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del operator

Table of Contents

1. Definition

The operator del in \( n \) dimensional euclidean space is defined as follows:

\begin{align*} \vec{\nabla} := \sum_{i = 1}^{n} \hat{e_{i}}\frac{\partial}{\partial e_{i}} \end{align*}

Where \( \frac{\partial}{\partial e_{k}}\) is the with respect to the \(k^{th}\) orthogonal axis, and \( \hat{e}_{k} \) is the orthogonal basis vector pointing in that direction. In three dimensional euclidean space using Cartesian coordinates, the del operator would look like:

\begin{align*} \vec{\nabla} = \begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{bmatrix} = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z} \end{align*}

The del operator is what is called a linear operator because it is consistent with operations pertaining to linear algebra.

2. Usage

The del operator is useful for representing the gradient, divergence, and curl of a given scalar or vector field.

2.1. Gradient

Multiplying the del operator by a scalar field yields a vector that is called the gradient of a function:

\begin{align*} \vec{\nabla}f = \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \end{bmatrix} = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k} \end{align*}

Where this vector points in the direction of the greatest rate of change, and has a magnitude corresponding with the slope. The reason why is somewhat intuitive, if you think about it a little.

2.2. Divergence

Taking the dot product of the del operator with a vector field yields a scalar function, which is called the divergence:

\begin{align*} \vec{\nabla} \cdot \vec{f} = \frac{\partial f_{x}}{\partial x} + \frac{\partial f_{y}}{\partial y} + \frac{\partial f_{z}}{\partial z} \end{align*}

Where \( f_{n} \) is the \( n \) component of \( \vec{f} \).

You can think of it as measuring the rate of change of the outwards or inwards direction of a vector field. In order to think about this more clearly, we can think about the two dimensional case with just x and y. Given a two-dimensional vector field, a two-dimensional divergence would look like this:

\begin{align*} \vec{\nabla} \cdot \vec{f} = \frac{\partial f_{x}}{\partial x} + \frac{\partial f_{y}}{\partial y} \end{align*}

and to explain this further, let's take a vector \( \vec{v} \), as well as two other vectors to compare it with, \( \vec{v_{up}}\), and \( \vec{v_{right}} \). Then, we take \( \vec{r} = \vec{f}(\vec{v}) \) and compare it to \( \vec{r_{up}} \) and \( \vec{r_{right}}\). We then compare the x component of the right vector with the original one, and we compare the y component of the up vector with the original one, by taking the difference. We then sum these differences, and what we are left with is a measurement of how spread apart the directions and magnitudes of these vectors are in this local area. If these \( \vec{r} \) vectors are infinitely close to each other, we can consider this comparison to be analogous to the divergence at that point. This argument naturally extends to three dimensions.

2.3. Curl

The curl of a vector field is defined as follows:

\begin{align*} \vec{\nabla} \times \vec{f} = \hat{i}(\frac{\partial f_{z}}{\partial y} - \frac{\partial f_{y}}{\partial z}) - \hat{j}(\frac{\partial f_{z}}{\partial x} - \frac{\partial f_{x}}{\partial z}) + \hat{k}(\frac{\partial f_{y}}{\partial x} - \frac{\partial f_{x}}{\partial y}). \end{align*}

Where the equation above is derived from the definition of the cross product. It represents the rate of change of a vector field "perpendicular" to the divergence of the field. In fact, if you have any field \( \vec{f} \), you can represent this field as an addition of a curl-less field and a divergence-less field. Another way to think of it is that you are measuring the strength of rotational component of the vector field about a certain axis.

2.4. directional derivative

The directional derivative is defined as follows:

\begin{align*} \vec{f} \cdot \vec{\nabla} = \sum_{i=0}^{n}f_{i}\frac{\partial}{\partial x_{i}} \end{align*}

Which represents a superposition of states which corresponds to the direction you want to take the derivative in.

2.5. Laplacian

The Laplacian is defined as follows:

\begin{align*} \nabla^{2}\vec{f} = \nabla \cdot \nabla\vec{f} \end{align*}

It returns a scalar field and is the multivariable analogue to the second derivative. Because both the divergence and gradient have been described, I feel it is trivial to understand the Laplacian.

2.6. Product Rules

The product rules pertaining to the del operator are consistent with that of linear algebra and single variable derivative rules. For example, \( \vec{\nabla} \times \vec{\nabla}f = 0\). You can show this yourself quite easily, so I find no need to go over it here. When in doubt, just assume the del works the same way as any old vector except you apply the product rule, and you will usually be correct.

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