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MA2507: ADVANCED CALCULUS (2016-2017)

Last modified: 28 Jun 2018 10:27

Course Overview

This is a course in multivariable calculus.  As the name suggests, it generalises familiar concepts from calculus (such as limits, derivatives, integrals and differential equations) to situations with many variables.

In addition to lectures and tutorials, there will be practical training through several computer sessions. Recommended to mathematicians and physicists.

Course Details

Study Type Undergraduate Level 2
Session Second Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus None. Sustained Study No
  • Dr Alexey Sevastyanov

Qualification Prerequisites


What courses & programmes must have been taken before this course?

  • MA1508 Calculus II (Passed)
  • MA1005 Calculus 1 (Passed)
  • Any Undergraduate Programme (Studied)

What other courses must be taken with this course?


What courses cannot be taken with this course?


Are there a limited number of places available?


Course Description

A) Several variables - Continuity and Partial Differentiation.
  • Functions of several variables, graphical surface representations, limits and continuity, partial derivatives, higher order partials, plane tangent, linear approximation, small errors.
  • Chain rule, polar coordinates, applications to some elementary PDEs.
  • Critical points, second derivative test, some discussion of global global max/min.
  • Taylor series and quadratic approximation.
B) Several variables - Multiple Integrals.
  • Revision of definite integral as area under curve, approximated by rectangles. Double integral as volume, approximated by rectangular pillars.
  • Iteration formulae for rectangles and for more general regions, change of order of integration via double integral.
  • Change of variable in double integrals (with emphasis on polars).
  • Triple integrals, cylindrical and spherical polar coordinates.
  • Applications to volumes, moments and centres of mass.
C) Ordinary differential equations.
  • Basic terminology, general solution, integral curves, initial and boundary conditions.
  • First order ODEs: linear equations, separable equations, brief treatment of homogeneous and Bernoulli equations, applications (eg. population problems and mixing problems).
  • Second order linear ODEs: basic theory for solution of equations with constant coefficients via CF + PI, applications; reduction of order and variation of parameters.
D) Introduction to computing software (4 practical sessions).
  • Simple arithmetic, operations, variables, booleans, conditionals, functions, procedures, plotting, functions in two variables, contour plots, parametric plots, basic algebra, differentiation, integration, programming, loops.

Contact Teaching Time

Information on contact teaching time is available from the course guide.

Teaching Breakdown

More Information about Week Numbers

Details, including assessments, may be subject to change until 31 August 2023 for 1st half-session courses and 22 December 2023 for 2nd half-session courses.

Summative Assessments

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour written examination paper, maximum resit (100%) and resit (80%) with in-course assessment (20%).

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.


In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.

Course Learning Outcomes


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