Vector Calculus: An Introduction

The aim of vector calculus is to study functions of several variables, or functions whose values are vectors. Such functions appear naturally in applications. As examples think of the temperature of a body varying with position, or the position of a particle travelling through the air. The first is a function of position described by three spatial components. The second is a function of time, but its value is a vector. We say it is vector valued. It also makes sense to study functions of several variables which are vector valued. Just think of the velocity of a fluid, which depends on position and time, and whose value can be thought of as a vector with three components. Hence we have to study a function of four variables, whose values are vectors with three components. There are many more models involving one or several quantities depending on a number of other quantities. Vector calculus provides a powerful mathematical tool to treat such problems.