\chapter{Multivariate and vector differential operators}




\section{Nabla symbol}
$\nabla$ is pronounced 'nabla'; it is a notational symbol used in denoting many multivariate and vector differential operators. It can be given the following interpretation.
This differential operator can be paired with notations of linear algebra (done to hint towards the similar calculations of differential operators and some linear maps of linear algebra) to create new differential operators.

Even though it is not a rigorously defined object (rather, a notation), its use in denoting differential operators assists in recalling their computation and properties.


\section{Gradient operator}

Fundamentally, the gradient operator on scalar field$f$ 


\begin{definition}[Gradient operator]
The \emph{Gradient operator} is a multivariate differential operator $\nabla$ such that $\nabla f$ is the unique vector whose dot product with $\mathbf{u}$ gives the directional derivative along $\mathbf{u}$.
\end{definition}

\begin{proposition}[Gradient operator of $\mathbb{R}^n$]
\[ \nabla f =  \begin{bmatrix}  \frac{\partial f}{\partial \mathbf{x}_1} \\ \frac{\partial f}{\partial \mathbf{x}_2} \\  \vdots \\  \frac{\partial f}{\partial \mathbf{x}_n}  \end{bmatrix} \]
\[ \nabla f_{i} = \frac{\partial f}{\partial \mathbf{x}_i}  \]
\end{proposition}


The idea behind the gradient operator is that it direction is towards the largest change in the function. Since partial derivatives calculate the amount of change along each axis, it is the linear combination of the increases along each axis; forming the direction of total change.

It also allows for an elegeant representation of the directional derivative.

\begin{proposition}
\[f'_{\mathbf{u}(\mathbf{x}) = \frac{\mathbf{u}}{ \| \mathbf{u} \|} \cdot \nabla f (\mathbf{x}) \]
\end{proposition}

The gradient operator is perhaps one of the most fundamental in vector analysis, and it is present in the \emph{gradient theorem}; a fundamental theorem of calculus for vector fields.






\section{Laplace operator}

\begin{definition}
The \emph{Laplace operator} or \emph{Laplacian} is a multivariate differential operator $\nabla^2$ (or sometimes written $\Delta$) defined on scalar fields as the following.
\[ \nabla^2 f = \sum^{n}_{i=1} \frac{\partial^2 f}{\partial \mathbf{x}_{i}^{2}} \]
\end{definition}

The vector differential operator called the \emph{divergence operator} allows for a particularly neat definition for the Laplacian.
\[\nabla^2 f = \nabla \cdot (\nabla f)\]
The $\nabla \cdot $ is this divergence operator; we will now commence our studies of such vector differential operators.


\section{Divergence operator}

On vector fields
- general def
- integral def
\[ \nabla \cdot \mathbf{F} =  \sum^{n}_{i=1} \frac{\partial f}{\partial \mathbf{x}_i}\]
\[ \nabla \cdot \mathbf{F} =  \lim_{\Delta V \to 0}\iint_{S} \mathbf{F} \cdot d\mathbf{S}\]
- Gauss' theorem





\section{Curl operator}
- general def
\[ \nabla \times \mathbf{F} = \hat{n} \lim_{\Delta s \to 0}\frac{1}{\Delta s} \oint_{\mathcal{C}} \mathbf{F} \cdot d\mathbf{r}\]
	<li>\(\hat{n}\) is the unit vector normal to the direction of counterclockwise rotation</li>
	<li>\(\mathcal{C}\) is a piecewise smooth (differentiable) closed (end point is the same as start point) curve</li>
	<li>\(\Delta s\) is area bound by \(\mathcal{C}\) (this equation is happens when the curve is infinitely tight)</li>

- basic properties
\[\nabla \times (\nabla f) = 0\]
\[\nabla \cdot (\nabla \times \mathbf{f}) = 0\]



%\chapter{Integral representation of oerators, do this without this new title}



\section{Integral theorems}

These theorems have long elementary proofs, mostly long calculations and considering functions that reminise the pullback; an object of differential geometry.

Fortunately, these theorems can be understood from (and indeed were inspired by) intuitive physical interpretations.

\subsection{Gradient theorem (Fundamental theorem of calculus for line integrals)}

\begin{theorem}[Gradient theorem]
\[ \int_{\mathbf{r}_0}^{\mathbf{r}_1} \nabla f \cdot d\mathbf{r} = f(\mathbf{r}_1) - f(\mathbf{r}_0) \]
\end{theorem}

This essentialy means that vector fields expressible as the gradient of some scalar field have path-independent integrals; all paths between the same two points return the same integral! This leads us to the idea of a \emph{conservative field}; vector fields where the integral between two points is 'conserved' by any path between said points.

\begin{definition}[Conservative vector field]
Vector field such that the line integral result is <b> purely dependent on the endpoints</b> of the line
\[\mathbf{F} \text{ is conservative } \iff\]
<ul>
	<li>\(\exists f : \nabla f = \mathbf{F}\)</li>
	<li>\(\nabla \times \mathbf{F} = \mathbf{0}\)</li>
	<li>\(\mathcal{C} \text{ is closed } \implies \int_{\mathcal{C}} \mathbf{F} \cdot d\mathbf{r}= 0\)</li>
</ul>
\end{definition}

We can invoke the gradient theorem for line integrals of some generic vector field if we can find some \emph{potential function} for the vector field; a scalar field whose gradient is the vecotr field in question. This is essentially 'reversing' the gradient operator, similar to how we 'reverse' the derivative for an antiderivative so we can apply FTC. 

\begin{definition}[Potential function]
\[f \text{ is the potential function of } \mathbf{F} \iff \nabla f = \mathbf{F}\]
\end{definition}



\subsection{Gauss' theorem}
There is a theorem asserting that the divergence of all points in a volume equals the flux integral of the volume's closed surface.j

The divergence of a vector field within a solid volume is equal to the 'flow' occuring through that volume's surface

Intuitively, this is because by thinking of the divergence of each infinitesimal volume element bounded by the surface, each infinitesimal volume elements has its flux 'cancelled out' by adjacent volume elements</p>

\begin{theorem}[Gauss' theorem]
\[ \iiint_{V} (\nabla \cdot \mathbf{F}) dV =  \iint_{S} \mathbf{F} \cdot d\mathbf{S} \]
\begin{itemize}
 \item $S$ is the closed differentiable surface
 \item $\mathbf{F}$ is the vector field
\end{itemize}
\end{theorem}




\subsection{Stoke's theorem}
<h3 class=cyan>Stoke's theorem</h3>
<p>Theorem asserting that the curl of all points on an open surface equals the line integral of the open surface's edge.</p>
<p> Intuitively, this is because by thinking of the curl of each infinitesimal surface element, each infinitesimal volume elements has its curl 'cancelled out' by adjacent surface elements</p>
<p>\(\displaystyle \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S} =  \oint_{\mathcal{C}} \mathbf{F} \cdot d\mathbf{r} \)</p>
<ul>
	<li>\(S\) is the open differentiable surface</li>
	<li>\(\mathcal{C}\) is the closed differentiable curve representing the edge of \(S\), oriented counterclockwise the surface's normal</li>
	<li>\(\mathbf{r}(t) : [t_0,t_1] \to \mathcal{C}\) is the position function along \(\mathcal{C}\)</li>
	<li>\(\mathbf{F}\) is the vector field</li>
</ul>

\subsection{Green's theorem}

Corrolary of Stoke's theorem, form of Stoke's theorem of a function projected in the $z=0$ plane.
\begin{theorem}[Green's theorem]
\[\displaystyle \iint_{S} ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) dxdy =  \oint_{\mathcal{\partial S}} P(x,y)dx + Q(x,y)dy \]
\end{theorem}


Green's theorem holds a special status in complex analysis where it is used to prove Cauchy's integral theorem. Among other uses, it finds use in proving the ancient isoperimetric inequality in differential geometry.







<h3 class=cyan>Vector operators</h3>
<h4 class=cyan>Nabla symbol</h4>
<p>Differential operator \(\nabla\) used as a notation for vector operators and hints towards their methods of calculation</p>
<p>\( \displaystyle \nabla = \begin{pmatrix}  \frac{\partial}{\partial x} \\  \frac{\partial}{\partial y} \\   \frac{\partial}{\partial z} \end{pmatrix} \)</p>
<h4 class=cyan>Gradient Gradiente 勾配</h4>
<p>Vector operator that <b>returns the vector of maximum change of a point in a scalar field</b> \(f\)</p>
<p>\(\displaystyle \nabla f = \begin{pmatrix} \frac{\partial f }{\partial x} \\ \frac{\partial f}{\partial y} \end{pmatrix}\)</p>
<ul>
	<li>\(f\) is the scalar field</li>
</ul>
<p>See <a href='/daigaku/2022_sum/m2.php'>Mathematics 2</a></p>

<h3 class=cyan>Divergence Divergenza 発散</h3>
<p>Vector operator that <b>returns the scalar quantity of flow in and out of a point in a vector field</b> \(\textbf{F}\)</p>
<p>To calculate based on intuition, look to the top, bottom, left and right of the point and note how regarding these adjacent vectors the point absorbs and emits</p>
<p>\(\displaystyle \nabla \cdot \textbf{F} = \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \textbf{F} = \frac{\partial F_{x}}{\partial x}+\frac{\partial F_{y}}{\partial y}+\frac{\partial F_{z}}{\partial z}\)</p>
<ul>
	<li>\(\textbf{F}\) is the vector field</li>
	<li>\(F_{n}\) is the vector field's \(n\) argument</li>
</ul>
<h4 class=cyan>Sink</h4>
<p>Points in vector fields with more <b>inward</b> flow</p>
<p>\((x,y) \text{ is a sink } \iff \nabla \cdot f(x,y) \lt 0\)</p>
<h4 class=cyan>Source</h4>
<p>Points in vector fields with more <b>outward</b> flow</p>
<p>\((x,y) \text{ is a source } \iff \nabla \cdot f(x,y) \gt 0\)</p>

<h3 class=cyan>Curl Rotore 回転</h3>
<p>Vector operator that <b>returns the vector normal to the direction of counterclockwise rotation with its magnitude representing the intensity of the rotation at a point in vector field</b> \(\textbf{F}\)</p>
<p>\(\nabla \times \textbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}  \\ F_x & F_y & F_z \end{vmatrix} = \begin{pmatrix} \frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}  \\   \frac{\partial F_{z}}{\partial x} - \frac{\partial F_{x}}{\partial z} \\   \frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}  \end{pmatrix}\)</p>
<ul>
	<li>\(\textbf{F}\) is the vector field</li>
	<li>\(F_{n}\) is the vector field's \(n\) argument</li>
</ul>


<h3 class=cyan>Laplacian Laplaciano ラプラス作用素</h3>
<p>Vector operator that <b>returns the scalar quantity of 'curvature' at a point in the scalar field</b> \(f\)</p>
<p>This works by capturing the gradient of the scalar field and finding the divergence of this gradient at some point</p>
<p>\(\displaystyle \nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\)</p>






