Synopsis for Introductory Calculus

Overview

These lectures are designed to give students a gentle introduction to applied mathematics in their first term at Oxford, allowing time for both students and tutors to work on developing and polishing the skills necessary for the course. It will have an `A-level' feel to it, helping in the transition from school to university. The emphasis will be on developing skills and familiarity with ideas using straightforward examples.

Learning Outcomes

At the end of the course students will be able to solve a range of Ordering Differential Equations and linear systems of first order Ordinary Differential Equations (ODEs).

Synopsis

General linear homogeneous ODEs: integrating factor for first order linear ODEs, second solution when one solution is known for second order linear ODEs. First and second order linear ODEs with constant coefficients. General solution of linear inhomogeneous ODE as particular solution plus solution of homogeneous equation. Simple examples of finding particular integrals by guesswork. Systems of linear coupled first order ODEs. Calculation of determinants, eigenvalues and eigenvectors and their use in the solution of linear coupled first order ODEs.\vspace*{5mm}

Parabolic, Spherical and Cylindrical polar coordinate systems. Introduction to partial derivatives. Chain rule, change of variable, Jacobians with examples including polar coordinate systems. Solving some simple partial differential equations.

Surfaces. Sketching simple quadrics. Gradient vector; normal to surface, directional derivative. Taylor's Theorem for a function of two variables (statement only). Critical points and classification using directional derivatives and Taylor's theorem. Informal (geometrical) treatment of Lagrange multipliers.

D. W. Jordan & P. Smith, Mathematical Techniques (Oxford University Press, 3rd Edition, 2003), \quad Chapters 1–4, 14–17.