This course will be conducted in English.

In this final module of Calculus (MATH4009), we will develop calculus on vector-valued functions (or vector fields in Physicists' terminalogy). These functions arise naturally from Physics and we will discuss how to make sense of integrals of them over curves and surfaces. We will discuss Stokes' and Divergence Theorem which can be regarded as (far reaching) extensions of the Fundamental Theorem of Calculus for line and surface integrals, and both theorems specialise to the Green's Theorem in a two-dimensional scenario. As an application, we will derive the Gauss' Law that describes the flux of an inverse square field across a closed surface.

Finally, to complete the discussion on limits of a function or a (infinite) sum of functions in the course of the study of Calculus, the definitions of limits of sequences and series are also introduced, which provide the theoretical basis of the introduction of a `power series'. `Power series' is a generalisation of polynomials and can be used to represent elementary as well as more general functions, which paves the way for more advanced analysis of functions, necessary in practical applications.

Definitions are discussed and the most important theorems are derived in the lectures with a view to help students to develop their abilities in logical deduction and analysis. Practical applications of Calculus in various fields are illustrated in order to promote a more organic interaction between the theory of Calculus and students' own fields of study. This course also provides discussion sections in which students are able to make their skills in handling calculations in Calculus more proficient under the guidance of our teaching assistants.