The module will build on the methods developed in MATH1006 (or MATH1008) but extend many of the ideas from ordinary functions to vector valued functions which, for example, may be used to describe forces or electromagnetic fields in 3-dimensional space. We will also look at the issue of solving differential equations, a topic of great importance in modelling the real world.
One of the pre-requisites for MATH2015
Aims and Objectives
Having successfully completed this module you will be able to:
- Identify and solve first order ODEs that are separable, linear or exact
- Express curves and surfaces in both parametric and implicit form
- Apply the divergence theorem and Stokes' theorem
- Evaluate partial derivatives and find critical points of functions of two variables
- Evaluate integrals of simple functions over simple regions of the plane and simple volumes
- Solve second order linear equations with constant coefficients
- Evaluate line integrals and fluxes of vector fields over curves and surfaces
- Evaluate the gradient of a scalar field and the divergence and curl of a vector field
Functions of two or more variables:
Evaluate partial derivatives, find critical points, and, for functions of two variables, classify them.
Multiple Integrals of a scalar function in (2 and 3 dimensions):
Evaluate integrals of simple functions over regions in plane bounded by graphs of simple functions, either directly or by change of coordinate system. Evaluate integrals over volumes bounded by planes, spheres and cylinders, using cylindrical and polar coordinates.
Gradients, divergences and curls.
Curves and line integrals:
Express, in simple cases, curves given parametrically. Evaluate lengths of curves in 2 and 3 dimensions. Evaluate integrals of scalar functions along curves with respect to arc-length. Evaluate
the integral of the tangential component of a vector field along a curve. Conservative fields.
Integration of normal components of a vector field or of a scalar field over surfaces described parametrically.
The divergence theorem and and Stokes' theorem and their application.
Types of ordinary differential equation. Solving simple differential equations, separation of variables, integrating factors and first order linear ordinary differential equations. Exact differential equations. Second order differential equations. Homogeneous linear ordinary differential equations with constant coefficients. Free and forced damped harmonic oscillator.
Learning and Teaching
Teaching and learning methods
Lectures, small group tutorials, private study. The method of delivery in lectures will be “chalk and talk”, using skelatal lecture notes.
|Total study time||150|
Resources & Reading list
Robert Wrede and Murray Spiegel. Schaum's Outlines: Advanced Calculus. McGraw-Hill.
Richard Bronson and Gabriel Costa. Schaum's Outlines: Differential Equations. McGraw-Hill.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External