xxVector Calculus

Vector Calculus

February 4, 2022

Contents

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.