\part{Multivariate and vector differential operators}


\chapter{Nabla symbol}
$\nabla$ is pronounced 'nabla'; it is a 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.
\chapter{Multivariate differential operators}

Differential operators on scalar fields
\section{Gradient operator}

Fundamentally, the gradient operator on $f$ 


\begin{definition}
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 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.


\chapter{Vector differential operators}

Differential operators on vector fields
\section{Divergence operator}


- 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
<h3 class=cyan>Gauss' theorem Teorema di Gauss 発散定理</h3>
<p>Theorem asserting that the divergence of all points in a volume equals the flux integral of the volume's closed surface.</p>
<p> 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>
<p>\(\displaystyle \iiint_{V} (\nabla \cdot \mathbf{F}) dV =  \iint_{S} \mathbf{F} \cdot d\mathbf{S} \)</p>
<ul>
	<li>\(S\) is the closed differentiable surface</li>
	<li>\(\mathbf{F}\) is the vector field</li>
</ul>




\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{Differential operators in different frames}





			<h4 class=cyan>Vector operators</h4>
			<ul>
				<li>\(\nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat{\theta} \)</li>
				<li>\(\nabla \cdot \textbf{F} = \frac{1}{r}\frac{\partial ( r F_{r})}{\partial r}+  \frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}\)</li>
				<li>\(\nabla^2 f = \frac{1}{r} \frac{\partial }{ \partial r} (r \frac{\partial f}{\partial r}) + \frac{1}{r^2}\frac{\partial^2  f }{\partial \theta^2}\)</li>
			<h4 class=cyan>Vector operators</h4>
			<ul>
				<li>\(\nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta} \hat{\theta}+\frac{\partial f}{\partial z} \hat{z}\)</li>
				<li>\(\nabla \cdot \textbf{F} = \frac{1}{r}\frac{\partial ( r F_{r})}{\partial r}+  \frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}+\frac{\partial F_{z}}{\partial z}\)</li>
				<li>\(\nabla \times \textbf{F} = ( \frac{1}{r} \frac{\partial F_{z}}{\partial \theta} - \frac{\partial F_{\theta}}{\partial z}) \hat{r} + ( \frac{\partial F_{r}}{\partial z} - \frac{\partial F_{z}}{\partial r}) \hat{\theta} + \frac{1}{r} (\frac{\partial (r F_{\theta})}{\partial r} - \frac{\partial F_{r}}{\partial \theta}) \hat{z}\)</li>
				<li>\(\nabla^2 f = \frac{1}{r} \frac{\partial }{ \partial r} (r \frac{\partial f}{\partial r}) + \frac{1}{r^2}\frac{\partial^2  f }{\partial \theta^2} + \frac{\partial^2 f}{ \partial z^2}\)</li>
			</ul>


			<h4 class=cyan>Vector operators</h4>
				\[ \nabla f = \frac{\partial f}{\partial \rho} \hat{\rho} + \frac{1}{\rho}\frac{\partial f}{\partial \theta} \hat{\theta}+ \frac{1}{\rho \sin (\theta)}\frac{\partial f}{\partial \varphi} \hat{\varphi}\]
				\[ \nabla \cdot \textbf{F} = \frac{1}{{\rho}^2}\frac{\partial ( {\rho} ^2 F_{\rho})}{\partial \rho}+  \frac{1}{\rho \sin (\theta) } \frac{\partial ( \sin (\theta) F_{\theta} )}{\partial \theta}+  \frac{1}{\rho \sin (\theta)} \frac{\partial F_{\varphi}}{\partial \varphi}\]
				\[ \nabla \times \textbf{F} = \frac{1}{\rho \sin (\theta)} ( \frac{\partial ( \sin (\theta) F_{\varphi}) }{\partial \theta} - \frac{\partial F_{\theta}}{\partial \varphi}) \hat{\rho} + \frac{1}{\rho} ( \frac{1}{\sin (\theta)}\frac{\partial F_{\rho}}{\partial \varphi} - \frac{\partial (\rho F_{\varphi}) }{\partial \rho}) \hat{\theta} + \frac{1}{\rho} (\frac{\partial (\rho F_{\theta})}{\partial \rho} - \frac{\partial F_{\rho}}{\partial \theta}) \hat{\varphi}\]
				\[ \nabla^2 f = \frac{1}{\rho^2} \frac{\partial }{ \partial \rho } (\rho^2 \frac{\partial f}{\partial \rho}) + \frac{1}{\rho^2 \sin ( \theta )} \frac{\partial }{\partial \theta } ( \sin ( \theta )  \frac{\partial f }{\partial \theta } )  +  \frac{1}{\rho^2 \sin ^2 (\theta  )} \frac{\partial ^2 f }{ \partial \phi} \]
			</ul>
			<h4 class=cyan>Volume element</h4>
			<p>\( dV = \rho ^2 \sin ( \theta ) d\rho d\theta d\phi \)</p>













\chapter{Integral representation}



\chapter{Introduction to the integral theorems}



\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.

<h3 class=cyan>Conservative field </h3>
<p>Vector field such that the line integral result is <b> purely dependent on the endpoints</b> of the line </p>
<p>\(\mathbf{F} \text{ is conservative } \iff\)</p>
<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>
<h4 class=cyan>Potential function</h4>
<p>\(f \text{ is the potential function of } \mathbf{F} \iff \nabla f = \mathbf{F}\)</p>







- 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>

- Green's theorem

<h3 class=cyan>Green's theorem</h3>
<p>Corrolary of Stoke's theorem, form of Stoke's theorem of a function projected in the xy-plane.</p>
<p>\(\displaystyle \iint_{S} ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ) dxdy =  \oint_{\mathcal{\partial S}} P(x,y)dx + Q(x,y)dy \)</p>









<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>







\chapter{That's all folks}

This book has discussed ways of how ideas in real analysis are extended into higher Euclidean spaces and how this is assisted by the means of linear algebra. 






