MATH2400
Mathematical Analysis
Juan Carlos Ponce Campuzano
I acknowledge the Traditional Owners and their custodianship of the lands on which we meet today and pay my respect to their Ancestors and their descendants.
Acknowledgement
of Country
Image: Digital reproduction of A guidance through timeΒ by Casey Coolwell and Kyra Mancktelow
What is Mathematical Analysis?
\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
\displaystyle \int_a^b f(x) dx=\lim_{n\to \infty} \sum_{n=0}^{n} f(x_i)\Delta x_i
\displaystyle f'(a) = \lim_{x\to a} \frac{f(x)- f(a)}{x-a}
What is Mathematical Analysis?
Essentially, it is the study of limts (and related theories) using the axiomatic properties of real numbers.
\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
\displaystyle \int_a^b f(x) dx=\lim_{n\to \infty} \sum_{n=0}^{n} f(x_i)\Delta x_i
\displaystyle f'(a) = \lim_{x\to a} \frac{f(x)- f(a)}{x-a}
This course is proof oriented instead of an application oriented course
What are Axioms?
Axioms
Definitions
All Mathematical Theories
Non-Euclidian Geometry
Group Theory
Algebra
Linear Algebra
Game Theory
Topology
Category Theory
\vdots
Euclidian Geometry
Calculus
Number Theory
What are Axioms?
Euclidian Geometry
Euclid's Elements (~2300 years old)
Definitions
Point
Straight line
Circle
Right angle (\(90^\circ\))
What are Axioms?
Euclidian Geometry
Euclid's Elements (~2300 years old)
Axioms
- It is possible to draw a straight line from any point to any other point.
- It is possible to extend a line segment continuosly in both directions.
- It is possible to describe a circle with any center and any radius.
- It is true that all righ angle are equal to one another.
- Two distinct intersecting lines cannot be parallel to the same line.
What are Axioms?
Euclidian Geometry
Euclid's Elements (~2300 years old)
Axioms
- It is possible to draw a straight line from any point to any other point.
- It is possible to extend a line segment continuosly in both directions.
- It is possible to describe a circle with any center and any radius.
- It is true that all righ angle are equal to one another.
- Two distinct intersecting lines cannot be parallel to the same line.
Proposition 47, Book I
What are Axioms?
Euclidian Geometry
Euclid's Elements (~2300 years old)
Axioms
- It is possible to draw a straight line from any point to any other point.
- It is possible to extend a line segment continuosly in both directions.
- It is possible to describe a circle with any center and any radius.
- It is true that all righ angle are equal to one another.
- Two distinct intersecting lines cannot be parallel to the same line.
Elisha S. Loomis (1940)
Mathematical Analysis
We are going to prove statements like:
0\cdot x = 0
\displaystyle \lim_{n\to\infty} \int_0^1 \frac{nx+ \sin(nx^2)}{n} \,dx = 0
Mathematical Analysis
Traditionall, a rigorous first corse in Analysis progresses (more or less) in the follwoing order:
In the other hand, the historical development of these subjects ocurred in reverse order:
sets,
real numbers
limits,
continuous functionns
derivatives
integration
Cantor 1874,
Dedekin
Cauchy 1823,
Riemann,
Weiestrass
Newton 1669
Leibniz 1684
Archimedes
3rd B.C.
Archimedes
c. 287 BC - c. 212 BC
0
1600
Liu Hui
c. 220 - c. 280
200
Hindu-Arabic numerical system
Algebra: Muhammad ibn Musa al-Khwarizmi
Analytic Geometry:
Descartes & Fermat
-200
Time line
Archimedes
c. 287 BC - c. 212 BC
1669
Newton
1684
Leibniz
1872
Dedekin
1874
Cantor
1854
Riemann
1755
Euler
...
0
1600
1900
1700
1800
2000
Cauchy
1823
Liu Hui
c. 220 - c. 280
1748
Agnesi
1861
Weiestrass
Time line
~300 years of mathematical development
(roughly)
Mathematical Analysis
Mathematical Analysis
13 Weeks
Good news!
You already know most of the content.
Lectures
Tutorials
How are we going to work in this course?
How are we going to work in this course?
- Every week you will practice with a set of problems.
It covers core material not covered in lectures!
- Internal students: on Campus
- External students: via Zoom π₯οΈΒ
Lectures
- We will cover the theory with examples.
Most weeks there will be at most
one complementary reading π π
Important:
How are we going to work in this course?
Available in Blackboard under Learning Resources
Most weeks there will be at most
one complementary reading π π
Important:
via weekly modules:
π
Why should I read the complementary material? π€
- Reading is, in general, a very useful cognitive exercise.
- In mathematical analysis, it can help you gain a deeper understanding of the concepts and theorems being presented, and develop your ability to think logically and rigorously.
- You are studing maths!
Most weeks there will be at most
one complementary reading π π
Important:
How are we going to work in this course?
Ingrid Deubechies
Maryam Mirzakhaniβ
You will gain some experience to read maths papers:
How are we going to work in this course?
Tutorials
Consult your timetable.
How are we going to work in this course?
π Tutorials begin in week 2.
Ask questions about assignments, content and
problem sheets (available in Blackboard).
π You should attempt problems beforehand.
- Internal students: on Campus
- External students: via Zoom π₯οΈΒ
How are we going to work in this course?
Link in Blackboard
- Post questions
- Try answering questions
- 3 Assignments - 36%
- Final exam - 64%
MATH2400 & MATH7400
Assessment
Assessment
Sumbission will be online through Blackboard
Under Assessment look for the folder corresponding
to your course code MATH2400 or MATH7400
Check the Submission Instructions βΌοΈ
Assignments
Assessment
Final Exam
Invigilated on-campus & over Zoom
Β for internal and external students, respectively
You will have two hours to
solve problems and formulate proofs
Final Exam
Assessment
In which you may encounter some beautiful expressions such as:
\displaystyle\lim_{\substack{h \to 0\\h\in \R^n}}
\frac{||f(x+h)-f(x) - Df(x)h||}{||h||} = 0
\displaystyle {\biggl( \sum_{k=1}^n x_k y_k \biggr)}^2
\leq
\biggl(\sum_{k=1}^n x_k^2 \biggr)
\biggl(\sum_{k=1}^n y_k^2 \biggr)
\displaystyle (f^{-1})'(y_0) = \frac{1}{f'\bigl( f^{-1}(y_0) \bigr)} = \frac{1}{f'(x_0)}
\displaystyle f'(p) \, e_j =
\sum_{k=1}^m
\frac{\partial f_k}{\partial x_j}(p) \,e_k
\displaystyle \limsup_{n \to \infty} \, x_n = \inf \{ a_n : n \in \N \}
\displaystyle \liminf_{n \to \infty} \, x_n \leq
\limsup_{n \to \infty} \, x_n
\displaystyle \lim_{n\to\infty} \int_0^1 \frac{nx+ \sin(nx^2)}{n} \,dx = 0
Final Exam
But before, we need to practice a lot π βοΈ
\displaystyle\lim_{\substack{h \to 0\\h\in \R^n}}
\frac{||f(x+h)-f(x) - Df(x)h||}{||h||} = 0
\displaystyle {\biggl( \sum_{k=1}^n x_k y_k \biggr)}^2
\leq
\biggl(\sum_{k=1}^n x_k^2 \biggr)
\biggl(\sum_{k=1}^n y_k^2 \biggr)
\displaystyle (f^{-1})'(y_0) = \frac{1}{f'\bigl( f^{-1}(y_0) \bigr)} = \frac{1}{f'(x_0)}
\displaystyle f'(p) \, e_j =
\sum_{k=1}^m
\frac{\partial f_k}{\partial x_j}(p) \,e_k
\displaystyle \limsup_{n \to \infty} \, x_n = \inf \{ a_n : n \in \N \}
\displaystyle \liminf_{n \to \infty} \, x_n \leq
\limsup_{n \to \infty} \, x_n
\displaystyle \lim_{n\to\infty} \int_0^1 \frac{nx+ \sin(nx^2)}{n} \,dx = 0
π
More Questions?
- For general inquiries: smp.student@uq.edu.au
- Questions about Tutorial sheets: In your Tutorial
- email: j.ponce@uq.edu.au
- Consultation hours on Mondays from week 2:
- In person: 1 pm, 67-741 Priestley Bld.
- On Zoom: 2 pm (link in BB).
MATH2400 overview
By Juan Carlos Ponce Campuzano
