Contents

Introduction

We will learn to differentiate and integrate functions (or maps) of the form

An element of ℝ^m or ℝⁿ is a vector so this subject is called vector calculus.

Examples of Maps

1. A function f : ℝ → ℝⁿ defines a curve in ℝⁿ. In physics, we might think of ℝ as time and ℝⁿ as physical space and write this as

   

   with x ∈ ℝⁿ. (Obviously we should take n = 3). Generalising, a map

   

   defines a surface in ℝⁿ, and so on.

2. In other applications, the domain ℝ^m might be viewed as physical space. For example, in physics a scalar field is a map

   

   for example temperature T(x) is a scalar field, as is the Higgs field.
   A vector field is a map

   

   for example the electric field E(x) and magnetic field B(x) are vector fields.

1 Curves

We consider maps of the form

Assign a coordinate t to ℝ and use Cartesian coordinates on ℝⁿ.

where e_i is an orthonormal basis such that e_i ⋅e_j = δ_ij. Note that summation convention is used here. (For ℝ³ we also use notation {e_i} = {,,}.)

The image of of the function f is a parametrised curve x(t), with t the parameter.

Examples

The choice of parametrisation is not unique, for example consider

This gives the same helix for all λ ∈ ℝ \{0}.
Sometimes the choice of parametrisation matters, for example if t is time and x(t) is position, then the velocity is proportional to λ. But we will see that some questions are independent of the choice of parametrisation.

1.1 Differentiating the Curve

A vector function x(t) is differentiable of t if, as δt → 0, we have

If (t) exists everywhere, the curve is said to be smooth.

In physics, dot is usually used for time derivatives, for example (t) and prime for spatial derivatives, for example f′(x).
In maths, these are used interchangeably.

Some notation: we write

The derivative is then