First-year syllabus extracts

The following are brief descriptions of three of Durham's current first-year mathematics honours courses, to give you an indication of the level of knowledge assumed in our second year.

MATH1061/1081 Calculus l (two terms)

Calculus is a fundamental part of mathematics and provides a foundation for all your future mathematical studies. This course will seek to consolidate and expand your knowledge of this topic and is designed to be completely accessible to the beginning calculus student. The three basic concepts of calculus will be covered, namely, limits, differentiation and integration. The emphasis of this module is on concrete methods for calculation, while the Analysis I module will revisit the above concepts and provide a deeper knowledge with a more formal approach.

First and second order ordinary differential equations are studied together with solution methods that are naturally associated with the techniques of integration.

Taylor and Fourier series are also covered, in preparation for their application in later modules.

Numerous exercises are provided to reinforce the material.

Outline of course

Aim: To master a variety of methods for solving problems and acquire some skill in writing and explaining mathematical arguments.

Term 1 (30 lectures)

• Elementary Functions of a Real Variable: Domain and range. Graphs of elementary functions. Even and odd functions. Exponential, trigonometric and hyperbolic functions. Algebraic combinations and composition. Injective, surjective and bijective functions. Theorem of inverse functions. Logarithm function as inverse of exponential function; inverse trigonometric functions.
• Limits and Continuity: Informal treatment of limits. Calculation of limits. Vertical and horizontal asymptotes. Continuity at a point and on intervals.
• Differentiation: Derivative as slope of tangent line. Differentiability and continuity. Product, quotient and chain rule. Implicit differentiation. Differential equations. Derivative as rate of change. Increasing and decreasing functions. Max-min problems.
• Integration: Antiderivatives. Fundamental theorem of calculus. Integration by parts and use of partial fractions to integrate rational functions. Integration of even/odd functions. Gaussian integration.
• Ordinary Differential Equations: First order: separable, exact, homogeneous, linear. Second order linear with constant coefficients, importance of boundary conditions, reduction to a set of first order equations, treatment of homogeneous and inhomogeneous equations, particular integral and complementary function.
• Taylor's Theorem: Taylor polynomials. Statement of Taylor's theorem with Lagrange remainder. Calculation of limits using Taylor's series.
• Fourier Series: Convergence, periodic extension, sine and cosine series, half-range expansion. Parseval's theorem.
• Multiple Integration: iterated sums, double and triple integrals by repeated integration, volume enclosed by surface, Jacobians and change of variables.

Term 2 (25 lectures)

• Functions of several variables: Plotting functions of two variables, sketches, contour maps. Continuity and differentiability. Chain rule. Cylindrical and Spherical Polar Coordinates. Taylor's Theorem for functions of more than one variable (statement only).
• Max/Min problems for functions of more than one variable: Stationary points, maxima, minima, saddle points via the Hessian matrix and its diagonalisation. Constrained variation and the use of Lagrange multipliers.
• Linear Differential Operators in one variable: 2nd order linear differential operators. Eigenvalue problems. Special polynomials as examples of eigenfunctions. Inner products on functions. Reality of eigenvalues and orthogonality of eigenfunctions for Hermitian operators. Fourier Series as an example.
• Linear PDEs and the Wave Equation: The wave equation in one dimension and its general solution. Principle of superposition for linear PDEs. Solution to the wave equation on a finite interval using the method of separation of variables. Generalisation to more than one spatial dimension.
• Fourier Transforms: Frequency analysis for non-periodic functions. Fourier transform as the limit of Fourier Series as the periodicity goes to infinity. Derivation of the inverse Fourier transform. Solving linear differential equations as an application.

MATH1071/1091 Linear Algebra I (two terms)

Techniques from linear algebra are used in all of mathematics. This course gives an introduction to all the major ideas in the topic. The things you learn in this course will be very useful for most modules you take later on.

The first term is concerned with the solution of linear equations and the various ways in which the ideas involved can be interpreted including those given by matrix algebra, vector algebra and geometry. This enables us to determine when a system of equations has a unique solution and gives us a systematic way of finding it. These ideas are then developed further in terms of the theory of vector spaces and linear transformations. We will discuss examples of linear transformations that are familiar from geometry and calculus.

Any linear map can be put into a particularly easy form by changing the basis of the space on which it acts. The second term begins with the solution of the eigenvalue problem which tells you how to find this basis. We then go on to generalise the notions of length, distance and angle to any vector space. These ideas may be used in a surprisingly large range of contexts. We show how all these ideas come together in the applications to geometry and calculus introduced in the first term.

Throughout the course we will also discuss examples for the notion of a group, which is one of the fundamental organizing objects in mathematics.

Outline of Course

Aim: To provide an introduction to all the major ideas in linear algebra.

Term 1

• Vectors in $ℝ^n$ (6 lectures): Vectors, addition and scalar multiplication in $ℝ^n$ with concrete examples in $ℝ^2$ and $ℝ^3$. Scalar product, vector product, triple product. Equations of lines and planes, linear systems of equations in 3 variables. Examples: scalar and vector equations of lines and planes in $ℝ^3$.
• Linear Systems and Matrices (5 lectures): Arbitrary linear systems of equations, Gauss-Jordan elimination. Solutions of linear equations as generalisations of lines and planes in $ℝ^3$. Multiplication and inversion of matrices. Gauss-Jordan elimination using matrix notation.
• Determinants and Groups I (6 lectures): Determinants and explicit methods for their calculation(row and column expansion). Properties of determinants. Examples: areas of parallelograms, volumes of parallelepipeds.
• Vector spaces (7 lectures): Vector spaces and subspaces over $ℝ$. Examples: lines and planes in $ℝ^3$. Linear independence, spanning sets, bases and coordinates, dimension. Vector spaces of polynomials. Affine subspaces. $\mathbf{C}^n$ as a vector space.
• Linear mappings (6 lectures): Definition of linear mapping, matrices as linear mappings in ℝn (examples: dilations, projections, reflections, rotations in $ℝ^2$ and $ℝ^3$). Differentiation and integration as a linear mapping (example: polynomials). Representation of linear mappings by matrices. Composition of linear mappings and matrix multiplication. Kernel (row and column), rank and image of a linear mapping.

Term 2

• Change of basis and diagonalisation (7 lectures): Change of basis and of coordinates for linear maps. Eigenvalues and eigenvectors. Explicit calculation with characteristic polynomial. Diagonalisation by change of basis.
• Inner product spaces (8 lectures): Definition and examples: $ℝ^n$, $\mathbf{C}^n$, polynomials. Cauchy-Schwarz inequality. Orthonormal bases and Gram-Schmidt procedure. Orthogonal and unitary matrices. Examples: projection, reflections and distances in $ℝ^2$ and $ℝ^3$. Orthogonal complement of a subspace. Diagonalisation of symmetric matrices by orthogonal matrices.
• Linear differential operators (3 lectures): 2nd order linear differential operators. Special polynomials as eigenfunctions.
• Groups II (4 lectures): More examples of linear groups: $\mathrm{O}(n)$, $U(n)$. Modular arithmetic. Matrix realisation of symmetry groups of polygons ($\mathbf{Z}_n$, dihedral groups). Axioms of groups. Examples: symmetric groups, $\mathrm{GL}(n)$, $\mathrm{SL}(n)$.

MATH1607 Dynamics I (one term)

Dynamics concerns evolution with time. In this course we study a model of time-development called "classical mechanics". This applies to the world around us and describes the motion of everyday objects via "forces". It was invented by Isaac Newton in the 17th century, when it stimulated revolutions in astronomy, physics and mathematics. Today it is a cornerstone of applied science.

This introductory course treats firstly the motion of point particles, and then the motion of a certain extended body - a flexible stretched string.

We use what you have covered in Calculus I (ordinary differential equations, partial differentiation, Fourier series) and Linear Algebra I (vectors). It is vital to be familiar with this material!

The Dynamics course leads on naturally to the second-year courses Mathematical Physics II and Analysis in Many Variables II.

Outline of course

• Equations of motion, force, mass, momentum, projectiles, Lorentz force on charged particles in constant electromagnetic fields.
• Concepts of energy and angular momentum.
• Simple harmonic motion and oscillations about a stable equilibrium; damped oscillations and resonance.
• Central forces and the use of energy and angular momentum to study orbits.
• Waves on strings, including the derivation of the wave equation and its general solution.
• Rigid bodies and moments of inertia.