\ifdefined\inmaster\else\def\subonly{\jobname}\input{calc3}\fi

\chapter{Differential forms in one dimension}
\label{cha:1dforms}

In your previous calculus classes, you may or may not have encountered \emph{differential forms} by name.
However, you've certainly met them, even if you didn't realize it.
When you write
\[ \int_{x=a}^b f(x) \,\dx \]
the thing being integrated, ``$f(x) \,\dx$'', is a differential form.

You may have been told that the ``$\dx$'' is simply a notation that indicates which variable we're integrating, but this is a lie.
In multivariable calculus, we can no longer maintain this fiction: we have to treat differential forms as honest objects.
Fortunately, as we will see, we also have some advantages over one-variable calculus in understanding differential forms.

In this chapter, we will learn about differential forms in one dimension.
We will use them to rephrase, and improve on, the concepts of one-variable calculus.
Then in later chapters we will generalize them to higher dimensions.

\section{Differential forms and their orders}
\label{sec:1d-differential-forms}

If $\dx$ isn't just a notation indicating the integration variable, what is it?
If you encountered differentials in one-variable calculus, you may have been told that $\dx$ is a small change in $x$ (sometimes denoted $\Delta x$), or even an ``infinitesimal'' change in $x$.
Formally, however, it is just another variable.

\begin{defn}
  A \textbf{differential 1-form (in one dimension)} is a function of two variables, one of which is the ``differential'' of the other and denoted with a $\dd$ in front.
  Thus, if the first variable is $x$, the second is $\dx$.
  On the other hand, if the first variable is $t$, then the second is $\dt$, and so on.
\end{defn}

We usually denote differential forms by lowercase Greek letters such as the following.
\begin{itemize}
\item $\omega$ (omega, not to be confused with the English letter $w$)
\item $\eta$ (eta, not to be confused with the English letter $n$)
\end{itemize}
Thus, we may write $\omega(x,\dx)$ to emphasize that $\omega$ is a function of $x$ and $\dx$.

\begin{egs}\label{egs:1d-differential-forms}
  The following are all differential 1-forms:
  \begin{gather*}
    x^2 \,\dx\\
    \half({x-2}+e^x)\,\dx\\
    1 + \dx + \half \dx^2 + \third \dx^3\\
    \sin(x + \dx) - \sin(x)\\
    |\dx|
  \end{gather*}
\end{egs}

Note that although $\dx$ is two letters, we regard it as one symbol standing for one variable.
In particular, an expression such as $\dx^2$ means $(\dx)^2$, not $\dd(x^2)$.
(We will give a different meaning to $\dd(x^2)$ in \cref{sec:differentials}.)

The first two of \cref{egs:1d-differential-forms} are special in the following way.

\begin{defn}
  A \textbf{linear} differential 1-form (in one dimension) is one defined by an expression such as
  \[ f(x) \, \dx \]
  where $f$ is a function only of the variable $x$.
\end{defn}

Most mathematicians only use the term ``differential form'' for linear ones.
However, the more general ones are quite useful.
In later chapters we will encounter still more general kinds of ``differential forms''.

Although formally $\dx$ is just another variable, we think of it as a \emph{small change} in the value of $x$.
Thus, a differential 1-form is a function whose input consists of a number $x$ together with a small change $\dx$ in the value of that number.
Specifically, by \emph{small} we mean \emph{small relative to the values generally taken by $x$}.
For instance, if $x=3$, then a possible small change would be $\dx = 0.01$.
In this case, a differential form such as $\dx^2$ is an \emph{even smaller} change, since $\dx^2 = 0.0001$.
We say that $\dx$ is a \emph{first order} form while $\dx^2$ is \emph{second order}.
Similarly, we say that $x$ itself is \emph{zeroth order}, as is any function value $f(x)$ not involving $\dx$.

Every linear form is first-order: when we multiply a zeroth order form like $f(x)$ by a first order one like $\dx$, the result is first order.
For instance, if $f(x) = 4$ and $\dx = 0.001$, then $f(x)\,\dx = 0.004$, which is about the same size as $\dx$.
A form such as $|\dx|$ is also first-order: no matter whether $\dx$ is positive or negative, its absolute value is about the same size.

If a form is a polynomial in $\dx$ (with ``coefficients'' that are arbitrary functions of $x$), such as $\half x\,\dx + 3x^2\,\dx^2 - e^{x^2}\, \dx^3$, then its order is the smallest exponent of $\dx$ that appears.
For instance, if $x=3$ and $\dx=0.001$, then $x\,\dx - \dx^2 = 0.002999$, which is about the same size as $\dx$; thus $x\,\dx - \dx^2$ is first-order.
On the other hand, still with $x=3$ and $\dx=0.001$, we have $\frac{1}{3}\dx^2 + x \,\dx^3 \approx 0.000000336$, which is about the same size as $\dx^2 = 0.000001$; thus $\frac{1}{3}\dx^2 + x \,\dx^3$ is second-order.

The concept of order can also be applied to forms that are not polynomials in $\dx$.
For instance, if $\dx=0.01$, then $\sin(\dx) \approx 0.0099998$, which is about the same size as $\dx$.
Thus, it is reasonable to expect that $\sin(\dx)$ is also first-order.
On the other hand, still with $\dx=0.01$, we have $\cos(\dx) - 1 \approx -0.00005$, which is much smaller than $\dx$ and about the same size as $\dx^2 = 0.0001$.
Thus, it is reasonable to expect that $\cos(\dx) - 1$ is second-order.

We can also see the orders of $\sin(\dx)$ and $\cos(\dx)-1$ by looking at their power series expansions.
Recall that the Taylor series of $\sin$ at $0$ gives
\[ \sin(\dx) = \dx - \frac{\dx^3}{3!} + \frac{\dx^5}{5!} - \cdots. \]
This is an ``infinite polynomial'' whose lowest-order term has degree $1$; thus the same rule for orders of polynomials says that it is first-order.
Similarly, the Taylor series of $\cos$ at $0$ gives
\[ \cos(\dx) - 1 = - \frac{\dx^2}{2!} + \frac{\dx^4}{4!}- \frac{\dx^6}{6!} + \cdots \]
whose lowest-order term has degree $2$, so that $\cos(\dx)-1$ should be second-order.

It is possible to give a precise definition of the word ``order'' in general, but it is a bit complicated and not necessary for our purposes, so we will not do so.
All that we will need is a definition of when a differential form is \emph{smaller than first order} (e.g.\ second-order or third-order).
This is because in calculus, we are concerned with \emph{approximations to first order}, also known as \emph{linear approximations}; and forms that are smaller than first order do not affect these approximations.
Thus, we will call them \emph{negligible} (to first order).

\begin{defn}\label{def:negligible}
  A differential 1-form $\omega$ is called \textbf{negligible} at a point $x$ if
  \[ \lim_{\dx\to 0} \frac{\omega(x,\dx)}{\dx} = 0. \]
\end{defn}

Recall that definitions in mathematics introduce a \emph{new} meaning for a word.
Thus, \emph{from now on} (in this class) the word ``negligible'' \emph{will mean} what this definition says it does.
Our previous discussion was only motivational; the definition is the ``gold standard'' for the meaning of the word.

\begin{eg}
  The differential form $\dx^2$ is negligible at any point, since
  \[ \lim_{\dx\to 0} \frac{\dx^2}{\dx} = \lim_{\dx\to 0} \dx = 0. \]
  Similarly, if $f$ is any function of $x$, then $f(x)\,\dx^2$ is negligible at any point, since
  \[ \lim_{\dx\to 0} \frac{f(x)\,\dx^2}{\dx} = f(x)\, \lim_{\dx\to 0} \dx = 0. \]
\end{eg}

\begin{eg}
  More generally, if $n > 1$, then $\dx^n$ is negligible at any point, since
  \[ \lim_{\dx\to 0} \frac{\dx^n}{\dx} = \lim_{\dx\to 0} \dx^{n-1} = 0. \]
  (The final limit is equal to zero only if $n>1$.)
  Similarly, $f(x)\,\dx^n$ is also negligible.
\end{eg}

\begin{eg}
  On the other hand, for a \emph{linear} differential 1-form $f(x)\,\dx$, we have
  \[ \lim_{\dx\to 0} \frac{f(x)\,\dx}{\dx} = \lim_{\dx\to 0} f(x) = f(x). \]
  Thus, a linear differential 1-form is negligible only if $f(x) = 0$.
\end{eg}

\begin{eg}
  The differential 1-form $|\dx|$ is also not negligible, for we have
  \[ \frac{|\dx|}{\dx} =
  \begin{cases}
    1 &\quad \dx>0\\
    -1 &\quad \dx<0
  \end{cases}\]
  which does not have a limit as $\dx\to 0$.
\end{eg}

\begin{eg}
  If $\omega$ and $\eta$ are negligible at $\ptx$, then we have
  \[ \lim_{\dx\to 0} \frac{\omega(x,\dx) + \eta(x,\dx)}{\dx} =
  \left(\lim_{\dx\to 0} \frac{\omega(x,\dx)}{\dx}\right) + \left(\lim_{\dx\to 0} \frac{\eta(x,\dx)}{\dx}\right)
  = 0 + 0 = 0.
  \]
  Thus, the sum of two negligble 1-forms is again negligible.
  This implies that a polynomial in $\dx$, with coefficients that are functions of $x$, is negligible if all the exponents of $\dx$ are at least $2$.
  For instance, $\frac{1}{x}\,\dx^2 + e^x \,\dx^3$ is negligible.
\end{eg}

\begin{eg}
  Suppose that $\lim_{\dx\to 0} \omega(x,\dx) = L$ exists (and is finite), while $\eta$ is negligible at $x$ so that $\lim_{\dx\to 0} \frac{\eta(x,\dx)}{\dx}= 0$.
  Then we have
  \[ \lim_{\dx\to 0} \frac{\omega(x,\dx) \, \eta(x,\dx)}{\dx} = \left(\lim_{\dx\to 0} \omega(x,\dx)\right) \left(\lim_{\dx\to 0} \frac{\eta(x,\dx)}{\dx}\right) = L \cdot 0 = 0 \]
  Thus, the product $\omega\,\eta$ is also negligible at $x$.
\end{eg}

Notice that all the examples of negligible forms are what we would intuitively call second-order or higher.
We will continue to use the informal terminology of ``first-order'', ``second-order'', and so on when discussing forms.
However, in definitions and theorems we will use only the term ``negligible'' that we have defined precisely.

\section{Integrating 1-forms}
\label{sec:1d-integration}

When we ``integrate a function $f$'' in one-variable calculus from $a$ to $b$, we divide up the interval $[a,b]$ into a large number $N$ of small subintervals $[x_{i-1},x_i]$ of width $\dx_i$, with $x_0 = a$ and $x_N=b$:
\begin{center}
  \begin{tikzpicture}[yscale=0.3,xscale=1.5]
    \draw[<->] (-5,0) -- (5,0);
    \draw (-3,-1) -- +(0,2) node[above] {$a=x_0$};
    \draw (-1.5,-1) -- +(0,2) node[above] {$x_1$};
    \draw (-1,-1) -- +(0,2) node[above] {$x_2$};
    \draw (0,-1) -- +(0,2) node[above] {$x_3$};
    \node at (.75,2) {$\cdots$};
    \draw (1.5,-1) -- +(0,2) node[above] {$x_{N-2}$};
    \draw (2.5,-1) -- +(0,2) node[above] {$x_{N-1}$};
    \draw (4,-1) -- +(0,2) node[above] {$x_N=b$};
  \end{tikzpicture}
\end{center}

and consider the \emph{Riemann sum}
\[ \sum_{i=1}^N f(\rstag x i)\, \dx_i \]
where $\rstag x i$ is some point in $[x_{i-1},x_i]$.
Then we take a limit as the subinterval widths $\dx_i$ go to zero.

If we regard this instead as integrating the differential form $\omega(x,\dx) = f(x)\, \dx$, we can rewrite the Riemann sum as
\[ \sum_{i=1}^N \omega(\rstag x i,\dx_i). \]
In other words, the differential form specifies, for each value $\rstag x i$ of $x$ and each subinterval width $\dx_i$, the appropriate (approximate) contribution $\omega(\rstag x i,\dx_i)$ to the integral.
We simply add up these contributions, and then take the limit.

We can now generalizes this to integrate \emph{any} differential form in one dimension.
We simply define
\[ \lint{[a,b]} \omega = \lim_{\dx_i \to 0} \sum_{i=1}^N \omega(\rstag x i,\dx_i). \]
if this limit exists.

Note that when integrating a differential form, we do not write a ``$\dx$'' at the end of the integral: we simply write $\lint{[a,b]} \omega$.
The $\dx$ at the end of an ordinary integral is \emph{part} of the differential form being integrated.

Furthermore, rather than writing $a$ and $b$ at the top and bottom of the integral in the usual way, we write the interval $[a,b]$ as a subscript.
If we want to emphasize the variable being integrated, then we can write $\lint{x\in [a,b]} \omega$; as in one-variable calculus this is useful when integrating by substitution.

\begin{eg}\label{eg:onevar-linint}
  

  On the other hand, we can also consider also the case when $b<a$.
  In one-variable calculus, you probably learned that
  \begin{equation}\label{eq:onevar-integral-revorient}
    \itint x b a {f(x} = - \itint x a b {f(x)}
  \end{equation}
  but you might not have been given a principled reason why.
  The reason is that when $b<a$, the intermediate points $x_i$ that ``cover the distance from $a$ to $b$'' have to be \emph{decreasing} in order to get \emph{from} the greater number $a$ \emph{to} the lesser number $b$:
  \[ a = x_0 > x_1 > \cdots > x_{N-1} > x_N = b.\]
  In particular, $x_i < x_{i-1}$ for all $i$, so their difference $\Delta x_i = x_i - x_{i-1}$ is \emph{negative}.
  Apart from this minus sign, however, the Riemann sums for $\itint x a b {f(x)}$ will be the same as those for $\itint x b a {f(x)}$, so that \cref{eq:onevar-integral-revorient} holds.
\end{eg}

\begin{eg}\label{eg:onevar-absint}
  As a first new example, consider the form $f(x) \, |\dx|$ in one dimension, where $f$ is some continuous function.
  If $a<b$, then in our partition of the interval $[a,b]$ the intermediate points $x_i$ \emph{increase} from $a$ to $b$:
  \[ a = x_0 < x_1 < \cdots < x_{N-1} < x_N = b. \]
  In particular, $x_i > x_{i-1}$ for all $i$, so their difference $\Delta x_i = x_i - x_{i-1}$ is positive.
  Thus, the Riemann sums for $\int{[a,b]} f(x) \, |\dx|$ are
  \[ \sum_{i=1}^N f(\rstag x i)\, |\Delta x_i| = \sum_{i=1}^N f(\rstag x i)\, \Delta x_i\]
  which are exactly the same as the Riemann sums for $\itint x a b {f(x)}$.
  Thus, if $a<b$, then $\lint{[a,b]} f(x) \, |\dx| = \itint x a b {f(x)}$.

  However if we instead consider the integral $\lint{[a,b]} f(x) \, |\dx|$ when $a<b$, then the Riemann sums will be the \emph{same} as those for $\lint{[b,a]} f(x) \, |\dx|$, because of the absolute value.
  Thus in contrast to \cref{eq:onevar-integral-revorient}, we have
  \begin{equation}\label{eq:onevar-absint-revorient}
    \lint{[b,a]} f(x)\,|\dx| = \lint{[a,b]} f(x) \,|\dx|.
  \end{equation}
  In conclusion, therefore, we can say that
  \begin{equation}
    \lint{[a,b]} f(x) \,|\dx| =
    \begin{cases}
      \displaystyle\itint x a b {f(x)} \qquad\text{if } a<b\\[10pt]
      \displaystyle\itint x b a {f(x)} \qquad\text{if } b<a.
    \end{cases}\label{eq:onevar-absint}
  \end{equation}
  In other words, the differential form $f(x)\,\dx$ notices the ``direction'' (or \emph{orientation}) of the interval over which it is integrated, and gives a different result whether we integrate it ``from $a$ to $b$'' or ``from $b$ to $a$'' --- whereas the form $f(x)\,|\dx|$ does not notice the direction.

  (This is not, of course, to say that the integral $\lint{[a,b]} f(x) \,|\dx|$ is always a positive number: the function $f$ might take on negative values.)
\end{eg}

\begin{eg}
  Consider the form $\dx^2$.
  If we have a division into subintervals $[x_{i-1},x_i]$ such that $\Delta x_i < \varepsilon$ for all $i$, then the corresponding Riemann sum is
  \begin{align*}
    \sum_{i=1}^N (\Delta x_i)^2 &< \varepsilon \sum_{i=1}^N (\Delta x_i)\\
    &= \varepsilon (b-a).
  \end{align*}
  In the limit, $\varepsilon \to 0$, and thus the Riemann sums also go to zero.
  So we have
  \[ \lint{[a,b]} \dx^2 = 0. \]
  Intuitively, we may say that an ordinary integral $\lint{[a,b]} f(x) \,\dx$ obtains a zeroth-order result by adding up a large number of small first-order changes; but if we try to add up the \emph{same} number of \emph{second-order} changes, the result will be negligible.
\end{eg}

More generally, we have the following:

\begin{thm}\label{thm:int-negligible-onevar}
  If $\omega$ is negligible, then for any closed interval $[a,b]$ we have
  \[ \lint{[a,b]} \omega \;=\; 0.\]
\end{thm}

\begin{adv}
  If $\omega$ is not assumed \emph{a priori} to be integrable, then the conclusion of \cref{thm:int-negligible-onevar} requires a slightly more powerful definition of integration than is usually taught in one-variable calculus, called ``Henstock integration''.
  Roughly, we need to be more flexible with the exact meaning of what it means for the $\Delta x_i$ to all approach zero at once.
  Since we will not be making ``$\Delta x_i\to 0$'' very precise anyway, and all our examples will be integrable, we henceforth ignore this issue.
\end{adv}

TODO: In the middle of reorganization\dots

In particular, it follows that
\begin{equation}\label{eq:int-apx1-equal}
  \text{If }\omega\apx1\eta, \text{then} \displaystyle\lint{[a,b]} \omega = \lint{[a,b]} \eta.
\end{equation}
For example, recall that if $f$ is differentiable, then
\begin{align*}
  f(x+\dx)-f(x) &\apx1 \df. %\\
  % &= f'(x) \,\dx
\end{align*}
Therefore, we have
\[ \lint{[a,b]} \df = \lint{[a,b]} \Big(f(x+\dx) - f(x)\Big). \]
(You may be tempted to add a $\dx$ on the right-hand side of this equation, but that would be wrong: $f(x+\dx) - f(x)$ alone is the \emph{differential form} that we are integrating.)

Now note that for any partition of $[a,b]$ into subintervals, we have $x_{i-1} + \Delta x_i = x_i$, by definition.
Thus, if we choose the left-hand endpoints (that is, $\rstag x i = x_{i-1}$), then the Riemann sum for $\lint{[a,b]} (f(x+\dx) - f(x))$ will be
\begin{align*}
  \Big(f(x_1) - f(x_0)\Big) + \Big(f(x_2)-f(x_1)\Big) + \cdots + \Big(f(x_N) - f(x_{N-1})\Big)
  &= f(x_N) - f(x_0)\\
  &= f(b) - f(a).
\end{align*}
Therefore,
\begin{equation}
  \lint{[a,b]} \df = f(b) - f(a).\label{eq:differential-ftc-eval}
\end{equation}
Recalling that $\df = f'(x)\,\dx$, we see that this is half of the Fundamental Theorem of Calculus.
\begin{stewart}(The textbook calls this half the ``Evaluation Theorem''.)\end{stewart}

The other half of the Fundamental Theorem of Calculus says that every continuous function has an antiderivative: i.e.\ if $f$ is continuous, then there is a function $F$ such that $F'(x) = f(x)$.
This is equivalent to saying that any \emph{linear} differential form $f(x)\,\dx$ is the differential $\dd F = F'(x)\,\dx$ of some function.
(It follows from this and \cref{eq:differential-ftc-eval} that one such function $F$ is $F(x)= \int_a^x f(x)\,\dx$, and this is often included as part of the Fundamental Theorem.)
Therefore, at least in principle, we can evaluate the integral of \emph{any} linear differential form using \cref{eq:differential-ftc-eval}.
(Although, as you probably learned in your previous calculus classes, in practice there are many functions whose antiderivatives have no simple formula.)

This is not the case for nonlinear differential forms.
Forms such as $|\dx|$ and $\dx^2$ and $e^\dx$ are not the differential of any function, so we cannot evaluate their integrals using \cref{eq:differential-ftc-eval}.
Of the nonlinear forms, the ones which arise most in practice are those of the form $f(x)\,|\dx|$, and those we can integrate using \cref{eq:onevar-absint}.
In many other cases, we can apply \cref{thm:int-negligible-onevar} or \cref{eq:int-apx1-equal}.