The University of Southampton

MATH1007 Mathematical Methods For Physical Scientists 1b

Module Overview

The aim of the module is to provide students with the necessary skills and confidence to apply a range of mathematical methods to problems in the physical sciences. We 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 also look at the issue of solving dfifferential equations, a topic of great importance in modelling the real world.

Aims & Objectives

Aims

Aim

Having successfully completed this module, you will be able to:

  • 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
  • Evaluate the gradient of a scalar field and the divergence and curl of a vector field
  • Express curves and surfaces in both parametric and implicit form
  • Evaluate line integrals and fluxes of vector fields over curves and surfaces
  • Apply the divergence theorem and Stokes' theorem
  • Identify and solve first order ODEs that are separable, linear or exact
  • Solve second order linear equations with constant coefficients

Syllabus

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.

Vector Calculus:
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.

Surfaces:
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. 

Differential equations
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 & Teaching

Learning & teaching methods

Lectures, small group tutorials, private study. The method of delivery in lectures will be “chalk and talk”, using skelatal lecture notes.

ActivityDescriptionHours
LectureThree lectures a week. The method of delivery in lectures will be “chalk and talk”. The lecture notes are skeletal in nature; complete versions of each chapter will be posted on Blackboard once the chapter has been completed. 36
TutorialEach student will be allocated to one weekly tutorial. The self-marking of the previous week's problem sheet will be checked, and help given on next week's sheet. Solutions to all problem sheets will be available on the module Blackboard site at the appropriate times.12

Assessment

Assessment methods

MethodHoursPercentage contribution
Weekly problem sheets, self-marked by students, with tutorial leaders checking the self-marking.-10%
Four coursework sheets.-10%
Exam2 hours hours80%

Referral Method: By examination

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