# 2M1: Functions of two variables and PDEs

This part of the course, dealing with functions of two variables and partial differential equations (PDEs), is taught during weeks 5,7,8,9 and 10 in Renold H11 on Monday 11 a.m. by Prof. Matthias Heil. This page provides online access to the lecture notes, example sheets and other handouts and announcements. Most of the material will be taught in "chalk and talk" mode. If OHP transparencies are used, copies will be made available (after the lecture) on this page. Please consult the service course page for details on how to get hold of material for the other parts of the course.

Please note that the lecture notes only summarise the main results and will generally be handed out after the material has been covered in the lecture. You are expected take notes during the classes.

NOTE: The various links on this page (probably) won't work from the mirrored version on the school's service course pages. Please go to the original at http://www.maths.man.ac.uk/~mheil/Lectures/2M1/index.html before trying to download any of the handouts.

## Syllabus:

• Functions of two variables and practical examples. Quick reminder of ordinary and partial derivatives; reminder of Taylor expansions and maxima/minima for functions of one variable. Generalisation to 2D. Criteria for minima/maxima/saddle points. 2D Taylor series.
• What are PDEs? Where do they arise? What are their solutions? Examples and their physical/engineering origin: 1D advection, 2D Laplace, 1D unsteady heat and 1D linear wave equations. Boundary and initial conditions. How to verify that a function is a solution of a PDE. Solving PDEs by separation of variables.

## Do-able questions

 Date Topics covered Do-able questions 26/10/09 Review of functions of a single variable (plots; derivatives; stationary points; Taylor series). Motation for fcts of multiple (two) variables; partial derivatives; plots; necessary conditions for a stationary point. Example Sheet I: Q1 and the first half of Q2. 09/11/09 Classification of stationary points; 2D Taylor series. Example. All questions on Example Sheet I. 16/11/09 Reminder of ODEs: What are they? BC/IC and BVP/IVP. Physical examples. What are PDEs? How do we check if a candidate solution solves a given BVP/IVP? Examples of PDEs, their physical background and suitable BC/IC: 1D Advection, 2D Laplace, 1D unsteady heat, 1D linear wave. Question 1 on Example Sheet II. 23/11/09 Separation of variables: A step-by-step guide, illustrated for the linear wave equation. All questions on Example Sheet II. 30/11/09 Separation of variables for the unsteady heat equation. All questions on Example Sheet II.

## Assessment:

The course will be examined in a two hour exam in January/February 2010.

Please note a few corrections for previous handouts (the files above have already been corrected).