Differentials as "infinitesimals" versus as linear mappings or differential forms. 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.
 A: The answer to my question, (or resolution to the contradiction) is that there exist (at least) three different meanings to an expression of the form $d\mathfrak{x}$.


*

*$d\mathfrak{x}\in\mathbb{R}^{n}$ is an arbitrary value, or the linear mapping of such.  For example 
$$df_{\mathfrak{x}}=\partial_{i}f\left[\mathfrak{x}\right]dx^{i}\in\mathbb{\mathbb{R}}.$$

*$d\mathfrak{x}$ is part of an integration symbol, for example:  $\int\mathfrak{f}\cdot d\mathfrak{x}$.  Where it tells us how the Riemann sum is created, prior to taking the limit as the mash of the partition goes to zero.

*As a field of linear mapping resulting from taking the limit of the first difference as the argument increment goes to zero.  For example:
$$\Delta f_{tan}=df_{\mathfrak{x}}\left[\Delta\mathfrak{x}\right]=\frac{\partial f}{\partial x^{i}}dx^{i}\left[\Delta\mathfrak{x}\right]=\frac{\partial f}{\partial x^{i}}\Delta x^{i}\in\mathbb{R}.$$


A superior practice is to distinguish differential values from differential forms, for example
$$\Delta f_{tan}=\tilde{d}f_{\mathfrak{x}}\left[\Delta\mathfrak{x}\right]=\frac{\partial f}{\partial x^{i}}\tilde{d}x^{i}\left[\Delta\mathfrak{x}\right]=\frac{\partial f}{\partial x^{i}}\Delta x^{i}$$
Using that notation and asserting $d\mathfrak{x}\equiv{\Delta\mathfrak{x}}$ results in
$$d f=\tilde{d}f_{\mathfrak{x}}\left[d\mathfrak{x}\right]=\frac{\partial f}{\partial x^{i}}\tilde{d}x^{i}\left[d\mathfrak{x}\right]=\frac{\partial f}{\partial x^{i}}d x^{i}.$$
Unfortunately, many people who know about differential forms do not know what they are talking about regarding standard practices in calculus.  See the 1953 edition of Thomas's Calculus and Analytic Geometry, Article 4-8.  That's not where I first learned what a differential is, and how it relates to integration, but it is how I learned it.
This has been one of the most difficult topics for me to sort out in mathematics.  Largely because the authors I consulted did not understand what they were contrasting (their version of) differential forms with, and the mathematicians and physicist I ask, arrogantly condescended that I simply didn't understand that "summing infinitesimals and dividing by zero" was wrong.
