School of Mathematics and Statistics (SoMaS)

## MAS140 Mathematics (Chemical)

 Both semesters, 2011/12 20 Credits Lecturer: Dr Nick Gurski Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

This course is taught in both semesters.

This course is taught together with MAS143 and MAS149 in Semester 1 and MAS151 and MAS152 in Semester 2.

Prerequisites: A-level mathematics (or equivalent)
No other modules have this module as a prerequisite.

## Outline syllabus

Semester 1:
• Functions of a real variable: The concept of a function, limits, continuity.
• Differentiation: Basic rules of differentiation, maxim and minima, curve sketching
• Partial differentiation: 1st and 2nd derivatives, geometrical interpretation.
• Hyperbolic functions: Definitions, derivatives and inverse hyperbolic functions.
• Series: Taylor and Maclaurin series, l'Hôpital's rule.
• Complex numbers: Basic manipulation, Argand diagram, de Moivre's theorem, Euler's relation.
• Vectors: Vector algebra, dot and cross products, differentiation.
Semester 2:
• Integration: Indefinite integrals of simple functions. Simple substitutions. Standard forms involving inverse trigonometric and inverse hyperbolic functions. Examples using completing the square and partial fractions. Integration by parts. Definite integrals: properties, evaluation, application to area.
• Matrices and linear equations: Definition of an m x n matrix. Special matrices (identity, zero, square etc.). Matrix algebra. Transpose. Symmetric and skew-symmetric matrices, and the decomposition of square matrices. Determinants. Inverse of a non-singular matrix. Use of matrices to solve systems of linear equations (homogeneous and nonhomogeneous). Gaussian elimination. Eigenvalues and eigenvectors.
• Ordinary differential equations: First order differential equations: variables separable, linear with integrating factor, general solution, solution satisfying given initial conditions. Second order linear differential equations with constant coefficients: auxiliary equation, complementary function. Particular integral for polynomials, exponentials, trigonometric functions and products of polynomials and exponential/trigonometric functions on right-hand side. Use of eigenvalues and eigenvectors in the solution of systems of linear differential equations.

## Office hours

I will hold office hours in J7, Hicks building, Monday 3-4pm and by appointment.

## Aims

• To reinforce previous knowledge
• To develop new, basic mathematical concepts needed to support Engineering subjects at Levels 1 and 2.
• To provide a foundation for the Level 2 Mathematics course for Chemical Engineers.

## Learning outcomes

Semester 1:
• Evaluate simple limits using algebraic techniques
• Differentiate, find maxima and minima and apply this technique to curve sketching
• Find first and second order partial derivatives
• Understanding of hyperbolic functions
• Apply l'Hôpital's rule
• Manipulate complex numbers
• Know the basic properties of vectors.
Semester 2:
• Evaluate indefinite and definite integrals using the techniques of substitution and integration by parts.
• Manipulate matrices, evaluate determinants and find the inverse of a non-singular square matrix.
• To apply matrix methods to the solution of systems of simultaneous linear equations.
• Find eigenvalues and corresponding eigenvectors of a square matrix
• Solve first order ordinary differential equations which are (i) variables separable, (ii) linear
• Solve second order linear homogeneous ordinary differential equations with constant coefficients.
• Solve second order linear inhomogeneous ordinary differential equations with constant coefficients, using a trial technique for the particular integral.
• Use eigenvalues and eigenvectors to solve systems of linear first order ordinary differential equations.

## Teaching methods

The teaching method is traditional using (i) lectures interspersed with examples, (ii) tutorial classes where students attempt problem sheets. Students are also encouraged to do some reading round the subject.

40 lectures, 20 tutorials

## Assessment

Two-hour written examination in each Semester.
Format of each exam: Part A (50%) compulsory questions, Part B (50%) two from three longer questions.

## Full syllabus

Semester 1:

Lectures 1-3: Functions of a real variable.
• Domain and range of a function. Simple graphs. Even and odd functions.
• Binomial theorem. Simple examples of indeterminate forms 0/0, ∞/∞, 0×∞, ∞×∞. The limit (sinx)/x. One-sided limits. Continuity.
Lectures 4-6: Differentiation.
• Examples on rules for differentiation.
• Inverse functions. Inverse circular functions and their derivatives. Logarithm and exponential functions.
• Maxima and minima. Points of inflexion. Simple curve sketching.
Lectures 7-8: Partial differentiation.
• 1st and 2nd order partial derivatives.
• Geometrical interpretation of 1st order partial derivatives.
Lectures 9-10: Hyperbolic functions.
• Definitions, properties and graphs.
• Inverse hyperbolic functions: derivatives and graphs.
Lectures 11-12: Series.
• Taylor and Maclarin series.
• L'Hôpital's Rule.
Lectures 13-17: Complex numbers.
• Basic algebraic properties. Complex conjugate.
• The Argand diagram. Modulus and argument. Multiplication and division using polar form.
• Simple loci.
• De Moivre's theorem.
• Euler's relation. The relationship between hyperbolic and circular functions.
Lectures 18-21: Vectors
• Magnitude, unit vector, vector algebra
• Component form, i, j, k, scalar product.
• Vector product, determinant form.
• Differentiation of vectors.
Semester 2:
Lectures 1 - 6: Integration
Indefinite integrals of simple functions. Simple substitutions. Standard forms with inverse trigonometric and hyperbolic functions. Examples using completing the square and partial fractions. Integration by parts. Definite integrals: properties, evaluation, application to area.
Lectures 7 - 14: Matrices
Definition of an mxn matrix. Special matrices (identity, zero, square, symmetric etc). Matrix algebra. Transpose. Determinants. Inverse of a non-singular matrix. Use of matrices to solve systems of linear equations (homogeneous and non-homogeneous). Gaussian Elimination. Eigenvalues and eigenvectors.
Lectures 15 - 20: Differential equations
First order differential equations: variables separable, integrating factor, general solution, solution satisfying given initial conditions. Second order linear differential equations with constant coefficients: auxiliary equation, complementary function. Particular integral for polynomials, exponentials, trigonometric functions and products of polynomials and exponential/trigonometric function on right hand side. Use of eigenvalues and eigenvectors in the solution of systems of ordinary linear differential equations.
Lectures 21 - 22: Revision
Optional revision lectures tailored to the needs of those who choose to attend.