MMath Mathematics / Course details

The unit introduces the basic ideas of complex numbers relating them to the standard rational and transcendental functions of calculus. The core concepts of limits, differentiation and integration are revised. Techniques for applying the calculus are developed and strongly reinforced. Vectors in two and three dimensions are introduced and this leads on to the calculus of functions of more than one variable, vector calculus, integration in the plane, Green's theorem, Stokes' theorem and Gauss' theorem.

The course unit unit aims to provide a firm foundation in the concepts and techniques of the calculus, including real and complex numbers, standard functions, curve sketching, Taylor series, limits, continuity, differentiation, integration, vectors in two and three dimensions and the calculus of functions of more than one variable.

On successful completion of this module students will be able to: 

• Manipulate complex numbers and use them to relate trigonometric, hyperbolic and exponential functions.
• Evaluate complex roots and represent complex numbers in the complex plane, using results such as Euler's formula and de Moivre's theorem.
• Sketch polynomial, rational, inverse and some standard functions of a single variable, expressed in Cartesian or polar coordinates.
• Evaluate and interpret limits and derivatives of algebraic functions, including functions expressed in implicit or parametric form.
• Select and deploy methods for evaluating integrals of functions of a single variable.
• Construct and manipulate Taylor series of scalar functions of one and two variables.
• Manipulate vectors using tools such as scalar and vector products, deploying these quantities to solve geometric problems.
• Construct, deploy and interpret derivatives of scalar- and vector-valued functions of more than one variable.
• Construct, evaluate and interpret simple integrals of functions of two variables, using tools such as the Jacobian and Green's theorem.
• Construct and evaluate line integrals for conservative and non-conservative vector fields.
• Locate and classify extrema of functions of two variables, using discriminants and Lagrange multipliers.

1. Introducing Complex Numbers:  imaginary numbers; complex numbers; standard and polar form; complex plane; algebra of complex numbers; complex conjugate. 

2. Functions: general definition of function; modulus function; polynomials; sketching real, complex and parametric functions; combining functions; inverse functions; trigonometric functions; properties of functions: continuity; symmetry; periodicity; monotonicity.

3. Limits: basic concept of a limit and definition of limit of a function; infinity; discontinuity; left and right limits; limits of combinations of functions.

4. Power series: notion of power series as the limit of a sum of polynomials; convergence; exponential function defined as power series; power series expressions for trigonometric functions; hyperbolic functions and their relationship to trigonometric functions.

5. More Complex Numbers:  Euler's formula and de Moivre's theorem; exponential form of complex numbers; roots of unity; fundamental theorem of algebra; transformations of complex numbers.

6. Differentiation: fundamental definition of derivative; l'Hopital's rule; sums, products, quotients and chain rule; derivatives of inverse functions; turning and critical points; Taylor series; radius of convergence; derivatives of implicit and parametric functions; logarithmic differentiation.

7. Integration: definite and indefinite integrals; fundamental theorem of calculus; proper and improper integrals; techniques for integration; lengths of curves, surfaces and volumes of revolution.

8.  Vectors in two and three dimensions:  Cartesian coordinates, position vectors; vectors as directed line segments (magnitude, direction); addition, subtraction; multiplication by a scalar; unit vectors; inner/dot product;  angle between two vectors; orthogonality; components and projections;  extension to higher dimensions; vector/cross product in three dimensions;  scalar triple product and volume; vector representation of points, lines and planes and relations between them.  Orthogonal coordinate systems (polar, parabolic, elliptic, cylindrical and spherical).

9. Functions of more than one variable:  partial derivative (definition, notation evaluation); chain-rule, higher partial derivatives including mixed derivatives; higher dimensional Taylor expansion; gradient vector (grad); critical or turning points (maxima, minima, saddle-points); relationship with contour lines and gad; identifying critical points; derivative test and use of the discriminant; turning points with constraints;  Lagrange multipliers; div, curl and related identities.

10. Multiple Integrals: area, integration of a scalar-valued function in the plane; choice of order of integration; area integral in polar coordinates;  Jacobian and change of variable; directed path or line integral of a vector valued function in the plane; path-dependence/independence; conservative vector fields; potential function.

Supervision attendance and participation; Weighting within unit 10%

Coursework; In-class test in week 6, weighting within unit 10%

Examination; Weighting within unit 80%

Feedback seminars will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

James Stewart, Calculus, Early Transcendentals, International Student Edition, Thomson (any recent edition).

[This text covers almost every aspect of what you will be learning, with many examples. You should ensure that you can have easy access to a copy.

Useful background material can be found in:

Hugh Neill and Douglas Quadling. Cambridge Advanced Mathematics Core 3 and 4.

[This text describes well and clearly what should be known from A-level. It (as for other A-level texts) provides an introduction to what should be known before entering a university calculus course.]

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  Please see Blackboard / course unit related emails for any further updates