This lecture.
In this lecture:
Inverse functions. Review of the basic concepts.
Calculus of inverse functions:
o Continuity of the inverse.
o Derivative formula for the inverse function.
Inverse trigonometric functions.
Inverse functions
Comment
This f
This lecture.
This lecture we are going to focus on the calculus of the
trigonometric functions.
Trigonometric functions: the definitions?
Recall the main idea of these functions
Trigonometric functions: the definitions?
The precise definition (wont be te
This lecture.
This lecture we are going to focus on the concept of the
derivative of a function.
Recall why we needed limits
Remember the main motivation for introducing limits?
We discovered we need limits because when we tried to write
down formulas for
Andrew Kricker
[email protected]
SPMS building, 4th floor,
office number MAS 04-18.
The textbook
It is:
Calculus
By James Stewart
The international, 7th,
metric edition.
I recommend buying this because:
This course will be based closely on this boo
This lecture.
The definition of and topics related to the important concept
of Continuity.
The definition and first examples.
Important classes of continuous functions:
Polynomials
Rational functions
Trigonometric functions
Exponential functions
U
The precise definition of limit.
Recall the idea of limit
?
Recall the intuitive definition we gave earlier.
Right now well:
Play with this definition a bit to get a feeling for what it says.
Then well obtain the precise mathematical definition by
simpl
This lecture: Part 1.
Recall:
Example.
=0.15
=0.20
=0.25
=0.30
=0.35
=0.40
=0.45
=0.50
=0.55
=0.60
=0.65
=0.70
=0.75
=0.80
=0.85
=0.90
Functions can only have one limit
Now that we have a rigorous definition of limit, we can set about
building up the theo
This lecture:
Horizontal asymptotes: a first example.
Horizontal asymptotes
Horizontal asymptotes
A round of the game:
+
M=2
Lets watch a few rounds
Player A.
Player B.
Lets watch a few rounds
Player A.
Well consider how to go about
demonstrating this a b
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singap
PH3201: Statistical Mechanics I
Problem Set 5. Due: Friday, Apr 15th 2016
Read Chapter 30 plus relevant lecture notes.
1. Blundell 30.5
Show that Bose-Einstein condensation does not occur in two dimensions.
2. Fluctuations! This problem investigates mean
PH3201: Statistical Mechanics I
Problem Set 4. Due: Friday, Apr 1st 2016
Read Chapter 29, 30, plus relevant lecture notes.
1. Blundell 29.3
By rewriting the Fermi-Dirac function
f (E) =
as
1
f (E) =
2
1
e(E)
+1
.
1
1 tanh (E ) ,
2
show that f (E) is symme
NANYANG TECHNOLOGICAL UNIVERSITY
PH3201 PRACTICE MIDTERM EXAMINATION
Total: 100 points. Time allowed: 105 minutes
This exam contains three problems. Do all the work you want graded in the exam books
provided. Write legibly. If I cant read it, it doesnt co
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singap
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singap
NANYANG TECHNOLOGICAL UNIVERSITY
PH3201 MIDTERM EXAMINATION
Friday, Feb 26th, 2015
Total: 100 points. Time allowed: 90 minutes
This exam contains three problems. Do all the work you want graded in the exam book provided.
In order to earn credit for a prob
NANYANG TECHNOLOGICAL UNIVERSITY
SEMESTER 2 PRACTICE EXAMINATION 2014-2015
PH3201 Statistical Mechanics I
Apr/May 2015
Time Allowed: 2 1/2 Hours
INSTRUCTIONS TO CANDIDATES
1. This examination paper contains FOUR (4) questions and comprises FIVE
(5) pages.
PH3201: Statistical Mechanics I
Problem Set 2. Due: Friday, Feb 12th 2016
Please submit assignment to your TA at the start of tutorial.
Read Chapter 19, 20, 21 plus relevant lecture notes.
1. Method of Lagrange multipliers.
Find the minimum value of quadr
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singap
PH3201: Statistical Mechanics I
Problem Set 1. Due: Friday, Jan 29 2016
Read Section 1.4, Chapter 3, Chapter 4 plus relevant lecture notes.
1. (Understanding Stirling.) Give a few proofs of Stirlings formula for n!:
(a) Write log n! =
Pn
1
log n. Approxim
PH3201: Statistical Mechanics I
Problem Set 3. Due: Friday, Mar 18th 2016
Read Chapter 22.1 to 22.6, 23, and 24.
1. Blundell 22.2
The fugacity z is defined as z = e . Using eqn 22.15, show that
z = n3th
for an ideal gas, and comment on the limits z 1 and
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ATTENTION: The Singap
Polyas Enumeration Theorem
F. Oggier
August 10, 2016
These notes have been made using the following references [1, 2, 3, 4].
1
Counting Necklaces
Consider a decorative ornament that consists of n coloured bead, arranged on
a circular loop of strings. This
Tutorial 1
PH3201 - Statistical Mechanics
Semester 2, 2015/2016
1. (Poisson distribution) Blundell 3.3.
2. (3-level system) Consider a system consisting of two particles, each of which can be in any
one of three states of respective energies 0, , and 3. T
PH3201
Tutorial 3 solution
Question 1: Blundell 21.2
A state variable F is extensive if, after multiplying all the extensive variables
(N, V) in the expression for F by a scale factor and leaving all the intensive
variables (P, and T) in F unchanged, the
Tutorial 5
PH3201 - Statistical Mechanics
Semester 2, 2015/2016
Below are 2 problems from midterm in 2015. This tutorial will serve as a midterm review of
sorts.
1. ([35 points] Anisotropic well.) The Hamiltonian for a particle of mass m in an anisotropic
PH3201
Tutorial 1 solution
Question 1: Blundell 3.3.
Poisson distribution with parameter m is given as
P (x) =
em mx
x!
for any x = 0, 1, 2,
a)
X
X
em mx
P (x) =
x=0
x!
x=0
= e
m
X
mx
x=0
m m
= e
= 1
1
e
x!
It is important as sum of all probabilities mus
Tutorial 2
PH3201 - Statistical Mechanics
Semester 2, 2015/2016
1. (Method of Lagrange multipliers.) You have an ellipse (x/a)2 + (y/b)2 = 1. Find the
inscribed rectangle of maximum area. Show that the ratio of the area of the maximum-area
rectangle to th