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\begin{document}

\title{Leibniz's notation}
\author{@masaka}
\date{January 12th, 2013}
\maketitle

\tableofcontents

\section[Introduction]{ Introduction}
The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates – see Development of the quarrel below) over who had first invented calculus. It is a question that had been the cause of a major intellectual controversy over who first discovered calculus, one that began simmering in 1699 and broke out in full force in 1711.


\section{Leibniz's notation}

Leibniz notation centers around the concept of a differential element.
The differential element of $x$ is represented by $dx$.
You might think of $dx$ as being an infinitesimal change in $x$. It is important
to note that $d$ is an operator, not a variable. So, when you see $\frac{dy}{dx}$,
you can't automatically write as a replacement $\frac{y}{x}$.

We use $\frac{df(x)}{dx}$ or $\frac{d}{dx}f(x)$ to represent the derivative of a
function $f(x)$ with respect to $x$.
$$ \frac{df(x)}{dx} = \lim_{Dx \to 0} \frac{f(x+Dx) - f(x)}{Dx} $$
We are dividing two numbers infinitely close to 0,
and arriving at a finite answer. $D$ is another operator that can be
thought of just a change in $x$. When we take the limit of $Dx$ as $Dx$ approaches 0,
we get an infinitesimal change $dx$.

Leibniz notation shows a wonderful use in the following example:
$$ \frac{dy}{dx} = \frac{dy}{dx} \frac{du}{du} = \frac{dy}{du} \frac{du}{dx} $$
The two $du$s can be cancelled out to arrive at the original derivative.
This is the Leibniz notation for the Chain Rule.

Leibniz notation shows up in the most common way of representing an integral,
$$ F(x) = \int f(x) dx $$
The $dx$ is in fact a differential element. Let's start with a derivative that
we know (since $F(x)$ is an antiderivative of $f(x)$).
\begin{eqnarray*}
\frac{dF(x)}{dx} & = & f(x) \\
dF(x) & = & f(x)dx \\
\int dF(x) & = & \int f(x)dx \\
F(x) & = & \int f(x) dx
\end{eqnarray*}
We can think of $dF(x)$ as the differential element of area. Since $dF(x) = f(x) dx$,
the element of area is a rectangle, with $f(x) \times dx$ as its dimensions. Integration is
the sum of all these infinitely thin elements of area along a certain interval. The result: a finite number.


One clear advantage of this notation is seen when finding the length $s$ of a curve.
The formula is often seen as the following:
$$ s = \int ds $$
The length is the sum of all the elements, $ds$, of length. If we have a function
$f(x)$, the length element is usually written as $ ds = \sqrt{1+[\frac{df(x)}{dx}]^2} dx $. If we
modify this a bit, we get $ ds = \sqrt{[dx]^2 + [df(x)]^2} $. Graphically, we
could say that the length element is the hypotenuse of a right triangle with one
leg being the $x$ element, and the other leg being the $f(x)$ element.


There are a few caveats, such as if you want to take the value of a
derivative. Compare to the prime notation.
$$ f'(a) = \left. \frac{df(x)}{dx} \right |_{x=a} $$

A second derivative is represented as follows:
$$ \frac{d}{dx} \frac{dy}{dx} = \frac{d^2y}{dx^2} $$
The other derivatives follow as can be expected: $\frac{d^3y}{dx^3}$, etc.
You might think this is a little sneaky, but it is the notation. Properly using
these terms can be interesting. For example, what is $\int \frac{d^2y}{dx} $? We
could turn it into $\int \frac{d^2y}{dx^2} dx$ or $\int d\frac{dy}{dx} $.
Either way, we get $\frac{dy}{dx}$.



\section{History}

The Newton-Leibniz approach to infinitesimal calculus was introduced in the 17th century. While Newton did not have a standard notation for integration, Leibniz began using the $$\int$$ character. He based the character on the Latin word summa ("sum"), which he wrote $\int$umma with the elongated \emph{\textbf{S}} commonly used in Germany at the time. This use first appeared publicly in his paper De Geometria, published in Acta Eruditorum of June 1686, but he had been using it in private manuscripts at least since 1675.
In the 19th century, mathematicians ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. A number of 19th century mathematicians (Cauchy, Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above. Nonetheless, Leibniz's notation is still in general use. Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. In physical applications, one may for example regard $f(x)$ as measured in meters per second, and $dx$ in seconds, so that $f(x)dx$ is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with dimensional analysis.
In the 1960s, building upon earlier work by Edwin Hewitt and Jerzy {\L}o\'{s}, Abraham Robinson developed rigorous mathematical explanations for Leibniz' intuitive notion of the "infinitesimal," and developed non-standard analysis based on these ideas. Robinson's methods are used by only a minority of mathematicians. Jerome Keisler wrote a first-year-calculus textbook based on Robinson's approach.

\section{the difference between Leibniz's notation and Lagrange's notation}
There are two main types of notation used to denote the derivative of a function

\textbf{Lagrange’s Notation} is to write the derivative of the function $f(x)$ as
$$f'(x).$$

\textbf{Leibniz’s Notation} is to write the derivative of the function $f$ as
$$\frac{df}{dx}.$$
Two other notations are worth mentionin

\textbf{Newton’s Notation} is to write the derivative of y using a dot
$$\dot{y}.$$

\textbf{Euler’s Notation} is to use a capital D i.e.
$$D_{x}f(x).$$
The Lagrange and Leibniz notation will be considered in some situations involving
differentiation. It may be that the comments are influenced too much
by the particular methods of teaching received by the author. Any further comments are welcome.
\begin{enumerate}
  \item Functions of a single variable
  \begin{enumerate}
    \item Basic
    \begin{center}
    \begin{tabular}{|c|c|c|}
      \hline
      % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
       & Lagrange & Leibniz \\\hline
      Function& $f(x)$ & $f$ \\\hline
      Derivative & $f^{'}(x)$ & $\frac{df}{dx}$ \\\hline
      2nd Derivative & $f^{’’}(x)$ & $\frac{d^{2}f}{dx^{2}}$ \\\hline
      Higher Derivative & $f^{n}(x)$ & $\frac{d^{n}f}{dx^{n}}$ \\\hline
      Integral & & $\int f(x)dx $ \\
      \hline
    \end{tabular}
    \end{center}
   \emph{ Comments} For the higher derivatives the (n) is a little cumbersomeand can possible be mistaken for an index. An integral is rarely seen without a $dx$ so there is no entry in the Lagrange Column.
    \item Differention Rules
    \begin{center}
    \begin{tabular}{|c|c|}
      \hline
      % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
      Product Rule & \\\hline
      Lagrange & $[u(x)v(x)'=u(x)v'(x)+u'(x)v(x)]$ \\\hline
      Leibniz & $\frac{d}{dx}[uv]=u\frac{dv}{dx}+v\frac{du}{dx}$ \\
      \hline
    \end{tabular}
    \end{center}
    \emph{Comments} The Leibniz notation is probably more common here.

    \begin{center}
    \begin{tabular}{|c|c|}
      \hline
      % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
      Chain Rule &  \\\hline
      Lagrange & $f[g(x)]'=f'[g(x)]\times g'(x)$\\\hline
      Leibniz & $\frac{d}{dx}[f(g(x))]= \frac{df}{dg} \times \frac{dg}{dx} $\\
      \hline
    \end{tabular}
    \end{center}
   \emph{ Comments} Neither set looks ’comfortable in its entirety. The most comfortable may be a mixture such as $\frac{d}{dx}[f(g(x))]=f'[g(x)]\times g'(x).$




    \item Integration
    \begin{center}
    \begin{tabular}{|c|c|}
      \hline
      % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
      Integration by parts &  \\\hline
      Lagrange & $\int u(x)v'(x)dx=u(x)v(x)-\int v(x)u'(x)dx$ \\\hline
      Laibniz & $\int u\frac{dv}{dx}=uv-\int v\frac{du}{dx}dx$ \\
      \hline
    \end{tabular}
    \end{center}
    \emph{Comments} The Lagrange notation is certainly more common here.
    \begin{center}
    \begin{tabular}{|c|c|}
      \hline
      % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
      Arc Length &  \\\hline
      Lagrange & $\int_{a}^{b}\sqrt{1+[f'(x)^{2}]}dx$\\\hline
      Leibniz & $\int_{a}^{b}\sqrt{1+[(\frac{df}{dx})^{2}]}dx$\\
      \hline
    \end{tabular}
    \end{center}
    \emph{Comments} Not a great deal to choose between the two.




  \end{enumerate}
  \item Functions of Two Variables
  \begin{enumerate}
    \item Basic
    \begin{center}
    \begin{tabular}{|c|c|c|}
      \hline
      % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
       & Lagrange & Leibniz \\\hline
      Function & $f(x,y)$ & $f$ \\\hline
      Derivative & $f_{x},f_{y}$ & $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}$ \\\hline
      2nd Derivative & $f_{xx},f_{xy},f_{yy}$ & $\frac{\partial^{2}f}{\partial x^{2}},\frac{\partial^{2}f}{\partial xy},\frac{\partial^{2}f}{\partial y^{2}}$ \\\hline
      Higher Derivative & $f_{xxxyyyy}$ & $\frac{\partial ^{5}f}{\partial x^{3}\partial y^{4}}$ \\
      \hline
    \end{tabular}
    \end{center}
    \emph{Comments} A long series of subscripts can start to look a bit clumsy.

    \item Others
    \begin{center}
    \begin{tabular}{|c|c|}
      \hline
      % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
      Partial Differential Equation &  \\\hline
      Lagrange & $x^{2}f_{x}-2xyf_{y}=1$ \\\hline
      Leibniz & $x^{2}\frac{\partial f}{\partial x}-2xy\frac{\partial f}{\partial y}=1$ \\
      \hline
    \end{tabular}
    \end{center}
    \emph{Comments} Again, the Liebniz notation is certainly more common here.

    \begin{center}
    \begin{tabular}{|c|c|}
      \hline
      % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
      Checking PDE solution &  \\\hline
      Lagrange & $f_{x}=2xyg'(x^{2}y)+\frac{1}{x^{2}}$ \\\hline
      Leibniz & $\frac{\partial f}{\partial x}=2xy\frac{\partial g(x^{2}y)}{\partial x^{2}y}+\frac{1}{x^{2}}$
       \\
      \hline
    \end{tabular}
    \end{center}
    \emph{Comments} The Leibniz notation is having difficulty here with terms such as$\frac{\partial g(x^{2}y)}{\partial x^{2}y}$ being extremely clumsy.
    \end{enumerate}
\end{enumerate}

\begin{thm}
If we have the following conditions:
\begin{enumerate}
\item $f(x)$ is continuous on $[a,b]$,
\item $f(x)$ is derivable on $(a,b)$,
\item $f(a)$ and $f(b)$ have the same value,
\end{enumerate}
Then there exists $\xi\in(a,b)$ such that $f'(\xi)=0$.
\end{thm}

\section{Conclusion}
In general, the Leibniz notation rests more comfortably with these examples.However, there were several cases where the Lagrange notation had a slight advantage. For the final case of checking the solution of a partial differential equation, this was a large and significant advantage.For the final case of checking the solution of a partial differential equation, this was a large and significant advantage.
\end{document}