\part{Real Multivariate and vector functions}



\chapter{Multivariate functions}

The goal of this chapter is to translate knowledge from real analysis to higher dimensions of Euclidean spaces. Real analysis studied functions of the form $f : \mathbb{R} \to \mathbb{R}$, however we now want to consider $f : \mathbb{R}^m \to \mathbb{R}^n$; we'll be looking at functions between real Euclidean spaces, possibly with different dimensions. One consequence is that the absolute value as a metric generalizes to the norm.

It's worth mentioning that many of the ideas addressed in this book are standards for any multidimensional generalization of analysis. We first look at three types of functions that are used across all types of multidimenional analysis.

\begin{definition}
A \emph{vector valued function}  is a function $\mathbf{r} : X^n \to \mathbb{R}^m$ that maps elements of a space to a Euclidean space.
\end{definition}

\begin{definition}
A \emph{vector field} is a vector valued function $\mathbf{f} : X^n \to \mathbb{R}^n$ that assigns each vector in some space $X^n$ to a vector in a real space of the same dimension. It is a vector valued function where the domain and image spaces have the same dimension.
\end{definition}


\begin{definition}
A \emph{scalar field} is a vector valued function $f : X^n \to \mathbb{R}$ that assigns a real number to each pont in some space $X^n$.
\end{definition}


Though these definitions apply to functions whose domains are arbitrary spaces $X$, this book will only \emph{vector valued functions with Euclidean domains} $f : \mathbb{R}^m \to \mathbb{R}^n$, so the following variants of these 3 definitions will be frequent objects of interest within this book.

\begin{itemize}
\item Real multivariate functions $f : \mathbb{R}^n \to \mathbb{R}, n > 1$; scalar fields with Euclidean domain.
\item Real vector functions $\mathbf{f} : \mathbb{R}^m \to \mathbb{R}^n, m,n > 1$; vector valued functions with Euclidean domain.
\item Real vector fields $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^n, n > 1$; vector fields with Euclidean domain.
\end{itemize}

There are special classes within these functions, such as \emph{differentiable curves and surfaces}; we'll delve into this theory soon.

After having learned the theory of this book, functional analysis can be used to explore these functions on more general spaces called \emph{manifolds}.

\begin{example}
The following function $f : \mathbb{R}^2 \to \mathbb{R}$ is an example of real multivariate function.
\[ f(x,y)=  x^2 -y^2 \]
\end{example}
\begin{example}
The following function $\mathbf{r} : [0,6\pi] \to \mathbb{R}^3$ is an example of real vector function (it's also a 'differentiable curve') that maps a single variable to a $\mathbb{R}^2$ vector. This is a helix that makes 3 revolutions.
\[ \mathbf{r}(t)= \begin{bmatrix} \cos (t) \\ \sin (t) \\ t \end{bmatrix}\]
\end{example}
\begin{example}
The following function $\mathbf{f} : \mathbb{R}^2 \to \mathbb{R}^2$ is an example of a real vector field. It returns a vector that is orthogonal to the input vector.
\[ \mathbf{f}(\mathbf{x})= \begin{bmatrix} \mathbf{x}_2 \\ -\mathbf{x}_1  \end{bmatrix} = \mathbf{x}_2 \mathbf{i} -\mathbf{x}_1 \mathbf{j} \]
\end{example}



\chapter{Multivariate limits}
In terms of theory and rigor, limits are defined in an extremely similar way to real analysis. In terms of reality and intuition, the primary difference is that in real analysis, it suffices to check that both the left and right limits exist and are equal. In higher dimensions, there are an uncountably infinite directions in which a limit may be approached. 

limit in $\mathbb{R}^n$

\begin{definition}
For a real vector function $\mathbf{f}$, its \emph{limit at $\mathbf{p}$} is a vector $\mathbf{L}$ such that for any positive $\varepsilon$, we can find a positive $\delta$ so that whenever $\|\mathbf{x}-\mathbf{p} \| < \delta$ we have  $\| \mathbf{f}(\mathbf{x}) -\mathbf{L} \| < \varepsilon$. Basically, as $\mathbf{x}$ converges to $\mathbf{p}$, $\mathbf{f}(\mathbf{x})$ converges to $\mathbf{L}$.

\[ \lim_{\mathbf{x} \to \mathbf{p}} \mathbf{f}(\mathbf{x}) = \mathbf{L} \iff \forall \varepsilon \in (0,\infty) ( \exists \delta \in (0,\infty) [ \| \mathbf{x}-\mathbf{p}\| < \delta \implies \| \mathbf{f}(\mathbf{x})-\mathbf{L} \| < \varepsilon ] )\]
\end{definition}

\begin{proposition}
for real vector functions limits areclosed under
	addition
	subtraction
	scalar multiplication	
	dot product
	cross product
	norm
\end{proposition}

\chapter{Multivariate continuity}


\begin{proposition}
\[ \mathbf{f} \text{ is continuous at } \mathbf{x}_0 \iff lim_{\mathbf{x} \to\mathbf{x}_0} \mathbf{f}(\mathbf{x}) = \mathbf{f}(\mathbf{x}_0)  \]
\end{proposition}

\begin{proposition}
Let $\mathbf{f} : \mathbb{R}^m \to \mathbb{R}^n$. $\mathbf{f}$ is continuous at $\mathbf{x}_0$ iff each component function $\mathbf{f}_{i}$ continuous at $\mathbf{x}_{0}$.
\end{proposition}


\begin{proposition}
for real vector functions continuity is closed under
	addition
	subtraction
	multiplication
	composition
\end{proposition}



\chapter{Multivariate differentiation}



In real analysis, the derivative is a tool that describes the 'tangent' or 'best linear approximator' at a point of a function.

For multivariate Functions, there are several directions to approach domain elements by and the function may increase at different rates for different directions taken; tangents in one direction may not be tangents in another. 

One idea is to only consider the derivative along a specific direction to a domain element; this is what \emph{directional derivatives and partial derivatives} strive to do, and it will prove an adequate stating point for now.




\section{Directional derivative}

Partial derivatives can take the derivative with respect to any variable, however does there exist a way of taking the derivative of a multivariate function with respect to any arbitrary 'direction', rather than the directions dictated by the variables (usually axis')? If one represents their direction by $\mathbf{u}$, the \emph{directional derivative} acts as a framework to take the derivative along $\mathbf{u}$.


We start by considering the derviatives at a point with respect to a specific direction specified by a unit vector. These are the \emph{directional derivatives}.
\begin{definition}
	Let $\mathbf{f}: \mathbb{R}^m \to \mathbb{R}^n$  be a real vector function and $\hat{\mathbf{u}}$ be a unit vector, the \emph{directional derivative of $\mathbf{f}$ at $\mathbf{x}_0$} is defined as the following. It represents the derivative in the sense of real analysis along $\mathbf{u}$.
\[ \nabla_{\mathbf{u}} \mathbf{f} (\mathbf{x}_0) = \lim_{h \to 0} \frac{\mathbf{f}(\mathbf{x}_0 + h \mathbf{u}) - \mathbf{f}(\mathbf{x}_0) }{ h \|\mathbf{u}\| }\]
\end{definition}


\section{Partial derivative}

A special case of directional derivatives is when we take the directional derivative by purely along a single variable. These are the \emph{partial derivatives}; the derivative of a multivariate function with respect to a single variable.

\begin{definition}
The \emph{partial derivative of $\mathbf{f}$ at $\mathbf{x}_0$ with respect to $\mathbf{x}_k$} is defined as the following.
\[ \mathbf{f}_{\mathbf{x}_k} (\begin{bmatrix}\mathbf{x}_1 \\ \vdots \\ \mathbf{x}_k \\ \vdots \\ \mathbf{x}_m \end{bmatrix}) = \lim_{h \to 0} \frac{\mathbf{f}( \begin{bmatrix}\mathbf{x}_1 \\ \vdots \\ \mathbf{x}_k +h \\ \vdots \\ \mathbf{x}_m \end{bmatrix} ) - f(\begin{bmatrix}\mathbf{x}_1 \\ \vdots \\ \mathbf{x}_k \\ \vdots \\ \mathbf{x}_m \end{bmatrix}) }{ h }\]
\end{definition}

This is essentially differentiation in the sense of real analysis where other variables behave as constants.


We've used the notation $f_{x}$ to describe the partial derivative of $f$ with respect to $x$, however most notations of analysis support for partial derivatives.
\emph{Leibniz notation} expresses partial derivatives in the following way.
\[ \frac{\partial f}{\partial x} \]
\[ \frac{\partial^2 f}{\partial x \partial y}\]

\emph{Lagrange notation} expresses partial derivatives in the following way.
\[ f_x \]
\[ f_{xy} \]

\emph{Euler-Arbogast notation} expresses partial derivatives in the following way.
\[ \partial_x f \]
\[\partial_{xy}  f \]

\emph{Newton's notation} does not support partial derivatives.



\section{The Jacobian matrix and Hessian matrix}

Now that the notions of partial derivatives are accessible, we discuss 2 interesting matrixes; they will be useful tools later on.


\begin{definition}
The \emph{Hessian matrix of $f$} is the matrix of second-order partial derivatives of a real multivariate function.
\[ \mathbf{H}_{f} \]
\[ (\mathbf{H}_{f})_{ij} = \frac{\partial^2 f}{\partial \mathbf{x}_{i} \partial \mathbf{x}_{j}} \]
\end{definition}

\begin{definition}
The \emph{Jacobian matrix of $\mathbf{f}$} is the matrix of first-order partial derivatives for each function within a real vector function.
\[ \mathbf{J}_{\mathbf{f}} \]
\[ (\mathbf{J}_{\mathbf{f}})_{ij} = \frac{\partial f_{i}}{\partial \mathbf{x}_j} \]
\end{definition}





\chapter{Total derivative}

Though a point may have directional derivatives defined for all directions, it is still possible for the function to be discontinuous at that point! Just because a function is differentiable and continuous through at a point on a certain path,

A more holistic approach that captures the geometry of multidimensional space is needed for deeper insight of differentiability of such functions.

In an informal sense, differentiable functions express the fact that a function can be approximated as a linear function on a small enough scale; if we 'zoom in' infinitely on the function, it is linear. Let's try and formulate this in the sense of real analysis.

\[ f(x_0+h) \approx f(x_0) + f'(x_0)h \]

\[ \lim_{h \to 0} \frac{ f(x_0+h) -f(x_0) }{h} =f'(x_0)\]
\[ \lim_{h \to 0} \frac{ f(x_0+h) -f(x_0) }{h} - f'(x_0) = 0 \]
\[ \lim_{h \to 0} \frac{ f(x_0+h) -f(x_0) - f'(x_0)h }{h} = 0 \]
\[ \lim_{h \to 0} \frac{ f(x_0+h) - [ f(x_0) + f'(x_0)h] }{h} = 0 \]

This proves that differentiable functions are such that $f(x_0 +h)$ can be represented by a linear approximate $f(x_0) + f'(x_0)h +r(h)$, where $r(h)$ is some remainder function that is dominated by $h$ in the sense that $\lim_{h \to 0}\frac{r(h)}{h}=0$; it is 'negligible' in the limit.

Differentiation from this perspective is easier to generalize; we don't necessarily need to define differentiability by expressing a direct Newton quotient, but by specifying that some linear function \emph{exists} that can approximate the function in this specific way. All we need to do now is generalize the idea of a real linear function from real analysis to real vector linear function!

Linear transforms are the generalization for higher dimension spaces, and these can be represented my matrixes; the analogue of out $f'(x_0)h$ will be $\mathbf{J}\mathbf{h}$, where $\mathbf{J}$ is some matrix that makes such a linear transform. In this vain, we can let our linear approximator be $\mathbf{f}(\mathbf{x}_0) + \mathbf{J}\mathbf{h}$.


\begin{definition}
A real vector function $\mathbf{f}$ is \emph{(totally) differentiable} iff there exists some matrix $\mathbf{J}$ where the following holds. The matrix $\mathbf{J}$ is the \emph{total derivative of $\mathbf{f}$}
\[  \lim_{\mathbf{h} \to \mathbf{0}} \frac{\| \mathbf{f}(\mathbf{x}_0 +\mathbf{h}) - ( \mathbf{f}(\mathbf{x}_0) + \mathbf{J}\mathbf{h} )  \|}{\| \mathbf{h} \|} = 0  \]
	\[  \mathbf{f} \text{ is differentiable at }\mathbf{x}_0 \iff  \exists \mathbf{J} [ \lim_{\mathbf{h} \to \mathbf{0}} \frac{\| \mathbf{f}(\mathbf{x}_0 +\mathbf{h}) - ( \mathbf{f}(\mathbf{x}_0) + \mathbf{J}\mathbf{h} )  \|}{\| \mathbf{h} \|} = 0 ] \]
\end{definition}

This is the ultimate definition for differentiability for any real vector function (I'll often drop the 'totally' part because I'm lazy). This condition is stronger than the notions of directional and partial derivatives; it entirely encapsulates the idea that the function can locally be approximated by a single linear transform.

We introduce a second equivalent totally differentiable function that mentions the use of a remainder function; this is extremely useful in proofs.

\begin{proposition}
A real vector function $\mathbf{f}$ is \emph{(totally) differentiable} iff there exists some matrix $\mathbf{J}$ and real vector function $\mathbf{r}$ where the following holds.
\[ \mathbf{f}(\mathbf{x}_0 + \mathbf{h} ) = \mathbf{f}(\mathbf{x}_0)+ \mathbf{J}\mathbf{h} + \mathbf{r}(\mathbf{h}) \]
\[  \lim_{\mathbf{h} \to \mathbf{0}} \frac{ \| \mathbf{r}(\mathbf{h}) \| }{ \| \mathbf{h \|}} = 0  \]
\end{proposition}

We now delve into the properties of totally differentiable functions.


\begin{proposition}
Let $\mathbf{f}$ be differentiable at $\mathbf{x}_0$, then $\mathbf{f}$ is continuous at $\mathbf{x}_0$.
\end{proposition}


\begin{proposition}
Differentiable implies directional derivative eists in all directions.
\end{proposition}


The following lemma will be used prove a theorem that will tell us exactly how to calculate the total derivative, however the lemma also has some standalone value since it links the total derivative to the directional derivative.

\begin{lemma}
	Let $\mathbf{f}$ be a function differentiable at $\mathbf{x}_0$ with total derivative $\mathbf{J}$, then $\nabla_{\mathbf{u}}\mathbf{f}(\mathbf{x}_0) =  \mathbf{J}^i(\mathbf{x}_0) \mathbf{u}$ , where $\mathbf{J}^i$ is the $i$th column of $\mathbf{J}$.
\end{lemma}

We now prove the following revelation.

\begin{theorem}
Let $\mathbf{f}$  be differentiable at $\mathbf{x}_0$, then the total derivative at $\mathbf{x}_0$ is the Jacobian matrix evaluated at $\mathbf{x}_0$ $\mathbf{J}_{\mathbf{f}}(\mathbf{x}_0)$ 
\end{theorem}

It was our Jacobain matrix all along; the Jacobian matrix represent this best linear transform at a point! If for a real function $f(x_0+h) \approx f(x_0) + f'(x_0)h$, then for a real vector function we have $\mathbf{f}(\mathbf{x}_0)$!

Using this fact, we can prove the following.
\begin{corollary}
	\[ \nabla_{\mathbf{u}} \mathbf{f} = \sum^{n}_{i=1} \mathbf{u}_{i} f_{\mathbf{x}_i} \]
\end{corollary}

We will have an even more elegant representation for this when we study differential operators.

\subsection{Properties of Jacobian matrix}

Now that we've discovered that the Jacobian matrix embodies the total derivative in a Euclidean space, it is fitting that we study this structure more closely. 

\begin{proposition}
	\[\mathbf{J}_{\mathbf{f}+\mathbf{g}} = \mathbf{J}_{\mathbf{f}} + \mathbf{J}_{\mathbf{g}} \]
	\[\mathbf{J}_{w \mathbf{f}} = w \mathbf{J}_{\mathbf{f}} + \mathbf{f}\mathbf{J}_{w} \]
\end{proposition}



\section{Multivariate chain rule}

We've already seen that composition of continuous functions are continuous in any space; however are compositions of differentiable functions differentiable? We've seen this to be true in real analysis, and the chain rule gives us a neat formula on how to compute derivatives with respect to the composed functions and their derivatives.

In higher dimensions the chain rule holds similarly in terms of the Jacobian matrix.

\begin{theorem}[Multivariate chain rule]
\[ \mathbf{J}_{\mathbf{f} \circ \mathbf{g}} = \mathbf{J}_{\mathbf{f}}(\mathbf{g})  \mathbf{J}_{\mathbf{g}} \]
\end{theorem}


We require a generalization of the chain rule for such compositions.
\begin{proposition}
\[ \frac{\partial \mathbf{f} }{\partial t } \sum \frac{\partial \mathbf{f}}{\partial \mathbf{x}_j } \frac{\partial \mathbf{x}_j }{\partial t}\]
\end{proposition}

\section{Clairaut's theorem}


\begin{theorem}[Clairaut's theorem]
\end{theorem}

This theorem has many names \emph{Clairaut's theorem, Young's theorem, Schwarz' theorem,} or simply the \emph{symmetry of second derivatives}.

\section{Multivariate inverse function theorem}



\begin{theorem}[Multivariate inverse function theorem]
	\[ \mathbf{J}_{\mathbf{f}^{-1}}(\mathbf{x}_0) = [\mathbf{J}_{\mathbf{f}}(\mathbf{x}_0)]^{-1}\]
\end{theorem}

\subsection{Multivariate inverse function theorem}
\subsection{Multivariate implicit function theorem}




\section{Multivariate Taylor's theorem}
\subsection{Multivariate Taylor's theorem}
\subsection{Multivariate Taylor series}

\section{Derivative tests}
\subsection{Properties of the Hessian matrix}

\begin{proposition}[Multivariate derivative test]
If a real multivariate function $f$ with continuous second derivatives has $\nabla f(\mathbf{x}_0)=\mathbf{0}$ and $\mathbf{H}_f (\mathbf{x}_0)$ is positive definite, then $\mathrm{x}_0$ is a local minimum of $f$.
\end{proposition}


\begin{proposition}
Let $f$ have continuous second-order partial derivatives on $U$. Then $f$ is convex on $U$ iff its Hessian matrix $\mathbf{H}_{f}$ is positive semi-definite.
\end{proposition}

\begin{proposition}
Let $f$ have continuous second-order partial derivatives on $U$. If the Hessian matrix $\mathbf{H}_{f}$ is positive definite on $U$, then $f$ is convex on $U$.
\end{proposition}

\subsection{Lagrangian multipliers}






\chapter{Multivariate integration}
\section{Multivariate integration}
\section{Fubini's theorem and Tonelli's theorem}

\subsection{Nested integration}


The theorems of Fubini and Tonelli assist immensely in the calculation and interpretation of multivariate integrals. They provide two different sufficient conditions for when one can swap the order of integral nesting.  It turns out we can do this for quite a rich class of functions, however proving these theorems requires us to interpret integration as Lebesgue integration rather than Riemann integration. If you are unfamiliar with the Lebesgue integral, this is covered at the end of Real Analysis.
\subsection{Fubini's theorem}

\begin{theorem}
Let $f$ be Lebesgue integrable on $X \times Y$, then we have the following.
\[ \iint_{X\times Y} f(x,y) d(x,y) =\int_{X} ( \int_{Y} f(x,y) dy )dx = \int_{Y} ( \int_{X} f(x,y) dx )dy \]
\end{theorem}
\subsection{Tonelli's theorem}

Though similar to Fubini's condition, Tonelli introduces a similar yet distinct condition that implies that nested integration can be swapped.


\begin{theorem}
Let $f$ be a nonnegative measurable function on $X \times Y$, then we have the following.
\[ \iint_{X\times Y} f(x,y) d(x,y) =\int_{X} ( \int_{Y} f(x,y) dy )dx = \int_{Y} ( \int_{X} f(x,y) dx )dy \]
\end{theorem}

\section{Fubini's theorem and Tonelli's theorem}








<h3 class=cyan>Vector-valued function</h3>
<p>Function \(\textbf{r} : X^n \to \mathbb{R}^m, m \geq 2\) that takes parameter vector or scalar as input and outputs a cartesian vector</p>
<h4 class=cyan>Vector field Campo vettoriale</h4>
<p>Vector-valued function  \(\textbf{F} : X^n \to \mathbb{R}^n\) that assigns a cartesian vector with same dimension as the space, to each point in said space \(X^n\)</p>
<p>\(\textbf{F} ( \textbf{x} )\)</p>
<h4 class=cyan>Position function</h4>
<p>Vector valued function \(\textbf{r}: [t_0 , t_1] \to \mathbb{R}^n \) that takes a scalar parameter \(t\) as input and outputs a cartesian vector, essentially representing a path in a space</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>


<h3 class=cyan>Coordinate system</h3>
<p>System of variables used to define a set of points in a space</p>
<h4 class=cyan>Point representation</h4>
<p>Each coordinate system has a set of equations to translate points in another system to said coordinate system</p>
<h4 class=cyan>Vector representation</h4>
<p>Each coordinate system has an orthonormal basis (frame) relative to a point in space that represents any vector from that point, e see <a href=/daigaku/2024_aut/la.php>Linear Algebra</a></p>
<p>Note that some orthonormal basises may be dependent on some \(\theta\) or \(\phi\) relational to the vector's base from the origin</p>
<h4 class=cyan>\(\mathbb{R}^2\)</h4>
<ul>
	<li>Cartesian</li>
	<li>Polar</li>
	<li>Parabolic</li>
	<li>Bipolar</li>
	<li>Elliptic</li>
</ul>
<h4 class=cyan>\(\mathbb{R}^3\)</h4>
<ul>
	<li>Cartesian</li>
	<li>Cylindrical</li>
	<li>Spherical</li>
</ul>


			<h3 class=cyan>Cartesian coordinates</h3>
			<p>Ordered 3-tuple \( (x,y,z) \) that  represents a point in a 3D space</p>
			<ul>
				<li>\( x \in \mathbb{R}\) is the translation along the x-axis</li>
				<li>\( y \in \mathbb{R}\) is the horizontal translation perpendicular to the x-axis</li>
				<li>\( z \in \mathbb{R}\) is the vertical translation</li>
			</ul>
			<h4 class=cyan>Vector transform</h4>
			<p>\( \text{span}\{ \hat{i}, \hat{j}, \hat{k}\} = \mathbb{R}^3 \)</p>
			<ul>
				<li>\( \hat{i}  = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \)</li>
				<li>\( \hat{j}  = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \)</li>
				<li>\( \hat{k}  = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \)</li>
			</ul>
			<h4 class=cyan>Volume element</h4>
			<p>\( dV = dxdydz \)</p>


			<h3 class=cyan>Polar coordinates</h3>
			<p>Ordered pair \( (r,\theta) \) that represents any point and vector in a 2D space</p>
			<ul>
				<li>\(r \in \mathbb{R}_{+}\) represents the modulus</li>
				<li>\(\theta \in [0,2\pi] \) represents the azimuthal angle (angle from x to y)</li>
			</ul>
			<h4 class=cyan>Point transfom</h4>
			<ul>
				<li>\(r=\sqrt{x^2 + y^2}\)</li>
				<li>\(\theta = \arctan (\frac{y}{x})\)</li>
			</ul>
			<ul>
				<li>\(x=r\cos (\theta)\)</li>
				<li>\(y=r\sin (\theta)\)</li>
			</ul>
			<h4 class=cyan>Vector transform</h4>
			<p>For some chosen angle \(\theta\)</p>
			<p>\( \text{span}\{ \hat{r}, \hat{\theta}\} = \mathbb{R}^2 \)</p>
			<ul>
				<li>\( \hat{r}  = \begin{pmatrix} \cos (\theta) \\ \sin (\theta) \end{pmatrix} \)</li>
				<li>\( \hat{\theta}  = \begin{pmatrix} -\sin (\theta) \\ \cos (\theta) \end{pmatrix} \)</li>
			</ul>
			<p>\( P_{ (r,\theta) \to (x,y)} = \begin{bmatrix} \cos (\theta) & -\sin(\theta) \\  \sin (\theta) & \cos (\theta) \end{bmatrix}\)</p>
			<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>
			</ul>
			<h4 class=cyan>Area element</h4>
			<p>\( dA = rdrd\theta \)</p>


			<h3 class=cyan>Cylindrical coordinates</h3>
			<p>Ordered 3-tuple \( (r,\theta,z) \) that represents any point and vector in a 3D space</p>
			<ul>
				<li>\(r \in \mathbb{R}_{+}\) represents the modulus projected along the x-y plane</li>
				<li>\(\theta \in [0,2\pi]\) represents the azimuthal angle (horizontally from +x-axis to +y-axis)</li>
				<li>\(z \in \mathbb{R}\) represents vertical translation</li>
			</ul>
			<h4 class=cyan>Point transform</h4>
			<ul>
				<li>\(r=\sqrt{x^2 + y^2}\)</li>
				<li>\(\theta = \arctan (\frac{y}{x})\)</li>
				<li>\(z=z\)</li>
			</ul>
			<ul>
				<li>\(x=r\cos (\theta)\)</li>
				<li>\(y=r\sin (\theta)\)</li>
				<li>\(z=z\)</li>
			</ul>
			<h4 class=cyan>Vector transform</h4>
			<p>For some chosen angle \(\theta\)</p>
			<p>\( \text{span}\{ \hat{r}, \hat{\theta}, \hat{z}\} = \mathbb{R}^3 \)</p>
			<ul>
				<li>\( \hat{r}  = \begin{pmatrix} \cos (\theta) \\ \sin (\theta) \\ 0 \end{pmatrix} \)</li>
				<li>\( \hat{\theta}  = \begin{pmatrix} -\sin (\theta) \\ \cos (\theta) \\ 0 \end{pmatrix} \)</li>
				<li>\( \hat{z}  = \begin{pmatrix} 0  \\ 0 \\ 1 \end{pmatrix} \)</li>
			</ul>
			<p>\( P_{ (r,\theta, z) \to (x,y,z)} = \begin{bmatrix} \cos (\theta) & -\sin(\theta) & 0 \\  \sin (\theta) & \cos (\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix}\)</p>
			<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>Volume element</h4>
			<p>\( dV = rdrd\theta dz \)</p>


			<h3 class=cyan>Spherical coordinates</h3>
			<p>Ordered 3-tuple \( (\rho,\theta,\phi) \) that represents any point and vector in a 3D space</p>
			<ul>
				<li>\(\rho \in \mathbb{R}_{+}\) represents the modulus</li>
				<li>\(\theta \in [0,\pi] \) represents the inclination angle (vertically from +z-axis to x-y-plane)</li>
				<li>\(\phi \in [0,2\pi]\) represents the azimuthal angle (horizontally from +x-axis to +y-axis)</li>
			</ul>
			<h4 class=cyan>Point transform</h4>
			<ul>
				<li>\(\rho=\sqrt{x^2 + y^2 + z^2}\)</li>
				<li>\(\theta= \arccos (\frac{z}{\rho})\)</li>
				<li>\(\phi= \arctan (\frac{y}{x})\)</li>
			</ul>
			<ul>
				<li>\(x=\rho \sin(\theta) \cos (\phi)  \)</li>
				<li>\(y=\rho  \sin (\theta)\sin (\phi) \)</li>
				<li>\(z=\rho \cos (\theta)\)</li>
			</ul>
			<h4 class=cyan>Vector transform</h4>
			<p>\( \forall \theta,\phi\in [0,2\pi] ,\text{span}\{ \hat{\rho}, \hat{\theta}, \hat{\varphi}\} = \mathbb{R}^3 \)</p>
			<ul>
				<li>\( \hat{\rho}  = \begin{pmatrix} \sin (\theta) \cos (\phi) \\ \sin (\theta) \sin (\phi) \\ \cos (\theta) \end{pmatrix} \)</li>
				<li>\( \hat{\theta}  = \begin{pmatrix} \cos(\theta) \cos (\varphi) \\ \cos(\theta) \sin (\varphi) \\ -\sin (\theta) \end{pmatrix} \)</li>
				<li>\( \hat{\varphi}  = \begin{pmatrix} -\sin (\varphi)  \\ \cos (\varphi)  \\ 0 \end{pmatrix} \)</li>
			</ul>
			<p>\( P_{ (\rho,\theta, \phi) \to (x,y,z)} = \begin{bmatrix} \sin (\theta) \cos (\phi) & \cos(\theta) \cos(\phi) & -\sin(\phi) \\  \sin(\theta) \sin(\phi) & \cos(\theta) \sin(\phi) & \cos( \phi ) \\ \cos(\theta) & -\sin (\theta) & 0 \end{bmatrix}\)</p>
			<h4 class=cyan>Vector operators</h4>
				<li>\(\nabla f = \frac{\partial f}{\partial \rho} \hat{\rho} + \frac{1}{\rho}\frac{\partial f}{\partial \theta} \hat{\theta}+ \frac{1}{\rho \sin (\theta)}\frac{\partial f}{\partial \varphi} \hat{\varphi}\)</li>
				<li>\(\nabla \cdot \textbf{F} = \frac{1}{{\rho}^2}\frac{\partial ( {\rho} ^2 F_{\rho})}{\partial \rho}+  \frac{1}{\rho \sin (\theta) } \frac{\partial ( \sin (\theta) F_{\theta} )}{\partial \theta}+  \frac{1}{\rho \sin (\theta)} \frac{\partial F_{\varphi}}{\partial \varphi}\)</li>
				<li>\(\nabla \times \textbf{F} = \frac{1}{\rho \sin (\theta)} ( \frac{\partial ( \sin (\theta) F_{\varphi}) }{\partial \theta} - \frac{\partial F_{\theta}}{\partial \varphi}) \hat{\rho} + \frac{1}{\rho} ( \frac{1}{\sin (\theta)}\frac{\partial F_{\rho}}{\partial \varphi} - \frac{\partial (\rho F_{\varphi}) }{\partial \rho}) \hat{\theta} + \frac{1}{\rho} (\frac{\partial (\rho F_{\theta})}{\partial \rho} - \frac{\partial F_{\rho}}{\partial \theta}) \hat{\varphi}\)</li>
				<li>\(\nabla^2 = \frac{1}{\rho^2} \frac{\partial }{ \partial \rho } (\rho^2 \frac{\partial f}{\partial \rho}) + \frac{1}{\rho^2 \sin ( \theta )} \frac{\partial }{\partial \theta } ( \sin ( \theta )  \frac{\partial f }{\partial \theta } )  +  \frac{1}{\rho^2 \sin ^2 (\theta  )} \frac{\partial ^2 f }{ \partial \phi} \)</li>
			</ul>
			<h4 class=cyan>Volume element</h4>
			<p>\( dV = \rho ^2 \sin ( \theta ) d\rho d\theta d\phi \)</p>







