]> 17.1 Introduction

## 17.1 Introduction

When differentiating functions of several variables, which are called scalar fields, the gradient of the function, contains the all the information about the tangent (hyper)-plane (or linear approximation or differentials) that there is.

When differentiating vector fields, which are vector valued functions of several variables, there are partial derivatives of every component in every direction; thus there are nine of them in three dimensions.

However the divergence and curl are the combinations of these that are particular importance in applications, and we concentrate our attention on them. These are written as

$div v ⟶ = ∇ ⟶ · v ⟶$
and
$curl ⟶ v ⟶ = ∇ ⟶ × v ⟶$

with

$∇ ⟶ = ∂ ∂ x i ^ + ∂ ∂ y j ^ + ∂ ∂ z k ^$