xxVector Calculus

Vector Calculus

February 4, 2022

Contents

x1 Curves xx1.1 Differentiating the Curve

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

. f : \underbrace{ℝ^{m}}_{domain} → \underbrace{ℝ^{n}}_{codomain}

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

Examples of Maps

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

. f : t ↦ x(t)

with x ∈ ℝ^{n}. (Obviously we should take n = 3). Generalising, a map

. f : ℝ^{2} → ℝ^{n}

defines a surface in ℝ^{n}, and so on.

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

. f : ℝ^{3} → ℝ

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

.f : \underbrace{ℝ^{3}}_{physical space} → \underbrace{ℝ^{3}}_{somethinge more abstract}

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

1 Curves We consider maps of the form

. f : ℝ → ℝ^{n}

Assign a coordinate t to ℝ and use Cartesian coordinates on ℝ^{n}.

. x = (x^{1}, …, x^{n}) = x^{i}e_{i}

where e_{i} is an orthonormal basis such that e_{i} · e_{j} = δ_{ij}. Note that summation convention is used here. (For ℝ^{3} we also use notation {e_{i}} = {x̂, ŷ, ẑ}.)

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

. x(t) = (cos λt, sin λt, λt).

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

.x(t+δt)−x(t) = ẋ(t)δ(t)+O(δt^{2}).

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

Note. “Big O” notation O(δt^{2}) means terms proportional to δt^{2} or smaller.

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

Some notation: we write

. δx(t) = x(t + δt) − x(t)

The derivative is then

. x ≡ \frac{dx}{dt} := lim_{δt→0} \frac{δx}{δt}.