MATH10121 - 2006/07
- Title: Calculus and Vectors
- Unit code: MATH10121
- Credits: 20
- Prerequisites: A-Level Mathematics
- Co-requisite units: None
- School responsible: Mathematics
- Member of staff responsible: Prof J Dold
The programme 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.
Brief Description of the unit
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 and Stokes' theorem.
On successful completion of this module students will have acquired an active knowledge and understanding of the main concepts and techniques of single and multivariable calculus.
Future topics requiring this course unit
Almost all Mathematics course units will rely on material covered in this course unit.
- Foundations: algebra of real and complex numbers; simple functions based on power-laws, their graphs and basic properties (including limits at zero and plus or minus infinity); derivative (slope) of a power-law function and its integral (area under a curve and indefinite integral); differentials as describing tangent lines; definitions, graphs, derivatives and limits of sine, cosine, exponential and hyperbolic sine and cosine; basic trigonometric and hyperbolic identities.
- Curve sketching: Cartesian and polar coordinates; simple conic sections; rational functions, turning points, asymptotic behaviour; parametric curves.
- Power series: notion of a power series as a limit of polynomials, its derivative and integral; Taylor series; series for exponential, sine, cosine and other functions; radius of convergence; truncation of power series, error terms and big-O notation.
- More on complex numbers: Euler's formula and de Moivre's theorem; polar form of complex numbers; roots of unity; polynomials with real coefficients; complex forms of sine and cosine, relationship to trigonometric and hyperbolic identities.
- More on functions: functions, domain and range; sums, products, quotients, composition; inverse functions; rational, algebraic and transcendental functions; standard functions and their inverses; transformation, scaling, shifting, change of variable; symmetry, periodicity, increasing, decreasing, monotonicity.
- More on limits and differentiation: basic notions of limit and continuity; discontinuities, left and right limits; limits of sums, products, quotients, compositions; finding some limits, l'H˘pital's rule; definition of derivative; derivatives of inverse functions; sums, products, quotients and chain rule; derivatives of implicit and parametric functions; logarithmic differentiation.
- More on integration: definite and indefinite integrals; fundamental theorem of calculus; proper and improper integrals; techniques for integration: linearity, integration by parts, partial fractions, substitution; lengths of curves, surfaces and volumes of revolution.
- Vectors in two and three dimensions: representation as directed line segments (magnitude, direction); choice of axes, components, Cartesian representation; basic properties, addition, subtraction; scalar and vector product; representation of lines, planes, curves and surfaces; polar representation and relation to complex numbers in 2 dimensions; cylindrical and spherical polar representation in 3 dimensions; other orthogonal coordinate systems.
- Functions of more than one variable: partial derivative, chain-rule, Taylor expansion; turning points (maxima, minima, saddle-points); Lagrange multipliers; grad, div, curl and some useful identities in vector calculus; integration in the plane, change of order of integration; Jacobians and change of variable; line integrals in the plane, path-dependence, path independence; Green's and Stokes' theorem in the plane.
The following text is strongly recommended:
- James Stewart, Calculus, Early Transcendentals, Thomson, 5th Edition, International Student Edition, 2003.
- [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 & 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.]
Teaching and learning methods
Four lectures and one supervision class each week.
- Supervision attendance and participation; Weighting within unit 10%
- Coursework; Weighting within unit 15%
- Two and a half hours end of semester examination; Weighting within unit 75%