Suppose we have a curve, formed by a function f that maps real numbers to real numbers, such that f is everywhere smooth over a subset D of its domain. Let's suppose that, for all x in D, there is a vector space that contains all vectors tangent to the curve at that point, called the tangent space. A differential df is, thus, a linear functional that maps elements of the tangent space at a point x to the set of real numbers. The quantity dx denotes the differential of the identity function. As such, the derivative may be seen as a "function of proportionality" that gives, intuitively, the ratio of how much faster f grows compared to the identity function at a point x. That is, ##df = \frac{df}{dx}dx##.(adsbygoogle = window.adsbygoogle || []).push({});

This has, roughly, been my definition of the derivative of a function. First of all, is this a correct way of thinking about it in terms of differential geometry (minus the formalisms, of course. For example, the curve might be better considered as a manifold)?

Second, does this justify, to some degree, the idea that we can "cancel" differentials? Id est, ##\frac{dy}{dx}=\frac{dy}{\not{du}}\frac{\not{du}}{dx}##?

Third, does this concept generalize? That is, could there be "derivatives" for higher differential forms, like ##\frac{dy\wedge dx}{dz\wedge dw}##?

I'm a little new to the ideas of differential geometry, but feel free to use formal terms to force me to look up information on them so I can learn more. Thank you in advance for your help.

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# Trying to understand derivatives in terms of differential forms

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