This module will be dedicated to transferring all basic notions of calculus of functions of one variable to functions of several variables including limits, continuity, differentiation and integration. The second part of the module extends the calculus of several variables to the description and analysis of vector and scalar fields, line and surface integrals.  There will be an emphasis on ideas and results that can be applied in many areas of mathematical modelling. But the main vector calculus theorems that will be presented are also of significance without regard to any such applications. This is because they can be viewed as being natural extensions – to cases involving more than one-dimension - of the fundamental theorem of calculus which relates the process of integration to that of anti-differentiation.

• Locate points of discontinuity of specified functions
• Find equations for tangent planes to surfaces
• Expand functions of two variables in Taylor Series
• Locate stationary points of functions of two variables
• Determine the nature of stationary points
• Determine maximum and minimum values of functions subject to constraints
• Evaluate double and triple integrals by expressing them as repeated integrals and by change of variable
• Appreciate the utility of various vector calculus conceptions in the description of vector and scalar fields that appear in mathematical models.
• Calculate and interpret various derivatives, line integrals, surface integrals and volume integrals for vector and scalar fields.
• Make use of simple non-Cartesian co-ordinates in the description of vector and scalar fields.
• Understand, verify and apply the main theorems presented in the module.

54 fifty-minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 100 hours private study including preparation of solutions to given exercises.

Skills:

The capability to understand and apply the methods of differential and integral calculus to problems involving more than one variable, analyse and describe a range of properties of vector and scalar fields, including their derivatives and integrals.

Transferable Skills:

Calculation, visualisation and interpretation skills from multivariable and vector calculus provide essential ingredients for many areas of mathematical modelling, as well as being of fundamental mathematical importance.

Formative assessment is carried out by means of regular exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures. 

The summative assessment comprises of coursework during the module and a written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knwledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length.  Section B has a choice of three from four equally weighted questions.

• Functions of two and more variables
• Limits and continuity.
• Partial derivatives and differentiability.
• Chain rule.
• Taylor’s theorem.
• Maxima and Minima
• Stationary points of functions.
• Classification of stationary points
• Constraints and Lagrange multipliers.
• Multiple integration
• Double and Triple integrals.
• Reduction to iterated integrals.
• Change of variables. Non-Cartesian co-ordinates
• Jacobians and their properties.
• Plane-polar co-ordinates. 
• Cylindrical and spherical polar co-ordinates.
• Essential Reading and Resource List
• Vector and scalar fields
• Parameterised curves and tangent vectors
• Gradient operator, directional derivatives, divergence, curl and Laplacian
• Vector identities
• Examples of mathematical models involving vector and scalar fields and their derivatives
• Line integrals
• Application to energy change
• Independence from parameterisation
• Double integrals: Green’s theorem in the plane (statement and illustration).
• Surfaces
• Tangent vectors, normal and unit normal vectors, tangent and normal planes and lines.
• Parameterisation of surfaces, including examples for the cylinder and sphere
• Surface integrals: calculation using parameterisation, Stokes theorem (statement and illustration).
• Volume integrals: divergence theorem (statement and illustration).

Salas, S. L., Etgen, G. J., and Hille, E. 2007. Calculus: one and several variables. 10th ed. Wiley

Kreyszig, E. 2011. Advanced Engineering Mathematics. 10th ed. Wiley.

Binmore, K.G. 1982. Mathematical analysis: a straightfoward approach. 2nd ed.  CUP

Speigel, M.R. Schaum's outline of theory and problems of advanced calculus.