I have learned two seemingly contradictory definitions of what a differential is.
The first one I call the infinitesimal concept. It goes like this.
For a continuous mapping $\mathfrak{f}:\left[t_{0},t_{L}\right]\to\mathbb{R}^{n}$, the first difference of $\mathfrak{f}$ at $t\in\left[t_{0},t_{L}\right]$ due to a displacement $\Delta t$ such that $t+\Delta t\in\left[t_{0},t_{L}\right]$ is defined to be
$\Delta\mathfrak{f}{}_{t}\left[\Delta t\right]\equiv\mathfrak{f}\left[t+\Delta t\right]-\mathfrak{f}\left[t\right]$.
I submit that a non-controversial definition of the derivative of $\mathfrak{f}$ at $t$ is
$\lim_{\Delta t\to0}\frac{\Delta\mathfrak{f}_{t}\left[\Delta t\right]}{\Delta t}\equiv\mathfrak{f}^{\prime}\left[t\right]$.
In Leibniz notation this is written
$\mathfrak{f}^{\prime}\left[t\right]=\frac{d\mathfrak{f}}{dt}\left[t\right].$
In the case of $n=1$, George B. Thomas introduces the free variable $dt\in\mathbb{R}$ and calls
$df\equiv\frac{df}{dt}dt$ the differential of $f$.
So the mapping $df:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ might be pedantically written as
$df\left[t,dt\right]\equiv\frac{df}{dt}\left[t\right]dt$.
Some basic epsilonics gives a feel for how this might be used.
$\lim_{\Delta t\to0}\left[\frac{\Delta f_{t}\left[\Delta t\right]}{\Delta t}-\frac{df}{dt}\left[t\right]\right]=0$.
$\varepsilon\equiv\frac{\Delta f_{t}\left[\Delta t\right]}{\Delta t}-\frac{df}{dt}\left[t\right]$.
$\Delta f_{t}\left[\Delta t\right]=\left(\frac{df}{dt}+\varepsilon\right)\Delta t$
$\Delta f_{t}\left[\Delta t\right]\approx\frac{df}{dt}\Delta t$.
Or since $dt$ is a free variable, and $\Delta{t}$ is only restrict so that $t+\Delta{t}$ is the the domain of $f$, this may also be written as
$\Delta f_{t}\left[ dt\right]\approx\frac{df}{dt}dt$.
Or simply as
$\Delta f\approx\frac{df}{dt}dt$.
It is natural to extend that notion to a multi-valued function on a subset of $\mathbb{R}$
$d\mathfrak{f}=\frac{d\mathfrak{f}}{dt}dt$.
Now consider the unit-speed path $\mathfrak{s}=\mathfrak{s}\left[s\right]$. The derivative $\frac{d\mathfrak{s}}{ds}=\mathfrak{\hat{t}}$ is the unit tangent vector.
Writing the position function in component form gives
$\mathfrak{s}\left[s\right]=\left\{ x\left[s\right],y\left[s\right],z\left[s\right]\right\} $.
So
$\frac{d\mathfrak{s}}{ds}=\mathfrak{\hat{t}}=\left\{ \frac{dx}{ds},\frac{dy}{ds},\frac{dz}{ds}\right\} $ and $ds\mathfrak{\hat{t}}=d\mathfrak{s}=\left\{ dx,dy,dz\right\} $.
The magnitude of $\hat{\mathfrak{t}}$ is$\left|\hat{\mathfrak{t}}\right|=1=\left(\frac{dx}{ds}\right)^{2}+\left(\frac{dy}{ds}\right)^{2}+\left(\frac{dz}{ds}\right)^{2}$, and that of $d\mathfrak{s}=\left\{ dx,dy,dz\right\} $ is $\left|d\mathfrak{s}\right|=ds=\sqrt{dx^{2}+dy^{2}+dz^{2}}$.
But when I read about differential forms I encounter statements such as
$g(dx,dx)=g(dy,dy)=g(dz,dz)=1$.
Where $g$ is the inner product operation and $dx,dy,dz$ are ``basis 1-forms''. See here:
http://physics.oregonstate.edu/coursewikis/GDF/book/gdf/inner
Edwards enjoins me to understand $dx^{i}$ as a projection mapping $dx^{i}\left[\mathfrak{v}\right]=v^{i}$.
Here's my understanding of what a differential means to Edwards.
The function $f\vert\mathbb{R}^{n}\supset\mathscr{U}\to\mathbb{R}$ is defined to be differentiable if and only if there exists a linear mapping $df_{\mathfrak{r}}\vert\mathbb{R}^{n}\to\mathbb{R}$ such that
$\lim_{\mathfrak{h}\to\mathfrak{0}}\frac{f\left[\mathfrak{r}+\mathfrak{h}\right]-f\left[\mathfrak{r}\right]-df_{\mathfrak{r}}\left[\mathfrak{h}\right]}{\left|\mathfrak{h}\right|}=0$.
The linear operator $df_{\mathfrak{r}}$ is called the differential of $f$ at $\mathfrak{r}$. Its associated matrix $f^{\prime}\left[\mathfrak{r}\right]$ is called the derivative of $f$ at $\mathfrak{r}$.
How do I resolved the contradiction between the naive understanding of $dx$ as an infinitesimal value and and the enlightened view that it is a linear operator?
I am editing to add further observations.
1) The definition of a differential given by Thomas does not involve “infinitesimals”, per se. In fact, the ultimately independent variable, called $dt$ above, is freer than its less attractive cousin, $\Delta t$. This is because $dt$ can have any real number value; whereas, $\Delta t$ must conform to the domain of definition of its function $f$.
2) Thomas never “devides by zero”. In fact, he is overly pedantic about that fact.
3) the expression $d\mathfrak{f}=\frac{d\mathfrak{f}}{dt}dt$ is a well defined mapping of the free variable $dt$, and the implied point of evaluation of $\frac{d\mathfrak{f}}{dt}$.
4) Thomas calls $d\mathfrak{f}=\frac{d\mathfrak{f}}{dt}dt$ the differential form of the derivative. With the empfasis on the form of the expression. Edwards calls the linear mapping $df_{\mathfrak{r}}$ a differential form, with the empfasis on the fact that it is dependent upon its point of determination.
5) If I follow Edwards, and treat $dx$ as the projection mapping $dx\left[\mathfrak{v}\right]=v^{x}$, but rename it to $\overset{\sim}{dx}$ and feed my $d\mathfrak{s}=\left\{ dx,dy,dz\right\}$ to it, I get $\overset{\sim}{dx}\left[d\mathfrak{s}\right]=dx$. Which provides some consonance.
