## Engineering Mathematics 3

 Unit code: CHEN20041 Credit Rating: 10 Unit level: Level 2 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

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#### Aims

The unit aims to:

Provide an introduction to the methods of integration and solution of ordinary differential equation systems arising from the mathematical modelling of chemical engineering applications.

#### Overview

First-, second- and higher-order ordinary differential equations. Role of initial and boundary conditions.

A range of solutions to first-order differential equations will be covered with and without constant coefficients including separation of variables, linear differential equations, Bernouilli differential equations, homogeneous differential equations and differential equations solvable for x or y.

Then solutions to second- and higher-order differential equations will focus on differential equations with y or x missing, linear differential equations with constant coefficients, reduction of an order technique, the inverse operator technique, the Cauchy/Euler equation,

Partial differential equations. Characterization of solutions. Separation of variables technique of solving partial differential equations.

Application of differential equations to Physical and Chemical Engineering examples.

Double and triple integrals and their applications for calculating surface areas and volumes. Cartesian, polar and spherical coordinates. Converting integrals from Cartesian to polar or spherical coordinates.

Vector algebra. Scalar and vector fields. Scalar and vector products of vectors. Vector calculus.

Gradient, divergence and curl operators. Gauss’s theorem. Green’s theorem.

#### Teaching and learning methods

A combination of 16 lectures and 8 interactive workshops (problem solving sessions).

In addition, podcasts of the lectures will be available on Blackboard.

Lecture notes with additional problems and exercises will be available from Blackboard.

Worked-out answers of the problem solving sessions will be provided afterwards via Blackboard.

#### Learning outcomes

Explain how both differential equations and integration can arise in the process of setting up mathematical models.

Approximate solutions of a differential equation.

Compare different methods and chose a suitable method.

Appreciate the accuracy and limitation of solutions.

Apply the ideas and concepts to systems of differential equations.

Apply differential techniques to an engineering problem.

Identify double, triple and contour integrals.

Find a suitable method to solve double, triple and contour integrals.

Convert double and triple integrals to different coordinate systems.

Convert Cartesian variables to either polar or spherical coordinates.

Apply multiple integrals techniques to an engineering problem.

Identify scalar and vector notation.

Utilize scalar and vector products in the forms of gradient, divergence and curl operators.

Apply scalar and vector notation to an engineering problem.

#### Assessment Further Information

 Assessment task Length Weighting within unit (if relevant) Written Exam that will cover a selection of short questions on all topics covered: all questions compulsory. 1.5 hours 100%

(1)        M. R. Spiegel, Advanced Mathematics for Engineers and Scientists, McGraw-Hill Companies, New York, 1971.

(2)        F. Ayres & E. Mendelson, Calculus, 5th Edition, McGraw-Hill Companies, Inc, New York, 2009.

(3)        G. Arfken, Mathematical Methods for Physicists, 3rd Edition, Academic Press, Inc, San Diego, 1985.

#### Study hours

• Lectures - 20 hours
• Tutorials - 4 hours
• Independent study hours - 70 hours

#### Teaching staff

Samuel De Visser - Unit coordinator