Let $C=\{(x,y):f(x,y)=0\}$ be the level set of a continuously differentiable function $f(x,y)$ of two variables. Using implicit differentiation, we get: $$\frac{dy}{dx} = - \frac{\displaystyle\frac{\partial f}{\partial x}}{\displaystyle\frac{\partial f}{\partial y}}$$ Thus, with a wink and a nudge, we define the differential of $f$ to be $$df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy. $$ I am aware of at least two different ways to make this notion rigorous:
1. Using multilinear algebra, interpret $dx$ and $dy$ as differential forms.
2. Using non-standard analysis, interpret $dx$ and $dy$ as infinitesimals (in the hypperreals).
Questions: Are these two interpretations mutually exclusive? Is there a third way to think of them that unifies both?
As far as I am aware, differential forms and infinitesimals are very different objects, so it seems like these two ways of understanding the concept of differential are intractable when considered together. The motivation for the notion of differential as stated above comes from p. 174, section 3.6.2., of Algebraic Geometry: A Problem Solving Approach.
Answer
The approach you ascribe to nonstandard analysis is actually the standard notion of differential; e.g. in single variable calculus, we can define a differential $\mathrm{d}f$ to be the bivariate function
$$ \mathrm{d}f(x,h) = f'(x) h $$
If you conventionally always use the variable $\Delta x$ for the second argument and drop it from the notation (and do the usual thing to adapt to notation in terms of dependent variables rather than functions), the result is differentials as defined in Keisler's book.
In calculus in Euclidean space, it's not hard to see that this definition of differential is basically identical to the exterior derivative, if using the definition of differential forms as linear functionals on tangent vectors, since a point in the tangent bundle is given as $(x,h)$ where $x$ denotes the base point and $h$ is an element of the tangent space.
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