Vector Calculus


February 4, 2022

Contents

Introduction

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

m n f : ◟ℝ◝◜◞ → ℝ◟◝◜◞ domain codomain

An element of ℝm 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

    f : t ↦→ x (t)

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

    f : ℝ2 → ℝn

    defines a surface in ℝn, and so on.

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

    f : ℝ3 → ℝ

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

    f : ℝ◟◝3◜◞ → ◟ℝ◝3◜◞ physical space somethinge more abstract

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