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The University of Southampton

MATH1009 Math Methods for Scientist 1b

Module Overview

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

Learning Outcomes

Learning Outcomes

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


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 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.

Independent Study102
Total study time150

Resources & Reading list

Robert Wrede and Murray Spiegel. Schaum's Outlines: Advanced Calculus. 

Richard Bronson and Gabriel Costa. Schaum's Outlines: Differential Equations. 



MethodPercentage contribution
Coursework 20%
Written assessment 80%


MethodPercentage contribution
Written assessment 100%

Repeat Information

Repeat type: Internal & External


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Recommended texts for this module may be available in limited supply in the University Library and students may wish to purchase reading texts as appropriate.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

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