• Semester 1 (Day)
• Semester 1 (External)

Pre-requisite: 64613/75613

Synopsis

This unit is broadly divided into two interrelated halves.

The  vector calculus half develops further the theory and applications
of  the  calculus  of  vectors from that  established  in  64613.  The
principles  of modelling material such as air or water are  introduced
in  one-dimensional dynamics. Then the application of vector  calculus
to  modelling the 3-D flow of a fluid material is developed along with
the  notions  of scalar and vector functions, directional derivatives,
divergence,  curl  and  Laplace's equation. The integral  theorems  of
Gauss  and  Stokes  highlight many facets  of  fluid  flow  and  other
physical fields.

The  differential equations (DE's) half of the course  begins  with  a
formal  study of infinite series, leading to power series and Taylor's
theorem n-dimensions. Application is made to the classification
of  extrema  in  constrained optimisation problems. The  reduction  of
general  DE's  to  systems  of linear, first-order  DE's  is  used  to
visualise  dynamics  using the phase diagram and  to  introduce  fixed
points,  and stability. Fourier analysis introduces the use of  series
expansion  and integral transform methods which are then  employed  in
the  solution of partial differential equations that appear in  models
of many physical systems.