ENGINEERING TRIPOS PART IB - 2012/2013
PAPER 7 - MATHEMATICAL METHODS (1)
Leader: Dr. T. Hynes
Timing: Weeks 1-3 and 6-8 Michaelmas term, 2 lectures/week; weeks 4-5
Michaelmas term, 1 lecture/week
Structure: 14 lectures
The course provides an elementary introduction to vector calculus and aims
to familiarise the student with the basic ideas of the differential calculus
(the vector gradient, divergence and curl) and the integral calculus (line,
surface and volume integrals and the theorems of Gauss and Stokes). The
physical interpretation of the mathematical ideas will be stressed throughout
via applications which centre on the derivation and manipulation of the common
partial differential equations of engineering. The analytical solution of
simple partial differential equations by the method of separation of variables
will also be discussed.
The aims of the course are to:
- Provide the necessary
background mathematics to ensure that students are confident in handling
partial differential equations in vector form while maintaining a tangible
physical appreciation of the manipulations involved.
As specific objectives, by the end of the course students should be able to:
- Differentiate and integrate
scalar functions of two or more variables including transformations to
other co-ordinate systems.
- Manipulate vector
differential equations including the gradient, divergence and curl
operators while retaining a physical appreciation of the mathematical
- Perform line, surface and
volume integrals and understand their various physical interpretations.
- Set up conservation
statements in both differential and integral form and be able to transform
from one to the other using Gauss's theorem.
- Appreciate the physical
significance of curl and its relationship to circulation via Stokes's
theorem in simple examples.
- Solve common PDE's
(particularly the Laplace, Poisson, heat conduction and wave equations)
with simple boundary conditions by the method of separation of variables.
A knowledge of the following Part IA lecture material on functions of more
than one variable will be assumed: representation of curves and surfaces
(including parametric representation); partial differentiation; total and
perfect differentials; Taylor series; maxima and minima.
The course will then consist of lectures on the following topics:
- Vector functions and
fields; field lines.
- Vector differentiation;
- The vector gradient and its
- Cylindrical and spherical polar
- The divergence and its
physical interpretation; solenoidal fields; conservation statements;
- Surface integrals; volume
integrals; Gauss's divergence theorem; integral-differential
transformations. Stokes's theorem.
- The curl and its physical
interpretation; irrotational fields; scalar potential; line integrals;
- Types of PDE and boundary
conditions; solution by separation of variables; examples of some common
PDE's (Laplace, Poisson, heat conduction, wave equation).
Please see the Booklist for Part IB Courses for references for this module.
Last updated: September 2012