MA2601 Calculus

The material in this page is for year 2009/10.

Teaching | There
will be 2 hours lectures and 1 hour tutorials per week for 10 weeks
(including revision on week 10). Notes will be provided at some stage after the lectures. Students are advised to attend lectures and take their own notes. Exercise sheets will be posted on-line and given to students during the tutorials. Solutions will be posted on-line later on. |

Assesment |
There
will be 2 courseworks
(contributing 5% each towards the final mark) The exam will take place in May 2010 together with Algebra. There will be 4 Calculus and 4 Algebra questions, each worth 20 points. The maximum mark will be achieved for correct answers to 5 questions, with at least 2 questions answered from each part. |

Past papers | Solutions
to past years papers can be found here. |

Exercise
sheets |
Exercise sheet 1 and Solutions Exercise sheet 2 and Solutions Exercise sheet 3 and Solutions Exercise sheet 4 and Solutions Exercise sheet 5 and Solutions Exercise sheet 6 and Solutions Exercise sheet 7 and Solutions In addition to these exercise sheets, interesting exercises with solutions are available from here. This is the web page for the 2nd year Calculus module that Actuarial Science students take. It is not the same module as the one you take, but exercise sheets 1 to 4 cover partial derivatives, Lagrange multipliers and double integrals. |

Coursework |
Coursework 1 and Solutions Coursework 2 and Solutions |

Module
notes |
THE COMPLETE MODULE NOTES THE REVISION LECTURE |

Some
interesting web-sites |
This
is a rather nice web-site which allows you to draw functions of two
variables. You can also use this site to visualize the partial derivatives. http://www-math.mit.edu/18.013A/HTML/tools/tools22.html This is another web-site which allows you to visualize the effect of a change of variables in two dimensions. http://cs.jsu.edu/~leathrum/Mathlets/jacobian.html#applettop |

MODULE
DESCRIPTION

AIMS: |
This
course has two main aims: first, the generalization and/or extension of
many concepts you have learned in your first year calculus to functions
of more than one real variable (e.g. continuity, limits,
differentiability etc.) and, second, learning new methods for solving
ordinary differential equations, which should also complement those you
have learnt last year. |

DESCRIPTION: |
In
the course we will basically study continuous
and differentiable functions of several real variables
(concentrating often on the example of functions of two real variables)
as well as linear ordinary differential equations. |

SYLLABUS: | 1. Introduction: what this course is
about 2. Functions of several real variables 2.1. Basic definitions and examples 2.2. Limits and continuity 2.2.1. Neighbourhood of a point in 1, 2 and 3-dimensions 2.2.2. Limits of functions of two variables 2.2.3. Continuity of functions of two variables 2.3. Differentiation of functions of several real variables 2.3.1. Definition of partial derivative for a function of two variables. 2.3.2. Chain rules. 2.3.3. Definition of differential. 2.3.4. Transformation of partial derivatives. 2.3.5. Extension to functions of more than two variables. 2.4. Local properties of functions of several variables 2.4.1. Taylor series expansions. 2.4.2. Classification of stationary values of functions of two variables. 2.4.3. Constraints via Lagrange multipliers. 2.5. Integration of functions of several variables 2.5.1. Standard coordinate systems in R² and R³ . 2.5.2. Integration in R² and R³ . 2.5.3. Change of variables and Jacobians. 3. Differential equations 3.1. Linear differential equations. 3.1.1. Second-order linear differential equations. 3.1.2. The method of variation of parameters. |

BIBLIOGRAPHY: |
[1]
Calculus, by T. Apostol
(Wiley). [2] Calculus: One and several variables, by S.N. Salas and E. Hille (Wiley). [3] Calculus: A complete course, by R.A. Adams (Addison Wesley). [4] Calculus, by H. Anton (Wiley). |