MAS153 Mathematics (Materials)

 Both semesters, 2021/22 20 Credits Lecturer: Prof Elizabeth Winstanley Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

There are no prerequisites for this module.

The following modules have this module as a prerequisite:

 MAS250 Mathematics II (Materials)

Outline syllabus

• Permutations and combinations
• Binomial theorem
• Vectors
• Hyperbolic functions
• Differentiation
• Integration
• Series
• Complex numbers
• Matrices

Aims

• To give students the necessary mathematical skills required to understand the scientific and engineering concepts introduced in the second year of their Material Science course.

Learning outcomes

• Be able to calculate permutations and combinations
• Be able to apply the binomial theorem to series expansions and limits
• Understand vector algebra and be able to apply it to simple geometrical problems and to resolving forces
• Be able to manipulate hyperbolic functions and their inverses, and to carry out differentiation and integration involving hyperbolic functions and their inverses
• Be able to integrate using partial fractions, and to apply Simpson's rule
• Be able to verify solutions of simple ordinary differential equations
• Understand the concept of convergence of infinite series, and be able to test it using the alternating series test and the ratio test
• Understand and be able to use complex algebra
• Understand matrix algebra and be able to apply it to the solution of systems of linear equations, and to be able to find matrix eigenvalues and eigenvectors

Teaching methods

Lectures, tutorials, independent study

40 lectures, 20 tutorials

Assessment

One three-hour exam at the end of the year (90%). Online tests (10%).

Full syllabus

Permutations and combinations

with simple probability applications.
The Binomial Theorem
Pascal's Triangle, binomial theorem for positive integer exponents, extension to arbitrary real exponents, application to limits.
Vectors
Definition, magnitude, elementary algebra, component form, unit vectors, scalar product, angle between vectors, application to resolving forces, equations of lines and planes, vector product, scalar and vector triple products, a simple application to circular motion and angular momentum.
Hyperbolic functions
Definition in terms of exponentials, identities, inverse hyperbolic functions.
Differentiation
Inverse functions, hyperbolic functions and their inverses. Applications including Newton's 2nd law of motion, simple harmonic motion, radioactive decay, energy states of a particle in a 1D box.
Integration
Hyperbolic functions, use of hyperbolic functions in integration by substitution, use of partial fractions, Simpson's rule.
Series
Convergence, alternating series test, ratio test, Maclaurin and Taylor series.
Complex Numbers
Definition, conjugate, algebra. Argand diagram, modulus and argument, polar form. The Euler relation, De Moivre's theorem, roots of complex numbers, simple loci.
Matrices
Definitions (including diagonal and identity matrices), algebra, determinants, inverse matrices, homogeneous and non-homogeneous systems of linear equations, Gaussian elimination, eigenvalues and eigenvectors.