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Continuous Mathematics:  2014-2015

Lecturer

Degrees

Preliminary ExaminationsComputer Science

Preliminary ExaminationsMathematics and Computer Science

Term

Overview

Many areas of mathematics, and applications of mathematics in the applied sciences, are underpinned by the concepts of calculus, i.e. differentiation and integration.

The first aim of this course is to provide an introduction to calculus in several variables that will form a basis for later courses. This theory will be demonstrated using examples drawn from maximising or minimising functions of one or more variables (with or without constraints), the solution of ordinary differential equations, and the solution of simple partial differential equations (including the reduction of partial differential equations to ordinary differential equations via a change of variables in special cases).

The second aim of the course is to introduce some computational techniques in calculus, for example numerical integration, the numerical solution of differential equations, and finding the solution of nonlinear equations in one or more variables. These techniques lend themselves to practical implementation, allowing demonstration of the theory developed during the course.

Learning outcomes

This is an introductory course in calculus. Students will learn:

1. how to differentiate a function of one or more variables using, for example, the chain rule, the product rule, and change of variables;

2. Taylor's theorem of one or more variables;

3. finding maxima and minima of functions of one or more variables either with or without constraints;

4. integration by parts;

5. simple methods for the solution of ordinary differential equations;

6. methods for numerical quadrature;

7. Fourier series representation of functions;

8. simple numerical methods for the numerical solution of ordinary differential equations;

9. simple methods for the solution of partial differential equations;

10. iterative methods for finding the solution of nonlinear algebraic equations.

Prerequisites

None

Synopsis

Lecture 1. Functions of several variables.  Partial differentiation.  The chain rule and the product rule.

Lectures 2-3. Taylor's theorem for a function of one variable and the connection with extrema. Taylor's theorem for a function of several variables. Maxima, minima and saddle points of functions of several variables.

Lectures 4-6. Solution of nonlinear equations in one dimension and higher dimensions using Newton's method.  The connection with optimisation of functions.  Lagrange multipliers for constrained optimisation.

Lecture 7. Integration of a function of one variable, integration by substitution, integration by parts. Calculation of the area under a curve by integration. Simple methods for numerical integration: the trapezium rule and Simpson's rule.

Lectures 8-9.  Fourier series representation of periodic functions.

Lectures 10-13. Ordinary differential equations. Initial value problems, boundary value problems. Solution of separable first order equations and constant coefficient second order problems. Simple numerical methods for first order equations. Solution of second order difference equations and their application to numerical methods for second order boundary value problems.

Lectures 14-16. Simple partial differential equations.  Separable solutions to the heat equation in one dimension. Reduction of partial differential equations to ordinary differential equations by change of variables.

Syllabus

Differentiation of functions of one or more variables: change of variables; the chain rule; and the product rule. Taylor's theorem of one or more variables (proof not required). Maxima, minima, saddle points of one or more variables. Integration of a function of one variable. Numerical integration: the trapezium rule and Simpson's rule (proof of error of Simpson's rule not required).  Fourier series representation of functions.  Ordinary differential equations: initial value problems; boundary value problems; separable solutions; solution of second order boundary value problems with constant coefficients. The forward and backward Euler methods for the numerical solution of initial value ordinary differential equations.  Numerical solution of the finite difference equations arising from second order boundary value problems.  Newton's method for the iterative solution of nonlinear equations in one and many dimensions.  Lagrange multipliers for constrained optimisation.

Reading list

The principal text for this course is

  • D.S. Sivia and S.G. Rawlings: Foundations of Science Mathematics, Oxford Chemistry Primers

 

Additional texts are:

  • Erwin Kreyszig: Advanced Engineering Mathematics. John Wiley & Sons.
  • Dominic Jordan and Peter Smith: Mathematical Techniques. Oxford University Press.

Feedback

Students are formally asked for feedback at the end of the course. Students can also submit feedback at any point here. Feedback received here will go to the Head of Academic Administration, and will be dealt with confidentially when being passed on further. All feedback is welcome.

Taking our courses

This form is not to be used by students studying for a degree in the Department of Computer Science, or for Visiting Students who are registered for Computer Science courses

Other matriculated University of Oxford students who are interested in taking this, or other, courses in the Department of Computer Science, must complete this online form by 17.00 on Friday of 0th week of term in which the course is taught. Late requests, and requests sent by email, will not be considered. All requests must be approved by the relevant Computer Science departmental committee and can only be submitted using this form.