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 $\vec{v}(\mathbf{r})$

The Gradient

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

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