Math IA, 2002-2003

Supervisor: Stephen Thompson

Introduction

I hope that we will have a good year learning Maths together. As the year passes by information about supervision work will be regularly posted here.

Supervision timetable

Location: Teaching room B Robinson College Cambridge

Monday Room J8:
5.00-6.00 p.m. Alastair Newman, Ed Parrott, Matt Davis

Wednesday Room J8:
5.00-6.00 p.m. Richard Knight,Oliver Sawers
6.00-7.00 p.m. Tom Diplock, David Grant

Exam terms supervision work

Week six
Questions from 2001 papers

Week five
Questions from 2000 papers

Week four
Questions from 1999 papers

Week three
Qu: 10,12,13,14,15,20,21,22,23
Matrix questions from 1998 papers

Week two
Qu: 1,2,3,4,5,6,8,9,11

Week one
Holiday work (1998 papers).

Lent terms supervision work

Week eight

Questions from the two sheets of tripos questions you have been given.

Week severn
Questions 18,19,20,21,22,24,25,26 from 2nd B course question sheet.

Week six
Questions 11,12,13,14,15,16 from 2nd B course question sheet.

Week five
Questions 1,2,4,6,7,8,9,10 from 2nd B course question sheet.
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week four
Questions 18,19,20 from B course question sheet.
three tripos question given out last supervision.
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week three
Questions 13,14,15,16,17 from B course question sheet.
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week two
2003 Christmas Exam in postscript or pdf format. Do corrections and the question you didn't attempt.
Questions 1,2,3,4,5,6,8,9,11 from B course question sheet.
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week one
Holiday work:
Questions: M 1,3,4,5; N1, and the tripos questions on the sheets you will have been given.
Please also revise this terms work in preparation for your exam after christmas

Michaelmas terms supervision work

Holiday work:
Questions: M 1,3,4,5; N1, and the tripos questions on the sheets you will have been given.
Please also revise this terms work in preparation for your exam after christmas

Week Severn Supervision: 1/12/02
Questions: L 1,2,3,4,5,6,7,9,10,11
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week Six Supervision: 25/11/02
Questions: J 19,20,22; K 3,4,5,7
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week Five Supervision: 18/11/02
Questions: I 1,2, 5; J 7,8,10,11,12,17,18
If you do not feel comfortable with complex numbers (you have only just meet them) then I recommend you try J1, J2, J3 and J5 also.
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week Four Supervision: 11/11/02
Questions: G3, H 1,2,3,4,5,6,7,8,9
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week Three Supervision: 04/11/02
Questions:F 1,3,4,5,6,7,8,9,10,11,12; G3
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week Two Supervision: 28/10/02
Questions: C 1,3,4,6,10,11,12; D 1; E 1,2,3
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Week One Supervision: 21/10/02
Questions: A 1,2,3,4,5,7; B 1,2,3,5,6
Put answers in my box in Robinson Porters lodge 24 hours before your supervision.

Book recommended:

K.F. Riley, M.P. Hobson, S.J. Bence: Mathematical Methods for Physics and Engineering; Cambridge University Press

Syllabus 2000-2001

Mathematical Methods I

(24 lectures, Michaelmas term)

Vector sum and vector equation of a line. Scalar product, unit vectors, vector equation of a plane. Vector product, vector and scalar triple products. Orthogonal bases. Cartesian components. Spherical and cylindrical polar coordinates. [5 Lectures]

Revision of single variable calculus. Elementary curve sketching. Idea of continuity and differentiability of functions. Orders of magnitude and approximate behaviour for large and small x . O notation. Leibnitz's formula. Integral as a sum, differentiation of an integral with respect to its limits or a parameter. Approximation of a sum by an integral. Stirling's approximation as an example. Schwarz's inequality. [5 Lectures]

Double and triple integrals in Cartesian, spherical and cylindrical coordinates. Examples to include evaluation of . [3 Lectures]

Power series. Statement of Taylor's theorem. Examples to include the binomial expansion, exponential and trigonometric functions, and logarithm. Newton-Raphson method. [2 Lectures]

Complex numbers and complex plane, vector diagrams. Exponential function of a complex variable. , complex representations of cos and sin. Hyperbolic functions. [2 Lectures]

Ordinary differential equations. First order equations : separable equations; linear equations, integrating factors. Second-order linear equations with constant coefficients; , as trial solution, including degenerate case. Superposition. Particular integrals and complementary functions. Constants of integration and number of necessary boundary/initial conditions. Particular integrals by trial solutions. [5 Lectures]

Differentiation of functions of several variables. Differentials, chain rule. Unconditional stationary values, maxima and minima. [2 Lectures]

Mathematical Methods II

(first 16 lectures, Lent term)

Elementary probability theory. Simple examples of conditional probability. Probability distributions, discrete and continuous, normalisation. Binomial distribution, ( p + q ) ⁿ, binomial coefficients. Normal distribution. Expectation values, mean, variance,

. [3 Lectures]

Exact differentials, illustrations including Maxwell's relations. Conditional stationary values, Lagrange multipliers, examples with two or three variables. Boltzmann distribution as an example. [4 Lectures]

Scalar and vector fields. Gradient of a scalar as a vector field. Line integral of a vector field. Conservative and non-conservative vector fields. Vector area. Surface integrals and flux of a vector field over a surface. Divergence of a vector field. as div grad. Curl. Divergence and Stokes's theorems. [6 Lectures]

Orthogonality relations for sine and cosine. Fourier series. Examples. [3 Lectures]

Computing Techniques and Applications

(6 lectures, Lent term)

Excel I

Numerical hygiene: convincing but erroneous answers, perils of using 32-bit numbers. Statistical analysis: standard error of the mean, linear regression. [1 Lecture]

Excel II

Differential equations: linear versus non-linear. Two case studies: wide amplitude simple pendulum, starting transient when switching on a LRC circuit. Graphical output. [1 Lecture]

Mathcad I

Vectors, tables, functions, plots. Simple iteration, seeded iteration. Newton-Raphson. Solve blocks. Matrices. [1 Lecture]

Mathcad II

Use of concurrent iterations to solve high-order differential equations. Symbolic computing (Maple) : differentiation, integration, series expansion, linear algebra. Case study : symbolic versus numerical inversion of Hilbert's matrix. [1 Lecture]

Assessed Exercise Briefing

Description of the problem. Outline of the solution. Assessment formalities. [1 Lecture]

Miscellany

Monte Carlo Techniques. Binary chop. Excel macros. Advanced features of Mathcad. [1 Lecture]

Mathematical Methods III

(12 lectures, Easter term)

Linear equations. Notion of a vector space; linear mappings. Matrix addition and multiplication. Use of the summation convention. Examples of applications. Determinant of a matrix. Statement of main properties of determinants. Inverse matrix. Equations

with non-zero solutions. Symmetric, antisymmetric and orthogonal matrices. Eigenvalues and eigenvectors for symmetric matrices. [6 Lectures]

Linear second-order partial differential equations; physical examples of occurrence, verification of solution by substitution, method of separation of variables (Cartesian coordinates only). [4 Lectures]

Elementary Analysis; idea of convergence and limits. Convergence of series; comparison and ratio tests. [2 Lectures]

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