Vector Calculus: An Introduction

These notes are based on a set of printed lecture notes for a second year course in vector calculus at the University of Sydney going first written in 2001 and used over many years. Students are expected to have have completed a course on differential and integral calculus as well as basic matrix theory. The emphasis is on providing an intuitive understanding of vector calculus with some rigour, but more emphasis is given to the the meaning of concepts.

Students now like to have content available online, so I have decided to transform the notes into an online format using the PreTeXt

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pretextbook.org

XML format. I have not had the time and energy to take advantage of all online features, so I consider this not a very good example of an online book! At some future time I may add some exercises and more examples.

The notes mainly present multiple integrals, line and surface integrals and the the integral theorems of vector calculus in two or three dimensions. I have made sure that all results generalise without much difficulties to higher dimensions. In particular this is the case for the definition of line and surface integrals, which immediately generalise to integrals over “\(k\)-dimensional sub-manifolds” of \(\mathbb R^N\text{.}\) This is in particular the case for the definition of surface integrals, where I give a definition involving a Jacobian determinant that is consistent with integrals of “surfaces” of any dimension. Still, the traditional notation and approaches are still included. We finally look at the main theorems of vector calculus: Green’s theorem, the divergence theorem and Stoke’s theorem for surfaces in three dimensional space.