ENGINEERING TRIPOS PART IA - 2012/2013
PAPER 4 - MATHEMATICAL METHODS
Michaelmas: Prof P. A. Davidson /Dr. J.P. Longley / Dr.G.N. Wells / Dr M.P. Juniper
Lent: Dr. J. Lasenby
Easter: Dr S. Singh
Michaelmas Term: 3 (standard course) or 2 (fast course) lectures per week, weeks 1-8;
Lent Term: 1 lecture per week, weeks 1-5 and 7-8; 2 lectures per week, week 6;
Easter Vacation: Programmed learning exercise;
Easter Term: 2 lectures per week, weeks 1-3; 1 lecture in week 4.
Structure: 40/32 lectures
The aims of the course are to:
- Instill fluency with the
basic mathematical techniques which are needed as tools for engineers.
- Revise, and teach afresh
where necessary, those parts of the A-level mathematics syllabuses which
are necessary for the first two years of the engineering course, and to
introduce those new mathematical techniques which are necessary for these
- Place emphasis throughout
upon the grasp of essentials and competency in manipulation.
At the end of the course students should be able to:
- Recognise the appropriate
mathematical tools and techniques (from the following syllabus) with which
to approach a wide variety of engineering problems.
- Specify a mathematical
model of a problem.
- Carry out appropriate
mathematical manipulations to solve the modelled problem.
- Interpret the significance
of the mathematical result.
Michaelmas term (24/16L)
The Michaelmas term course concerns revision and extension of concepts which
most students will have met at school. It will be given in two versions, a
standard course at a pace of three lectures per week and a fast course at a
pace of two lectures per week. Both will cover the same syllabus and employ the
same example sheets. The fast course is aimed primarily at those who have taken
double mathematics at A level and who have good mathematical fluency, the
standard course at those with less prior training. Examples papers will include
exercises to encourage students to practice mathematical skills learnt in their
1. Vectors (5/3L)
- Scalar and vector product.
- Moment of a force and
angular velocity vectors.
- Scalar and vector triple
- Examples of applications.
- Simple vector geometry,
vector equations of lines and planes.
- Determinant of 3x3 matrices
2. Functions and Complex Numbers (7/4L)
- Definitions and simple
properties of the hyperbolic functions.
- Statement of Taylor's
theorem, examples including trigonometric and hyperbolic function, exp,
- Simple ideas of series,
approximations, limits, L'Hopital's rule.
- Asymptotic behaviour of
functions for small and large argument.
- Revision of complex
arithmetic and representation in the Argand diagram. Idea of a complex
- De Moivre's theorem, use of
exp (iw t)
3. Introduction to Ordinary Differential Equations (ODE's)
- Linear equations of first
order, integrating factor, separation of variables.
- Second order ODE’s:
complementary functions, superposition and particular
- Linear difference
- Notions of a partial
4. Matrices (7/6L)
- Rules for calculating
inverse of 3x3 matrices.
- Change from one orthogonal
coordinate system to another, rotation matrix.
- Symmetric, antisymmetric
and orthogonal matrices.
- Eigenvalues and
eigenvectors for symmetric matrices.
- Orthogonality of
eigenvectors, expansion of an arbitrary vector in eigenvectors.
- Examples, including small
Lent Term (9L)
The course in the Lent and Easter terms introduces ideas which will be new
to most students, but which find application across the whole range of
5. Steps, impulses and linear system response (3L)
- Introduction to step and
impulse functions. Step and impulse response of linear systems represented
- Use of convolution to
obtain output given a general input.
6. Fourier series (4L)
- Fourier sine and cosine
series. Full and half range, consideration of symmetries, convergence and
- Complex Fourier series.
Physical interpretations, including effect of filtering a general periodic
7. Introduction to probability material listed in (8) below. (2L)
8. Probability(Programmed learning text, equivalent to four
lectures of material)
- Notion of probability.
- Permutations and
- Mean,variance and standard deviation of probability distributions.
- Discrete and continuous distributions.
- The Normal distribution and experimental errors
Easter term (7L)
9. Functions of Several Variables (4L)
- Differentiation of
functions of several variables.
- Chain rule, implicit
- Introduction to definition
- Stationary values,
- Taylor expansion of
10. Introduction to Laplace transforms (3L)
- Laplace transforms as a
means of solving ODEs with initial conditions (using tables of transforms
Please see the Booklist for Part IA Courses for module references.
Last updated: May firstname.lastname@example.org