Chapter 3
Differential Vector Calculus
3.1
Scalar and Vector Function
In this chapter, we will consider both scalar and vector functions.
A scalar function is a function
f : R3 ! R
x ! f (x) = f (x, y, z)
For example, f (x) = x2 + y 2 + z 2 or f (x) = x c
2
Standard Integrals
The only integrals you can analytically work out are given in Table 2. This is
derived from Table 1, using the Fundamental Theorem of Calculus, namely
Z
dF
F (x) = f (x)dx
= f (x).
dx
As for the standard derivatives, memorise them! Th
Chapter 5
Integral Vector Calculus: Surface
Integrals
See appendice at the end of Chapter 6 for an schematic overview of the integration part
5.1
5.1.1
Surface Integrals
Introduction
Surface integrals are a generalisation of the idea integration over a 2D
1
Vector Calculus
MT2506
Prof Ineke De Moortel
Dr Jean Reinaud
2015/2016
These notes are based on previous versions developed by Drs Jean Reinaud and Aidan Naughton.
2
Contents
1
2
3
Introduction to Vector Calculus
7
1.1
Introduction to Vector Calculus .
Chapter 2
Coordinate Systems
For many physical applications and problems, the optimal coordinate system will depend on your requirements. So far, we have only considered the Cartesian coordinate system. In this Chapter, we will
take a brief look at other
Chapter 1
Introduction to Vector Calculus
1.1
Introduction to Vector Calculus
Vector Calculus is the branch of mathematics that deals with differentiation and integration of vector
valued functions in two or three dimensions.
It has applications whenever
1
Standard Derivatives
Table 1 lists the standard derivatives that you really should memorise. Knowing
these will make the solutions to many differential equations much easier. These
are used in many subsequent modules, right the way up to level 5000 modu
1
('1)
(ii)
(iii)
M zsog
U
HONOURS M. A. AND HONOURS B. So. EXAMINATION
MATHEMATICS AND STATISTICS
Paper MT3808 Dynamical Systems
May 2000
Time allowed : Two hours
Attempt not more than THREE questions
The tent map Ta(a;) (a. 6 IR) is dened by
Purpose
Course
Lecturer
Academic Year
:
:
:
:
Example Solutions of Exam Questions
Dynamical Systems (MT3808/4808)
T. Neukirch
1999/2000 (Semester 2)
Question 1
Solution :
(i)
Fixed points: For x 1/2 we get
x = ax
=
x = 0.
x = a ax
=
x=
For x > 1/2 we get
Purpose
Course
Lecturer
Academic Year
:
:
:
:
Example Solutions of Exam Questions
Dynamical Systems (MT3808/4808)
T. Neukirch
2001/2002 (Semester 2)
Question 1
Solution :
(i)
Fixed points: We get
x = ax x3
x3 + (1 a)x = 0.
=
We conclude that
x=0
is a xed
Purpose
Course
Lecturer
Academic Year
:
:
:
:
Example Solutions of Exam Questions
Dynamical Systems (MT4508)
T. Neukirch
2003/2004 (Semester 2)
Question 1
Solution :
(i)
We have
r = r2
=
r = 0, 1 ,
and
= sin
=
sin = 0
=
= 0, .
for 0 < 2 . Since r = 0 i
MAY 2010 EXAMINATION DIET
SCHOOL OF MATHEMATICS & STATISTICS
MODULE CODE:
MT 4508
MODULE TITLE:
Dynamical Systems
EXAM DURATION:
2 hours
EXAM INSTRUCTIONS Attempt ALL questions.
The number in square brackets shows the
maximum marks obtainable for that que
Example Solutions
Course
Lecturer
Academic Year
:
:
:
Dynamical Systems (MT4508)
T. Neukirch
2009/2010 (Semester 2)
Question 1
Solution :
(a)
We expect two equilibria (one at x = 0), because there are two intersections
between y = x and the graph of the m
HONOURS M. A. AND HONOURS B. Sc. EXAMINATION
MATHEMATICS AND STATISTICS
Paper MT4508 Dynamical Systems
May 2004
Time allowed : Two hours
Attempt ALL questions
Consider the map f : R2 R2 dened in polar coordinates r 0, 0 < 2 by
1.
f (r, ) = (r 2 , sin ).
(