Last modified on 25 August 2009, at 03:17

# 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 FieldsEdit

#### DefinitionEdit

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, $T(\mathbf{r})$

#### DefinitionEdit

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 derivativeEdit

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 $a^T_{ij}=a_{ji}$

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

## DivergenceEdit

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

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

## CurlEdit

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 $(\nabla\times\mathbf{F})_{ij}=\partial_iF_j-\partial_jF_i$. 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, $\nabla\times\mathbf{F}=(\partial_2 F_3-\partial_3 F_2)\hat{x}+(\partial_1 F_3-\partial_3 F_1)\hat{y}+(\partial_1 F_2-\partial_2 F_1)\hat{z}$