Aims

• provide the mathematical foundations required for all the first semester and some of the second semester courses in the first year of the physics and astronomy programmes;
• prepare students for the second semester follow-on mathematics course PHYS1B46;
• give students practice in mathematical manipulation and problem solving.

Objectives

After completing this half-unit course, the student should be able to:

• solve simultaneous and quadratic equations with examples taken from physical situations;
• sum arithmetic, geometric and other simple series;
• appreciate the relation between powers, exponentials and logarithms and the more general concept of the inverse function in terms of a graphical approach;
• derive the values of the trigonometric functions for special angles;
• understand the relation between the hyperbolic and exponential functions;
• differentiate simple functions and apply the product and chain rules to evaluate the differentials of more complicated functions;
• find the positions of the stationary points of a function of a single variable and determine their nature;
• understand integration as the reverse of differentiation and to use this to evaluate integrals almost 'by inspection';
• evaluate integrals by using substitutions and integration by parts;
• understand a definite integral as an area under a curve;
• evaluate the Gaussian, Feynman, Gamma and Breit Wigner (Lorentzian) integrals and generate further definite integrals by differentiation w. r. t. a parameter;
• differentiate up to second order a function of 2 or 3 variables and be able to test when an expression is a perfect differential;
• change the independent variables by using the chain rule and, in particular, work with polar coordinates;
• find the stationary points of a function of two independent variables and to determine their nature;
• find the stationary points of function of two or more variables subject to constraints (Lagrange multipliers);
• manipulate real three-dimensional vectors, evaluate scalar and vector products, find the angle between two vectors in terms of components;
• construct vector equations for lines and planes and find the angles between them;
• express vectors, including velocity and acceleration, in terms of basis vectors in polar coordinate systems;
• understand the concept of convergence for an infinite series, be able to apply simple tests to investigate it;
• expand an arbitrary function of a single variable as a power series (Maclaurin and Taylor), make numerical estimates, and be able to apply L'Hopitals rule to evaluate the ratio of two singular expressions;
• represent complex numbers in Cartesian and polar form on an Argand diagram.
• perform algebraic manipulations with complex numbers, including finding powers and roots;
• apply de Moivres theorem to derive trigonometric identities and understand the relation between trigonometric and hyperbolic functions through the use of complex arguments. components;

Copyright © 2004-2006 UCL HEP group,   (last modified 15 Aug 2006)