Advanced Calculus - 2019 entry

This module introduces advanced methods of calculus. The advanced calculus topics in this module form the core knowledge base for mathematical sciences. They build on topics from Calculus and Geometry and introduce key methods relating to differential equations and partial differentiation. Differential equations are essential to many scientific and engineering problems. Partial differentiation is vital for modelling the real world, and you will use it for calculus problems in more than one dimension.

Pre-requisite module: “Calculus and Geometry” (ECM1901), or equivalent

This module aims to introduce you to advanced methods of calculus, building on the knowledge you acquired in Calculus and Geometry. It develops further the key concepts and skills that will form necessary background for later study in all branches of the mathematical sciences. The main emphasis of the module will be on practical methods and problem solving; however, all results will be stated formally and each sub-topic will be reviewed from a mathematically rigorous standpoint.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 Apply advanced techniques for integration;

2 Solve various types of ordinary differential equations;

3 Demonstrate an understanding of the fundamentals of calculus of several variables;

4 Show knowledge of geometric interpretations of various calculus techniques;

5 Appreciate the formal rigour that underpins calculus techniques;

Discipline Specific Skills and Knowledge:

6 Exhibit a clear grasp of fundamental mathematical concepts, manipulations and results in calculus and of their importance within branches of the mathematical sciences;

7 Evidence sufficient knowledge of those techniques to enable successful progression in mathematical sciences;

Personal and Key Transferable / Employment Skills and Knowledge:

8 Reason using abstract ideas;

9 Formulate and solve problems and communicate reasoning and solutions effectively both verbally and in writing;

10 Use learning resources appropriately;

11 Display self-management and time management skills.

- Integration and integration techniques: Riemann sums and formal proofs for existence of definite integrals, Fundamental Theorem of Calculus; integration by substitution; integration by parts; indefinite integrals [9 hours];

- Ordinary differential equations: first order equations and separation of variables; integrating factor method [3 hours];

- Differential equations with constant coefficients: complementary function and auxiliary equation; inhomogeneous equations [3 hours];

- Revision and mastery of integration and ODEs [3 hours];

- Partial differentiation: continuity for multivariable functions (with epsilon-delta proofs); directional and total derivative; multivariable Taylor series [3 hours];

- Calculus tools for optimisation: characterising critical points, maxima, minima and saddles; Lagrange multipliers for constrained problems [6 hours];

- Multiple integrals: Riemann sums; multiple integration in Cartesian, cylindrical and spherical polar coordinates [3 hours];

- Revision and mastery [3 hours].

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic Reading:

ELE - http://vle.exeter.ac.uk

Reading list for this module: