# Mathematical Methods of Physics/Gradient, Curl and Divergence

In this section we shall consider the vector space ${\displaystyle \mathbb {R} ^{3}}$ over reals with the basis ${\displaystyle {\hat {x}},{\hat {y}},{\hat {z}}}$.

We now wish to deal with some of the introductory concepts of vector calculus.

## Vector and Scalar Fields

#### Definition

Let ${\displaystyle C:\mathbb {R} ^{3}\to F}$ , where ${\displaystyle F}$  is a field. We say that ${\displaystyle C}$  is a scalar field

In the physical world, examples of scalar fields are

(i) The electrostatic potential ${\displaystyle \phi }$  in space

(ii) The distribution of temperature in a solid body, ${\displaystyle T(\mathbf {r} )}$

#### Definition

Let ${\displaystyle V}$  be a vector space. Let ${\displaystyle \mathbf {F} :\mathbb {R} ^{3}\to V}$ , we say that ${\displaystyle \mathbf {F} }$  is a vector field; it associates a vector from ${\displaystyle V}$  with every point of ${\displaystyle \mathbb {R} ^{3}}$ .

In the physical world, examples of vector fields are

(i) The electric and magnetic fields in space ${\displaystyle {\vec {E}}(\mathbf {r} ),{\vec {B}}(\mathbf {r} )}$

(ii) The velocity field in a fluid ${\displaystyle {\vec {v}}(\mathbf {r} )}$

## The Gradient

Let ${\displaystyle C}$  be a scalar field. We define the gradient as an "operator" ${\displaystyle \nabla }$  mapping the field ${\displaystyle C}$  to a vector in ${\displaystyle \mathbb {R} ^{3}}$  such that

${\displaystyle \nabla C=\left({\frac {\partial C}{\partial x}},{\frac {\partial C}{\partial y}},{\frac {\partial C}{\partial z}}\right)}$ , or as is commonly denoted ${\displaystyle \nabla C={\frac {\partial C}{\partial x}}{\hat {x}}+{\frac {\partial C}{\partial y}}{\hat {y}}+{\frac {\partial C}{\partial z}}{\hat {z}}}$

We shall encounter the physicist's notion of "operator" before defining it formally in the chapter Hilbert Spaces. It can be loosely thought of as "a function of functions"

#### Gradient and the total derivative

Recall from multivariable calculus that the total derivative of a function ${\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} }$  at ${\displaystyle \mathbf {a} \in \mathbb {R} ^{3}}$  is defined as the linear transformation ${\displaystyle A}$  that satisfies

${\displaystyle \lim _{|\mathbf {h} |\to 0}{\frac {f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )-A\mathbf {h} }{|\mathbf {h} |}}=0}$

In the usual basis, we can express as the row matrix ${\displaystyle f'(\mathbf {a} )=A=\displaystyle {\begin{pmatrix}{\tfrac {\partial f}{\partial x}}&{\tfrac {\partial f}{\partial y}}&{\tfrac {\partial f}{\partial z}}\\\end{pmatrix}}}$

It is customary to denote vectors as column matrices. Thus we may write ${\displaystyle \nabla f=\displaystyle {\begin{pmatrix}{\tfrac {\partial f}{\partial x}}\\{\tfrac {\partial f}{\partial y}}\\{\tfrac {\partial f}{\partial z}}\\\end{pmatrix}}}$

The transpose of a matrix given by constituents ${\displaystyle a_{ij}}$  is the matrix with constituents ${\displaystyle a_{ij}^{T}=a_{ji}}$

Thus, the gradient is the transpose of the total derivative.

## Divergence

Let ${\displaystyle \mathbf {F} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}}$  be a vector field and let ${\displaystyle \mathbf {F} }$  be differentiable.

We define the divergence as the operator ${\displaystyle (\nabla \cdot )}$  mapping ${\displaystyle \mathbf {F} }$  to a scalar such that

${\displaystyle (\nabla \cdot \mathbf {F} )={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}}$

## Curl

Let ${\displaystyle \mathbf {F} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}}$  be a vector field and let ${\displaystyle \mathbf {F} }$  be differentiable.

We define the curl as the operator ${\displaystyle (\nabla \times )}$  mapping ${\displaystyle \mathbf {F} }$  to a linear transformation from ${\displaystyle \mathbb {R} ^{3}}$  onto itself such that the linear transformation can be expressed as the matrix

${\displaystyle (\nabla \times \mathbf {F} )_{ij}={\frac {\partial F_{j}}{\partial x_{i}}}-{\frac {\partial F_{i}}{\partial x_{j}}}}$  written in short as ${\displaystyle (\nabla \times \mathbf {F} )_{ij}=\partial _{i}F_{j}-\partial _{j}F_{i}}$ . Here, ${\displaystyle x_{1},x_{2},x_{3}}$  denote ${\displaystyle x,y,z}$  and so on.

the curl can be explicitly given by the matrix: ${\displaystyle \nabla \times \mathbf {F} ={\begin{pmatrix}0&\partial _{1}F_{2}-\partial _{2}F_{1}&\partial _{1}F_{3}-\partial _{3}F_{1}\\\partial _{2}F_{1}-\partial _{1}F_{2}&0&\partial _{2}F_{3}-\partial _{3}F_{2}\\\partial _{2}F_{1}-\partial _{1}F_{3}&\partial _{3}F_{2}-\partial _{2}F_{3}&0\\\end{pmatrix}}}$

this notation is also sometimes used to denote the vector exterior or cross product, ${\displaystyle \nabla \times \mathbf {F} =(\partial _{2}F_{3}-\partial _{3}F_{2}){\hat {x}}+(\partial _{1}F_{3}-\partial _{3}F_{1}){\hat {y}}+(\partial _{1}F_{2}-\partial _{2}F_{1}){\hat {z}}}$