The course is conducted in English。Intensive courses。

In the final segment of Calculus (MATH4009), two main topics will be discussed. Firstly the course focuses on how calculus can be applied to 'vector fields.' Vector fields, which are vector-valued functions originated from Physics, will be thoroughly examined with an emphasis on understanding integrals over curves and surfaces. In particular, Green’s, Stokes', and Divergence Theorem will be presented as generalizations of the Fundamental Theorem of Calculus for, respectively, line and surface integrals. An application, we will derive the Gauss' Law, offering insights into the flux of an inverse square field across a closed surface. To conclude our study of Calculus, we will introduce the definitions of limits for sequences and series, addressing the theoretical foundations for the introduction of 'power series.' As a generalization of `polynomials', power series find application in representing both elementary and advanced functions, thereby laying the groundwork for a more sophisticated analysis of functions, essential in practical contexts. The key topics covered include: 1. Line integrals 2. Green’s Theorem and Conservative vector fields 3. Surface integrals and Flux 4. Stokes’ and Divergence Theorem 5. Series : definition and convergence tests 6. Power series : radius of convergence and their calculus 7. Taylor's theorem and its applications

On successful completion of this module students should be able to : (1) Parametrise curves and surfaces in Cartesian and other coordinates, including polar, cylindrical and spherical coordinates (2) Understand and be able to calculate line, surface integrals with respect to various coordinate systems. (3) Understand and prove properties of a conservative vector field (4) State the Green's, Divergence and Stokes' Theorems and use them to aid calculations (5) Apply these techniques to problems in mechanics (work done, circulation and flux) (6) Analyse convergence and divergence of sequences and series (7) Apply basic properties and calculus of a power series (8) State and apply the Taylor's Theorem to resolve problems about smooth functions (9) Approximate an infinite series by a partial sum and be able to estimate the error incurred

Assumed knowledge : - MATH4006, 4007, 4008, - Basic trigonometry, vector geometry, - Determinants of 2x2 and 3x3 matrices (knowledge in linear algebra will be useful but not necessary)

After each week of lectures, you are expected to - revise examples from the lectures, - complete relevant sections on WebWork, - complete weekly assessed/non-assessed assignment.

Stewart, Clegg, Watson, CALCULUS: EARLY TRANSCENDENTALS, Metric, 9th Edition

Instructor's lecture notes, J. Marsden, A Tromba, Vector Calculus (4th Edition), S. Lang, Calculus of Several Variables (3rd Edition).