epsilon\) => \(\int_{[a,b]}f\) is well defined Q.E.D. :::

:::spoiler the picture solution is not that correct should deal with closed interval more carefully and use \(max(|M|, |m|)\epsilon\) to proof eq 0 :::

Week 3-1¶

Practice Class¶

! 35.4 sufficient conditions for Riemann integrability on a bounded region¶

union of 2 content zero is content zero
the discontinuous parts are \(\delta S\) and \(Z\) so \(D\) is still intergrable

35.5 zero Reimann intergral¶

content zero and f bounded => intergrable and intergral = 0

end of class

Week 3-2¶

Inner Area and Outer Area¶

concept : lattice

!Theorem 33.3 norm Partition can bound difference to U, L, R¶

will be useful for future proof
\(\forall \epsilon >0 \exists \delta > 0\) s.t. if P is a partition with \(norm( P ) < \delta\), \(max\{|\int_a^bf(x)dx - R_f(P)|, \int_a^bf(x)dx - U_f(P)|, \int_a^bf(x)dx - L_f(P)|\} < \epsilon\)
proof sketch(U part)
- by theorem 32.1 equivalent condition for Riemann integrability
- => get a partition \(Q\) s.t. \(U-L < \frac{\epsilon}{2}\)
- let the norm-limited partition be \(P\)
- \(\lambda\) for \(P\) can be generated from \(\epsilon\) and \(|Q|\)
- key
  - refer to P's refinement \(P\cup Q\), and the fact that \(|Q|\) is finitely given by \(\epsilon\)
  - see \(P\cup Q\) as \(Q\) injected to \(P\)
- make \(\delta\) bounded by \(\frac{\epsilon}{4*M*|Q|}\)(teacher's version) or \(\frac{\epsilon}{2*(M-m)*|Q|}\)
- \(U_f(P) - U_f(P\cup Q) \le (M-m)\delta*|Q|\)
- \(U_f(P) \le U_f(P\cup Q) + \frac{\epsilon}{2}\)
- \(U_f(P) \le \int_a^bf(x)dx + \frac{\epsilon}{2} + \frac{\epsilon}{2}\)
  - by choice of \(Q\) and \(P\cup Q\) is refinement
- \(U_f(P) \ge \int_a^bf(x)dx - \epsilon\) by definition of intergral
- \(|\int_a^bf(x)dx - U_f(P)| \le \epsilon\)

Week 3-2¶

MVTs¶

Theorem 36.1¶

proof sketch:
- f is continuouson S and S compact
- Extreme Value Theorem => Max and Min Exists
  - i.e. \(\exists p, q \in S \ni f(p) = sup, f(q) = inf\)
  - https://en.wikipedia.org/wiki/Extreme_value_theorem
  - see the generalized version
- \(mg(x) \le f(x)g(x) \le Mg(x)\)
- since both f and g are continuous on S and S is Jordan measurable
- so \(m\int_Sg(x)d^kx \le \int_S f(x)g(x)d^kx \le M\int_Sg(x)d^kx)\)
- if \(\int_Sg(x)d^kx = 0\) always hold, take any \(c \in S\)
- else by corollary 10.4 (IVT for several variables)
- S is connected, f is continous on S
- \(m \le \frac{\int_S f(x)g(x)d^kx}{\int_S g(x)d^kx} \le M\)

Iterated Intergral and Funini's Theorem¶

definition of iterated intergrals¶

Week 4-1¶

IIS meeting => not attended

Week 4-2¶

Change of Valriables¶

Lebesgue's Criterion for integrablility(go back)¶

link

::: success - definition of oscillation of f of an interval/at a point x - \(\omega f(I) = sup\{|f(a)-f(b)|:a,b\in I\}\) - \(\omega _f(x) = inf\{\omega f(B(x, \delta)):\delta>0\}\) - trivial to see \(\omega _f(x)=0\) when f is continuous - exercise :::

::: info (integrable => measure 0 discontinuous points) - let discontinuous set be \(D\) - let \(N(\alpha) = \{x\in[a,b] : \omega f(x) \ge \alpha\}\) - since \(D = \cup_{k=1}^{\infty}N(\frac{1}{k})\) - the left is to proof \(N(\alpha)\) is \(measure\ zero\) - then by integrability for \(\alpha\) exist partition that \(diff \le \alpha\epsilon/2\) - done by definition of \(N(\alpha)\) - can show \(len(N(\alpha)) \lt \epsilon\) (integrable <= measure 0 discontinuous points) - let discontinuous set be D - let \(\epsilon = len(D)\) - let - devide into two parts - E, \(\omega f(J) >= \epsilon\) - still measure zero by subset - compact by closed and bounded - K = [a, b]\E, \(\omega f(J) < \epsilon\) - compact - both part is compact - => has finite subcover - by \(len(E) < \epsilon\) and \(\omega f(K) < \epsilon\)

:::

Week 4-3 (X)¶

Week 5-1 5-2¶

Change of Variables¶

Theorem 39.1¶

- part2 proof - \(\forall\ \bar{x} \in S,\ \exists\ r > 0 \ni B_r(\bar{x}) \subset S\) - \(h_i = (0, ... ,|h_i|, ...0)\) s.t. \(h_i < r\) - for each \(y \in T\), \(f(x, y)\) is \(C^1\), apply MVT - \(\exists t_i \ni f(\bar{x}+h, y) - f(\bar{x},y) = \nabla f(\bar{x}+t_ih,y)h\) - uniform continuity of \(\frac{\partial f(x, y)}{\partial x_i}\) on \(\bar{B_r(\bar{x})} \times T\)

39.2¶

proof, apply chain rule directly, y is i.d. variable, view as constant

39.3 the bounded convergence theorem(wait, graduate level class content)¶

39.4¶

apply 39.3

ch 40, 41 Improper Itergral¶

Type 1 : infinity
Type 2 : undefined
tests:
- comparison test

The course turn online¶

Week 9 - 2020/4/28 TA class¶

proof of 42.1¶

link
- MVT-2 of integral

Patition to 2 part need theorem¶

Week 12(2020/5/20) - Self study for exam 2¶

Ch40. - Inproper Integral I (S is not bounded)¶

Inproper Integral I (S is not bounded)
- comparison test
- 40.2 proof sketch:
  - two part(g>=f and limitied)
- 40.3 use definition of limit
- poly functions
  - by direct calculation
Care:
- condition, >0

Ch 41. - Inproper Integral II (f(S) is not bounded)¶

Inproper Integral II (f(S) is not bounded)
- silmilar to 40
Care:
- different conclusion on \(cx^{-p}\) at x -> 0

Ch 42. - Dirichlet's test / Absolute convergence / Cauchy principle value¶

42.1 Dirichlet's test for improper integral
- bounded and decay to 0
42.2 Define improper integral to Complex number
- absolute convergence for C
  - \(\sqrt{Re^2 + Im^2}\)
42.3 Absolute convergence implies convergence
- counter example for opposite direction
  - sinx/x
42.4 Cauchy principle value (P.V.) definition
- \(+- \infty\)
- \(|f(x->c)| = \infty\) and want \([a,c) \cup (c,b]\)
42.5 how to use
- If improper integral is convergent, P.V. = its value
- It's possible that improper integral is convergent but P.V. is not
42.6
- Let a < 0 < b and f ∶ [a,b] → R be continuous on [a,b] and differentiable at 0,
- then P.V.∫baf(x)/xdx exists

Ch 43. - Improper Integral on \(R^k\)¶

43.1 Limit indeppendent of choice of \(K_n\)
- may converge or diverge(only has case \(+\infty\))
43.2 Definition
- 43.1 shows 43.2 well-defined
43.3 Higher order 1/|x|^p
evaluation of function with no anti-derirative by higher dimension improper intergral + fubini

Ch 44. - Lebesgue measure / Lebesgue measurable (Definition)(Key!)¶

Key: cut in range not domain
Key: Preimage, Tiled
Key: closed/open/general/for Rk
Key: Countable(seq of)
See as a extension of Jordan measurable
Jordan measurable - Reimann Integrability <=> Lebesgue measurable - Lebesgue Integrability
Review of Jorndan Measurable

Lebesgue measurable but not Jordan measurable¶

Review of Jorndan Measurable¶

- Note: Riemann-Lebesgue Theorem

Ch 45. - Lebesgue Integral¶

define on function
- if for all integral I, pre-image(I) is L.M.
continous => f is lebesgue measurable
(similar to ) bounded convergence theorem (\(\forall x, f(x) = lim f_i(x)\))

45.5.2 Comment is important for complete understanding¶

https://drive.google.com/file/d/1-YLPdabFXtcVg3-nxqRzJbtVwzWPYGEN/view
- around 45:00

banach tarski paradox(supplement)¶

search => related to A.C. (Axiom of Choice)

Ch 46. - Riemann-Stieltjes integral¶

Noun: finer(is improvement of)
Key Concept: S(f, g, P)
EQ criterien (similar to R.I.)
cases (Step function, Bounded Variant)
intergrant f , inegrator g, partition P, dummy variable x

Ch 47-48¶

Arc Length / Work¶

C1 curve (vector-valued function)
dr = dg'(t), ds = d|g'(x)|

Line Integral / Work¶

real function \(f : U \supset C \to R~and~C1 curve C\)
- \(U~open\)
- \(C\subset U\subset R^k\)
- \(f:U \to R continous\)
vector function f
field f
- use dot product

differential form (a way to write work??)¶

Q

Rectifiable (48.4)¶

contious, not necissarily injective
define on C1 curve, Reiemann similar sum + sup
proof https://drive.google.com/file/d/1UzJXLamXOVOZJ_bJlGczmPjAc9w7DhLi/view

Review of FTC¶

Ch 49. - Green's theorem(2D proj.)¶

regular region
- is its interior's closure
  - i.e. remove single point
simple closed curve
piecewise smooth boundary with positive orientation(D on the left)

Use Green's to proof corollaries¶

\(e^{i\pi}\) rotation or (-b, a) of g => (-Q, P)

convervative filed¶

g is C2
gradient of g is f
results

proof of greens (Case)¶

https://drive.google.com/file/d/1N1tIY2vqZo4K9mbXG6-wf6WlJKmjP_aP/view?pli=1

Ch 50-51. - Surface Area and Surface Integral¶

Area¶

D->E
cross dg(x,y)dx dg(x,y)dy (D1)
special case of g(x,y) = (x,y,h(x))

Integral¶

integral>
- f: D->E->Real

Vector-valued¶

orientation of E
n
physic meaning : flux

Ch 52. - Vector derivatives¶

gradient
- apply for each dimension
curl
- cross
divergence
- dot
laplace
propositions

Ch 53. Gauss' Divergence Theorem¶

jordan measurable for low to high dimension
remark: divergence can be given a phiysic meaning by small ball and divergence theorem

Ch 54. Stroke's Theorem¶

cosistent positive orientatio
Right Hand Princile

Ch 55-56. Higher Dimension¶

skip

Week 12 - Exam 2¶

Week 15 - for exam 3¶

Week 15 - Series¶

https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch4.pdf

Definition of Convergence¶

definition is always important

Cauchy Criterien¶

a nice eq form

linearity of c. series¶

Absolute convergence(a.c.) and a+, a-¶

introduce a+, a-

Comparison Test / Ratio Test / Root Test (for a.c.)¶

comparison
ratio/root is to compare to geometric seq

Raabe’s test¶

Test Conclusion¶

Conditional convergence c.c.¶

Alternating sequence¶

convergence to 0 if
- decreasing
- alternating
- a_n to 0 b

Dirichlet’s test¶

bi be a bounded above complex sery
ai be a decrease to 0 sery
aibi is c.

Abel’s test¶

bi be a c. sery
ai be a decrease to 0 sery
aibi is c.

Rearrangement¶

definition
- bijection of indices
magic!
Note: Q, the statement that not abs convegent is not formal
- i.e. why \(a^+, a^-\), one diverge another dont => diverge

Rearrangement of a.c. series¶

prove by \(sup(\sigma^{-1}(\{1...n\}))\) exist as m
use this + cauchy

Rearrangement of c.c. series¶

any sum

Double Series!!!¶

define absolute convergence of double series
- by an ordering is a.c.
if a.c.(an ordering is )
- a.c. <=> all ordering is a.c.
- a.c. <=> iterated sum is a.c.
if not a.c. ( a+, a- discussion)
- \(a+ = \infty, a- \lt \infty\)
  - diverge to \(\infty\)
- \(a+ = \infty, a- = \infty\)
  - various ordering can converge to any real number
    - my proof(by expand to 1 d, use + - part to make S(if cannot => c.), and \(\sigma^{-1}\) can find k to dominate 0,0 to N,N)

Product of a.c. series is a.c.¶

Cauchy Product¶

ai, bi c. to A, B
if one of ai, bi is a.c.
- Cauchy product is c. with sum AB
else
- Cauchy product may d.

Week 15 - Sequence of functions¶

pt. uni. uni cauchy.¶

eq def of uni. conv¶

uni + cont. => target cont.¶

lim f uni and all fn cont. => cont.
cont. [] for all delta for above => () f cont.

Weierstrass approximation¶

cont. has poly seq uni. approx
http://www.math.univ-toulouse.fr/~lassere/pdf/2012INPsuitesD1bis.pdf

Dini's¶

for each x , fx is bounded monotone
x to lim f x is well defined and cont.
then lim f is uni approx

Sery of function¶

pt.
abs.
uni.

cauchy for uni conv¶

condition of swapping¶

limit and int/de
- R.I. uni.
- fn pt. C1, fn' uni.
seri
- same

Hint of final¶

one mutant one time proof (1/2)

abel

Arzela-Ascoli theorem, the Heine-Borel theorem for functional spaces

S compact C0 F closed bounded uni equconti

Last update: 2023-09-15

Introduction to Analysis II¶

week 1-1 (3/3/2020)¶

Reimann Intergral¶

partition and refinement¶

Week 1-2¶

Reimann Intergrability¶

refinement¶

boundedness and monotonicity => integrability¶

Week 1-3(3/5/2020)¶

part 1 (pdf 32)¶

Theorem 32.4 : continuity on a closed interval => integrability¶

Theorem 32.5 (the Intergrability Theorem)¶

Definition 32.6 (content zero and measure zero)¶

Theorem 32.7 (contious except content zero implies integrability)¶

Theorem 32.8 (the Riemann-Lebesgue theorem in one variable: Lebesgue’s criterion for Riemann-integrability)¶

Week 2-1(03/10/2020)¶

Theorem 33.1 (the fundamental theorem of calculus)¶

Week 2-2¶

the Generalized Mean Value Theorem for Integrals¶

Integration by parts¶

advanced practice problem¶

Week 2-3(2020/3/12)¶

Several variables¶

Jordan Measurable sets and Riemann Intergrability¶

content zero and Jordan measurable sets¶

content zero¶

! Jordan measurable¶

Case study¶

continuity except content zero implies Riemann integrability on a rectangle¶

sufficient conditions for content zero¶

sufficient conditions for Riemann integrability on a bounded region¶

zero Riemann integral and its corollaries¶

Inter and Outer Areas¶

HW1¶

Reference¶

My part - Theorem 32.7 - Continuity except content zero -> intergrability¶

Week 3-1¶

Practice Class¶

! 35.4 sufficient conditions for Riemann integrability on a bounded region¶

35.5 zero Reimann intergral¶

Week 3-2¶

Inner Area and Outer Area¶

!Theorem 33.3 norm Partition can bound difference to U, L, R¶

Week 3-2¶

MVTs¶

Theorem 36.1¶

Iterated Intergral and Funini's Theorem¶

definition of iterated intergrals¶

Week 4-1¶

Week 4-2¶

Change of Valriables¶

Lebesgue's Criterion for integrablility(go back)¶

Week 4-3 (X)¶

Week 5-1 5-2¶

Change of Variables¶

Theorem 39.1¶

39.2¶

39.3 the bounded convergence theorem(wait, graduate level class content)¶

39.4¶

ch 40, 41 Improper Itergral¶

The course turn online¶

Week 9 - 2020/4/28 TA class¶

proof of 42.1¶

Patition to 2 part need theorem¶

Week 12(2020/5/20) - Self study for exam 2¶

Ch40. - Inproper Integral I (S is not bounded)¶

Ch 41. - Inproper Integral II (f(S) is not bounded)¶

Ch 42. - Dirichlet's test / Absolute convergence / Cauchy principle value¶

Ch 43. - Improper Integral on \(R^k\)¶

Ch 44. - Lebesgue measure / Lebesgue measurable (Definition)(Key!)¶

Lebesgue measurable but not Jordan measurable¶

Review of Jorndan Measurable¶

Ch 45. - Lebesgue Integral¶

45.5.2 Comment is important for complete understanding¶

banach tarski paradox(supplement)¶

Ch 46. - Riemann-Stieltjes integral¶

Ch 47-48¶

Arc Length / Work¶

Line Integral / Work¶

differential form (a way to write work??)¶