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 $D \subseteq \mathbb{R}^n$ .

A vector-valued function defined on each points in $D \subseteq \mathbb{R}^n$ . More on this covered in the vector field note.

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}

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}