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This module aims to provide the required mathematical expertise for Earth science study and research.

After completing this course, the student should be able to:
•ÌýÌý Ìýunderstand the relation between the hyperbolic and exponential functions;
•ÌýÌý Ìýdifferentiate simple functions and apply the product and chain rules to evaluate the differentials of more complicated functions;
•ÌýÌý Ìýfind the positions of the stationary points of a function of a single variable and determine their nature;
•ÌýÌý Ìýunderstand integration as the reverse of differentiation;
•ÌýÌý Ìýevaluate integrals by using substitutions, integration by parts, and partial fractions;
•ÌýÌý Ìýunderstand a definite integral as an area under a curve and make simple numerical approximations;
•ÌýÌý Ìýdifferentiate up to second order a function of 2 or 3 variables and test when an expression is a perfect differential;
•ÌýÌý Ìýchange the independent variables by using the chain rule and work with polar coordinates;
•ÌýÌý Ìýfind the stationary points of a function of two independent variables and show whether these correspond to maxima, minima or saddle points;
•ÌýÌý Ìýevaluate line integrals along simple curves in three-dimensional space;
•ÌýÌý Ìýmanipulate real three-dimensional vectors, evaluate scalar and vector products, find the angle between two vectors in terms of components;
•ÌýÌý Ìýconstruct vector equations for lines and planes and find the angles between them, understand frames of reference and direction forÌý Ìý Ìý Ìý Ìý Ìý Ìý Ìý Ìý Ìý Ìý Ìýinterception using vectors;Ìý
•ÌýÌý Ìýexpress vectors, including velocity and acceleration, in terms of basis vectors in polar coordinate systems;
•ÌýÌý Ìýunderstand the concept of convergence for an infinite series and apply simple tests to investigate it;
•ÌýÌý Ìýexpand an arbitrary function of a single variable as a power series (Maclaurin and Taylor), make numerical estimates and apply l’Hôpital’s ruleÌý Ìý Ìý Ìýto evaluate the ratio of two singular expressions;
•ÌýÌý Ìýrepresent complex numbers in Cartesian and polar form on an Argand diagram;
•ÌýÌý Ìýperform algebraic manipulations with complex numbers, including finding powers and roots;
•ÌýÌý Ìýapply de Moivre’s theorem to derive trigonometric identities and understand the relation between trigonometric and hyperbolic functions usingÌý Ìý Ìý Ìýcomplex arguments.

This module is restricted to Earth Sciences students only who have A-Level Maths or equivalent.