Calculus 2: Multivariable and Differential Equations

BUEM001S5 (30 credits)

Aims

This module aims to develop the ideas and techniques of calculus introduced in Calculus 1: Single Variable to functions of more than one variable. It also covers exact and numerical solutions of ordinary differential equations, as well as modelling problems using differential equations.

Teaching and Assessment

Teaching for this module will take place throughout the year, with eight evenings in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term.

Of the final course mark, 80% is based on a three-hour exam in June and the other 20% is from assessed coursework.

Coursework  will consist of short, problem based assignments. You will have around three weeks to complete each one.

The examination in the Summer Term has two sections. Section A (worth 40%) consists of compulsory short questions. Section B (also worth 40%) contains several longer questions of which you must answer two.

Syllabus

Partial Differentiation
Limits, formal definition of the derivative, partial differentiation, tangent planes,  directional derivatives, stationary points, Lagrange multipliers, the chain rule, Taylor polynomial approximation of a function.

Integration
Double integrals, splitting the integral, changing the order of integration, polar coordinates, line integrals, Green’s Theorem.

Hyperbolic and Special Functions
Hyperbolic functions, sinh, cosh and tanh, gamma functions, beta functions, properties of hyperbolic and special functions, application of hyperbolic and special functions to evaluate certain integrals.

Ordinary Differential Equations
First order differential equations, variable separable, exact differential equations, integrating factors, homogeneous differential equations, some special families of first order differential equations, second order differential equations, homogeneous and non homogeneous differential equations with constant coefficients, some special families of second order differential equations numerical methods for finding approximate solutions of a differential equation.

Mathematical Models
Applications of Calculus including simple harmonic motion, damped and forced oscillations, population models, epidemiology, finance and economics.

Learning Outcomes

On successful completion of this module a student will be expected to be able to:

Knowledge and understanding of, and the ability to use, mathematical and/or statistical techniques. In particular:

• Techniques of calculus of more than one variable;
• Methods of solution of ordinary differential equations.

Knowledge and understanding of a range of results in mathematics.

Appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments. In particular:

• Knowledge of the theory underpinning the techniques of calculus;
• Ability to produce proofs of some results in calculus

Awareness of the use of mathematics and/or statistics to model problems in the natural and social sciences, and the ability to formulate such problems using appropriate notation. In particular:

• Modelling oscillating systems;
• Modelling problems in biology;
• Modelling problems in finance and economics.

Knowledge and understanding of the processes and limitations of mathematical approximation and computational mathematics. In particular:

• Approximating functions using Taylor series;
• Finding numerical solutions to differential equations;
• Estimating the error in numerical solutions to differential equations.

Knowledge and understanding of a range of modelling techniques, their conditions and limitations, and the need to validate and revise models. In particular:

• Modelling problems using differential equations.

A deeper knowledge of some particular areas of mathematics.

Ability to use a modern mathematical and/or statistical computer package with a programming facility, together with knowledge of other suitable packages.

Intellectual

Ability to comprehend conceptual and abstract material.
Develop a logical and systematic approach to problem solving.

Practical

Ability to use a range of software packages including word processing and spreadsheets.
Problem-solving skills, including the ability to assess problems logically and to approach them analytically.
Highly developed quantitative skills
Ability to transfer knowledge and expertise from one context to another.

Personal and Social

Ability to learn independently using a variety of media.
Ability to work independently with patience and persistence.
Time-management and organizational skills.
General IT skills, including word processing and spreadsheets.
Good communication skills, including the ability to write coherently.
Ability to complete a sustained and substantial task.
Ability to complete work in a limited time period.

Recommended books

(provisional list)

• Adams, RA, Calculus of Several Variables, Addison-Wesley
• Adams, RA, Calculus: A Complete Course, Addison-Wesley
• Harris, K, Discovering Calculus with Maple, Wiley
• Goldsmith C and D Nelson, Extensions of Calculus, Cambridge