Calculus and Analysis
Module code: MA1014
The branches of Mathematical Analysis, and differential and integral calculus in particular, form an essential part of the toolbox of any pure or applied mathematician, scientist or engineer. They lead on to the important topics of Differential Equations and of Complex Analysis, and form the foundation for Mathematical Modelling of real-life problems.
The history of analysis in Europe began in the 17th century, with the work of Newton, L’Hopital, and Leibniz, continued through the 18th with the work of Euler, of Lagrange, and of Taylor, and into the 19th with Fourier, Cauchy, Riemann, Bolzano, Weierstrass, Stokes and Green. Work of most of these pioneering analysts will be explained during the 1st year, except that of Stokes and Green which will be covered in 2nd year Calculus and Analysis!
The central (but hard!) idea behind all of Calculus and Analysis is the idea of a limit. We start the module with this. The idea of limit then gives us a specific and rigorous way to understand lots of intuitive ideas you might already know: what is meant by saying that a function is continuous, and what is meant by the gradient of a tangent line, what precisely is an integral, and so on. The same idea also lets us understand many new things -when does an infinite sequence of numbers converge to a limit, when does an infinite sum converge to a limit, when does a function have a Taylor series representation, and so forth.