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:PROPERTIES:
:ID: 4bfd6585-1305-4cf2-afc0-c0ba7de71896
:END:
#+title: del operator
#+author: Preston Pan
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* 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 [[id:3993a45d-699b-4512-93f9-ba61f498f77f][partial derivative]] 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.
* Usage
The del operator is useful for representing the gradient, divergence, and curl of a given
scalar or vector field.
** Gradient
:PROPERTIES:
:ID: 3587c3b4-c3d8-4ff1-b0ba-8eecb1ef0e4c
:END:
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.
** Divergence
:PROPERTIES:
:ID: 12a2d5b3-f98c-45e5-9107-5560288b5aa8
:END:
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.
** Curl
:PROPERTIES:
:ID: b25e0e44-c764-4f0a-a5ad-7f9d79c7660d
:END:
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.
** directional derivative
:PROPERTIES:
:ID: e255eb0a-246b-4a4b-8db8-ac0d15d9cc3c
:END:
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.
** Laplacian
:PROPERTIES:
:ID: 65004429-a6b7-41f2-8489-07605841da3d
:END:
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.
** Product Rules
The [[id:d1e245f4-0b04-450e-8465-a9c85fe57f7e][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 [[id:d1e245f4-0b04-450e-8465-a9c85fe57f7e][product rule]], and you will usually be correct.
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