Continuous Mathematics: 2018-2019
OverviewMany 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.
The second aim of the course is to introduce some computational techniques in calculus, for example numerical integration, the numerical solution of differential equations, Fourier analysis, 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 outcomesThis 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.
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.
Differentiation of functions of one or more variables using classical, complex, and vector/matrix formulations. The operators grad and div. Taylor's theorem, and classification of turning points of multivariate functions. Integration of simple multivariate functions. Numerical integration: the trapezium rule and Simpson's rule. Ordinary differential equations: initial value problems; boundary value problems; separable solutions; solution of simple second order boundary value problems. Fourier series representation of functions, and the Fast Fourier Transform. Introduction to techniques for the numerical solution of initial value ordinary differential equations. Newton's method for the iterative solution of nonlinear equations in one and many dimensions. Optimisation of multivariate functions, and the use of Lagrange multipliers for constrained optimisation.
The principal text for this course is
- D.S. Sivia and S.G. Rawlings: Foundations of Science Mathematics, Oxford Chemistry Primers
You may also find the accompanying book of worked problems useful.
Additional texts are:
- Erwin Kreyszig: Advanced Engineering Mathematics. John Wiley & Sons.
- Dominic Jordan and Peter Smith: Mathematical Techniques. Oxford University Press.