I have an upper school maths background so would appreciate answers that aren't too technical.
I'm trying to understand how the differential $df$ of a function can be more properly understood as a 1-form rather than (as I was taught at school) a scalar. I've found two arguments showing this, but I'm unable to reconcile them.
The first argument is from Lee's Introduction to Smooth Manifolds, p282-283 and involves a Taylor approximation. He says:
In elementary calculus, one thinks of $df$ as an approximation for the small change in the value of $f$ caused by small changes in the independent variables $x^{i}$. In our present context, $df$ has the same meaning, provided we interpret everything appropriately.
He then goes on to say this:
Which, on a good day, I can more or less follow.
The second (for me, hard to understand) argument is from Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1, p111-112. He talks about $df$ being a function on tangent vectors and then goes on to eventually state:
There's no mention here of a Taylor approximation, though I assume he's saying more or less the same thing, that $df$ can be better understood as a 1-form.
I'm not looking for a detailed explanation of Spivak but I would like to know, in broad terms, if they are indeed saying the same thing in different ways.
Yes, they are saying the same thing. They only have a bit different appoaches and notations. In Lee's formula, $v$ is a tangent vector, while in Spivak's formula it is $\frac{dc}{dt}.$