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## MATH1012 Mathematics 2

### 25 creditsClass Size: 290

Module manager: Dr Derek Harland; Dr Vincent Caudrelier
Email: D.G.Harland@leeds.ac.uk; V.Caudrelier@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2019/20

### Pre-requisites

 MATH1010 Mathematics 1

### This module is mutually exclusive with

 MATH1050 Calculus and Mathematical Analysis MATH1055 Numbers and Vectors MATH1060 Introductory Linear Algebra MATH1331 Linear Algebra with Applications MATH1400 Modelling with Differential Equations

This module is not approved as a discovery module

### Module summary

This module builds on the material of MATH1010 (Mathematics 1) in introducing fundamental areas of mathematics. Calculus is extended to functions of several variables. Further topics in linear algebra are developed, introducing concepts such as basis, dimension, eigenvectors. The study of ordinary differential equations is continued and extended to higher order equations and systems of differential equations. These fundamental techniques in applied mathematics will be applied to problems in Newtonian mechanics, which will serve as the main example of how mathematics is used to describe the physical world.

### Objectives

On completion of this module, students should:
- be able to derive and solve ordinary differential equations arising in applications, for example in the study of oscillators;
- model mechanical problems in both Cartesian and polar coordinate systems;
- solve problems based on Newton's Laws via principles of Work, Energy and Momentum;
- understand properties of linear algebra such as linear dependence, kernel, range and basis;
- be comfortable solving first order ordinary differential equations by a variety of methods;
- be able to compute eigenvalues and eigenvectors of matrices;
- be able to diagonalise matrices and perform a change of variables.

### Syllabus

- Review of vectors and matrices. Subspaces, bases and dimensions.
- Linear combinations and dependence. Kernel and range.
- Eigenvalues and eigenvectors. Diagonalisation.
- Introduction to ordinary differential equations. Solution of 1st order ODEs.
- Basic kinematics, phase space. Newton's laws of motion, forces (gravity, springs, viscous drag). Harmonic oscillator.
- Linear second order equations, supposition of solutions. Constant coefficient homogeneous differential equations.
- Undamped and damped harmonic oscillators. Phase portraits.
- Oscillators with external forcing. Inhomogeneous differential equations. Particular integrals.
- Forced oscillations and resonance. Impulse.
- Energy and work. Kinetic energy, potential energy, conservative and dissipative forces.
- Newton's law of gravitation. Circular motion. Polar coordinates. Angular velocity and momentum.
- Pendulums. Phase portraits.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 8 1.00 8.00 Lecture 55 1.00 55.00 Tutorial 11 1.00 11.00 Private study hours 176.00 Total Contact hours 74.00 Total hours (100hr per 10 credits) 250.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Weekly tutorials. Examples sheets marked and returned with feedback.

!!! In order to pass the module, students must pass the examination. !!!

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Written Work Example sheets and project work 20.00 Total percentage (Assessment Coursework) 20.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 00 mins 80.00 Total percentage (Assessment Exams) 80.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated