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