Field | Sahithyan's Notes

A field is a function that assigns a mathematical object, a scalar, vector, or tensor, to each point in its domain $D \subseteq \mathbb{R}^n$ .

A real-valued function defined on each points in $D \subseteq \mathbb{R}^n$ .

Gradient

For $\phi$ a field of $x$ , $y$ , $z$ , the gradient of $\phi$ is

\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}

$\nabla \phi$ is the gradient field of $\phi$ . $\nabla$ is the del operator, or nabla,

\nabla = \frac{\partial}{\partial x}\hat{i} + \frac{\partial}{\partial y}\hat{j} + \frac{\partial}{\partial z}\hat{k}

Where $\nabla \phi \neq \vec{0}$ :

$\nabla \phi$ points in the direction of the steepest increase of $\phi$ .
$|\nabla \phi|$ is the rate of increase of $\phi$ in that direction.
$\nabla \phi$ is perpendicular to the level surface $\phi = c$ through that point.

The rate of change of $\phi$ along a unit vector $\hat{u}$ is $\nabla \phi \cdot \hat{u}$ .

Examples:

The Laplacian operator is

\nabla^2 = \nabla \cdot \nabla = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}

For $\phi$ a field, the Laplacian of $\phi$ is

\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}

Physical meaning of $\nabla^2 \phi$ :

Compares the value of $\phi$ at a point with its average over a small sphere around that point.
$\nabla^2 \phi > 0$ means $\phi$ is lower than the average of its surroundings.
$\nabla^2 \phi < 0$ means $\phi$ is higher than the average of its surroundings.
$\nabla^2 \phi = 0$ means $\phi$ equals the average of its surroundings. Such a $\phi$ is harmonic.

Examples:

In the heat equation $\dfrac{\partial T}{\partial t} = \alpha \nabla^2 T$ , a point warms up when its neighbours are hotter on average.
In electrostatics, the potential $\phi$ satisfies $\nabla^2 \phi = 0$ in a charge-free region.

Find $\nabla \phi$ at $(1, 2, 1)$ for $\phi = x^2 y + y z^3$ .

Partial derivatives:

\frac{\partial \phi}{\partial x} = 2xy, \quad \frac{\partial \phi}{\partial y} = x^2 + z^3, \quad \frac{\partial \phi}{\partial z} = 3yz^2

Gradient field:

\nabla \phi = 2xy\,\hat{i} + (x^2 + z^3)\,\hat{j} + 3yz^2\,\hat{k}

At $(1, 2, 1)$ :

\nabla \phi = 2(1)(2)\,\hat{i} + (1 + 1)\,\hat{j} + 3(2)(1)\,\hat{k} = 4\hat{i} + 2\hat{j} + 6\hat{k}

Find $\nabla \phi$ for $\phi = x^2 + y^2 + z^2$ .

\nabla \phi = 2x\,\hat{i} + 2y\,\hat{j} + 2z\,\hat{k} = 2\vec{r}

Here $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ is the position vector.

Find $\nabla^2 \phi$ at $(1, 2, 1)$ for $\phi = x^2 y + y z^3$ .

Second partial derivatives:

\frac{\partial^2 \phi}{\partial x^2} = 2y, \quad \frac{\partial^2 \phi}{\partial y^2} = 0, \quad \frac{\partial^2 \phi}{\partial z^2} = 6yz

Sum:

\nabla^2 \phi = 2y + 6yz

At $(1, 2, 1)$ :

\nabla^2 \phi = 2(2) + 6(2)(1) = 16

Show that $\phi = \dfrac{1}{r}$ satisfies $\nabla^2 \phi = 0$ for $r \neq 0$ , where $r = \sqrt{x^2 + y^2 + z^2}$ .

Since $\dfrac{\partial r}{\partial x} = \dfrac{x}{r}$ :

\frac{\partial \phi}{\partial x} = -\frac{x}{r^3}

\frac{\partial^2 \phi}{\partial x^2} = -\frac{1}{r^3} + \frac{3x^2}{r^5}

By symmetry, the $y$ and $z$ terms are

\frac{\partial^2 \phi}{\partial y^2} = -\frac{1}{r^3} + \frac{3y^2}{r^5}, \quad \frac{\partial^2 \phi}{\partial z^2} = -\frac{1}{r^3} + \frac{3z^2}{r^5}

Sum:

\nabla^2 \phi = -\frac{3}{r^3} + \frac{3(x^2 + y^2 + z^2)}{r^5} = -\frac{3}{r^3} + \frac{3r^2}{r^5} = 0