## MATH1005 Core Mathematics

### 50 creditsClass Size: 299

Module manager: Dr Philip Walker
Email: P.Walker@leeds.ac.uk

Taught: Semesters 1 & 2 (Sep to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

Grade B in A-level Mathematics or equivalent.

### 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

Module replaces

MATH1010, MATH1012

This module is not approved as a discovery module

### Module summary

This module introduces students to several fundamental topics of mathematics.Calculus is an essential tool in many areas of mathematics. This module consolidatesbasic calculus material from A-level, extending it to more advanced techniques, suchas functions of several variables, and introducing elements of the analysis required toformalise the subject. These techniques lead to methods for solving simple ordinarydifferential equations, which are applied to problems in Newtonian mechanics. Linearalgebra provides a basis for wide areas of mathematics and this module provides theessential foundation for this topic. Students will complement theoretical work withprojects and assignments using a mathematical programming package.

### Objectives

Learning outcomes
.
On completion of this module, students should:
Calculus
•be able to differentiate functions of one variable and determine the location and nature of turning points;
•be able to compute the Taylor series of functions of one variable;
•be able to use a variety of methods to integrate simple functions;
•be aware of the analytical basis of calculus as expressed in rigorous definitions and theorems such as the Fundamental Theorem of Calculus;
•be comfortable with the calculus of several variables;

Linear algebra
•be able to solve systems of equations by row reduction;
•be able to manipulate matrices and vectors and understand their basic properties;
•understand properties of linear algebra such as linear dependence, kernel, range and basis;
•be able to compute eigenvalues and eigenvectors of matrices;
•be able to diagonalise matrices and perform a change of basis;

Ordinary differential equations and mechanics
•be able to use a variety of methods to solve a first-order differential equations and simple second-order differential equations;
•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;
Computational mathematics
•appreciate the value and limitations of computational methods, and be able to perform simple computational tasks using a mathematical programming package.

•have demonstrated problem solving and modelling, communication, and group-working skills.

### Syllabus

Calculus
•Functions and their inverses. Continuity and discontinuity. Graphs of functions.
•Differentiation. Calculations from first principles. Non-differentiability.
•Chain rule, product rule, extrema, Taylor series.
•Intermediate value, Rolle's and Mean Theorems.
•Functions of several variables.
•Partial derivatives, directional derivatives, multivariable chain rule.
•Stationary points of functions of two variables. Conditions for a stationary point. Criteria for extrema. Lagrange multipliers.
•Gradients of scalar functions. Tangent planes.
•Implicit differentiation. Change of variables. Solution of exact equations.
•Integration. Areas under curves. Riemann integration. Calculations from first principles.
•Definite and indefinite integrals. Integration techniques.
•Fundamental theorem of the calculus.
Linear algebra
•Systems of equations. Gaussian elimination. Echelon form.
•Vectors and matrices. Inverses. Transposes.
•Determinants. Computation.
•Vectors in 2 and 3 dimensions. Dot and cross products. Geometrical interpretation.
•Subspaces, bases and dimensions.
•Linear combinations and dependence. Kernel and range.
•Eigenvalues and eigenvectors. Diagonalisation.
Ordinary differential equations and mechanics
•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.
Computational mathematics
•Introduction to a mathematical programming package as a tool for numerical, graphical and symbolic computation.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 16 1.00 16.00 Lecture 99 1.00 110.00 Tutorial 22 1.00 22.00 Private study hours 363.00 Total Contact hours 148.00 Total hours (100hr per 10 credits) 511.00

### 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.

### 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.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) (S1) 2 hr 16.80 Standard exam (closed essays, MCQs etc) 3 hr 28.00 Standard exam (closed essays, MCQs etc) 2 hr 19.60 Online Time-Limited assessment 48 hr 7.20 Online Time-Limited assessment 48 hr 8.40 Total percentage (Assessment Exams) 80.00

In order to pass the module, students must achieve a passing standard on all the assessments