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The notes by Bachman recommended by Harrison Brown look pretty nice to me, but it seems to me that it is possible to clarify what he says even further by focusing on the simplest cases, namely the integral of a "constant function" over the simplest possible domain.

For the integral over an interval, the simplest case consists of a constant function. You can extend this case to the general case by the additive property of an integral and taking limits. But if you want an integral that is independent of the parameterization of the interval, this leads naturally to the idea that you don't want to integrate just a function $f(x)$ but a "1-form" $f(x) dx$.

This generalizes naturally to an integral of a constant function over a line segment sitting in $R^n$. If you want the concept of an integral that is independent of choice of a linear parameterization of the segment, as well as the linear co-ordinates on $R^n$, then this leads to naturally to the fact that what should be integrated is a "dual vector", i.e. a constant $1$-form. In fact, when developing these ideas, I suggest using an abstract real vector space $V$ as the ambient space instead of $R^n$.

When considering higher dimensions, I suggest focusing on linear embeddings of a $k$-dimensional cube and asking what gives a linear co-ordinate independent additive function of flat $k$-cubes embedded in $R^n$. I have not worked out the details myself, but I suspect that this leads naturally to the concept of constant $k$-forms.

My recollection is that there is a book "Advanced Calculus" by Harold Edwards that presents all of this, but I haven't looked at the book in a very long time.

In particular, it is worth noting that the question asked is really about algebra and not analysis. The analysis arises only when you want to extend the definition of an integral to a more general class of functions beyond constant ones.

My answer above does not address the exterior derivative. I will just add a brief comment about this and leave the details to the reader. My view of the exterior derivative is that, once you decide that exterior forms are indeed the natural objects of integration over a domain (but start with cubes!) in Euclidean space, it is the natural co-ordinate-free algebraic consequence of the fundamental theorem of calculus (or, if you insist, Stoke's theorem). That $d^2 = 0$ is the appropriate co-ordinate-free expression of the basic fact that "partials commute".

1

The notes by Bachman recommended by Harrison Brown look pretty nice to me, but it seems to me that it is possible to clarify what he says even further by focusing on the simplest cases, namely the integral of a "constant function" over the simplest possible domain.

For the integral over an interval, the simplest case consists of a constant function. You can extend this case to the general case by the additive property of an integral and taking limits. But if you want an integral that is independent of the parameterization of the interval, this leads naturally to the idea that you don't want to integrate just a function $f(x)$ but a "1-form" $f(x) dx$.

This generalizes naturally to an integral of a constant function over a line segment sitting in $R^n$. If you want the concept of an integral that is independent of choice of a linear parameterization of the segment, as well as the linear co-ordinates on $R^n$, then this leads to naturally to the fact that what should be integrated is a "dual vector", i.e. a constant $1$-form. In fact, when developing these ideas, I suggest using an abstract real vector space $V$ as the ambient space instead of $R^n$.

When considering higher dimensions, I suggest focusing on linear embeddings of a $k$-dimensional cube and asking what gives a linear co-ordinate independent additive function of flat $k$-cubes embedded in $R^n$. I have not worked out the details myself, but I suspect that this leads naturally to the concept of constant $k$-forms.

My recollection is that there is a book "Advanced Calculus" by Harold Edwards that presents all of this, but I haven't looked at the book in a very long time.

In particular, it is worth noting that the question asked is really about algebra and not analysis. The analysis arises only when you want to extend the definition of an integral to a more general class of functions beyond constant ones.