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.

• 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
• Understand the proofs of the theorems underlying the procedures described in the course.

22 - 50 minute lectures

5 - 50 minute exercise 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 50 hours private study including preparation of solutions to given exercises.

Skills:
Understand and apply the methods of differential and integral calculus to problems involving more than one variable.

Transferable Skills:
Formulating and solving problems by means of a suitable previously understood apparatus.

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 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 knowledge 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 but normally short.  Section B has a choice of two from three equally weighted questions.

• Functions of two and more variables
  • Limits and continuity.
  • Partial derivatives and differentiability. 
  • Tangent plane. 
  • 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. 
  • Jacobians and their properties. 
  • Plane-polar co-ordinates.  
  • Cylindrical and spherical polar co-ordinates. 

Please see Background Reading List for an indicative list.

Salas, S. L., Etgen, G. J., and Hille, E., Calculus: One and Several Variables, Wiley & Sons 

Binmore, K. G., & Davies, J., Calculus, Cambridge Univerity Press

Advanced Calculus, Spiegel, M., Schaum