MA2001: Calculus of Several Variables

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA2001
External Subject Code 100405
Number of Credits 10
Level L5
Language of Delivery English
Module Leader Dr Mikhail Cherdantsev
Semester Autumn Semester
Academic Year 2014/5

Outline Description of Module

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.

On completion of the module a student should be able to

  • Locate points of discontinuity of specified functions
  • Find equations for tangent planes to surfaces
  • Evaluate directional derivatives of given functions in given directions
  • 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 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.

How the module will be delivered

27 - 50 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 50 hours private study including preparation of solutions to given exercises.

Skills that will be practised and developed

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.

How the module will be assessed

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

The summative assessment is the 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.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 85 Calculus Of Several Variables 2
Written Assessment 15 Coursework N/A

Syllabus content

  • 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. 
    • Definite, indefinite and semi-definite cases for functions of two variables. 
    • Generalizations for functions of many variables. 
    • Constraints and Lagrange multipliers.
  • Multiple integration
    • Double integrals.
    • Reduction to iterated integrals. 
    • Change of variables. 
    • Jacobians and their properties. 
    • Plane-polar co-ordinates. 
    • Triple integrals. 
    • Cylindrical and spherical polar co-ordinates. 
    • Line integrals.

Essential Reading and Resource List

Please see Background Reading List for an indicative list.

Background Reading and Resource List

Functions of Two Variables, Dineen,  S., Chapman and Hall

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

Advanced Calculus, Spiegel, M., Schaum


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