# PX129 Tutorial

## Coordinator: Michael Pounds

## Weighting: 12 CATS

The tutor's mark is made up from marks for answers to the assessed weekly problems (50%) and from work associated with five worksheets (50%). The worksheets cover some background mathematical material assumed by other modules. The material covered includes complex numbers, vectors, matrices, multiple integration and integration along surfaces and contours.

*Please note that this module is only taken by students on the 3 year BSc (GF13) and the 4 year MMathPhys (FG33) programmes in Mathematics and Physics.*

**Aims:**

To cover some background mathematical material assumed by other modules, to give experience of learning by self-study and to develop the habit of keeping up with the problem sheets handed out in physics modules

**Objectives:**

You should become familiar with complex numbers, vectors and matrices, multiple integration and integration over lines, surfaces and volumes at a level necessary to cope with all first year physics modules and to start the second year core module, MA231 Vector Analysis . You should have attempted the questions on the physics problem sheets and handed in your answers to the designated questions for the core physics modules.

**Syllabus:**

**Worksheets:**

**Complex Numbers**Their construction from the reals; norm, argument, real and imaginary parts; addition, subtraction, multiplication and division; the Argand diagram and geometric view of complex numbers. de Moivre's theorem, exponential representation of a complex number in terms of its norm and its argument.

**Vectors**Vectors have magnitude and direction. Addition and subtraction, the null vector. Geometry of simple figures written in vector notation, equation of lines and planes, equation for centroid of a triangle. The dot product, the normal to a plane and alternative form for equations of planes, perpendiculars from points of a triangle to opposite sides meet at a point. Cross-product and the notion of an area in three dimensions as a vector. Equation of line of intersection of two planes. Triple scalar product, associative law, relation to volume of parallelopiped. Triple vector product

**Matrices**Motivation and definition. The 2 x 2 case: operations on vectors. Eigenvalues and eigenvectors. Diagonalizing matrices. Exponential of a diagonalizable matrix. Mention of the 3 x 3 and N x N cases.

**Multiple Integration**Integration of functions of more than one variable. The domain of integration and changing the order of integration. Computing the mass of an object with variable density. Changing variables and the Jacobian with particular reference to the transformation cartesian to polar coordinates

**Integration along Lines, Surfaces and Volumes**Notation for integration of both scalar and vector quantities over lines, surfaces and volumes. Integration along lines using parameterised curves, circulation around a contour. Infinitesimal surface element as a vector in 3D, use to compute flux across a surface. Volume integrals and revision of the Jacobian.

You should answer the questions on each of the worksheets and hand in your answers to your personal tutors as directed.

**Weekly Problem Sheets**

You will be asked to hand in written answers to designated problems from the problem sheets or attempt designated problems from the Mastering Physics package.

**Assessment:** Worksheets 50% + Physics Problems 50%

This module has a home page. Details of the assessment are included there.

**Leads to:** First and Second Year Physics modules, MA259 Multivariable Calculus.