# Mathematical Methods of Physics/Gradient, Curl and Divergence

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

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

## Vector and Scalar Fields

#### Definition

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

In the physical world, examples of scalar fields are

(i) The electrostatic potential $\phi$  in space

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

#### Definition

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

In the physical world, examples of vector fields are

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

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

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

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

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 $f:\mathbb {R} ^{3}\to \mathbb {R}$  at $\mathbf {a} \in \mathbb {R} ^{3}$  is defined as the linear transformation $A$  that satisfies

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

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

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

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

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

## Divergence

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

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

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

## Curl

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

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

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

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

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