A differential form is (technically) a function that we can calculate value at a point and AFAIK it has nothing to do with infinitesimals nor tends to anything. A course in precalculus, calculus, or even real analysis almost never gives an answer to "What is dx?". It is only until differential geometry, one gets to learn what it is. One should not learn these from Wikipedia but from a ...
The study of differential algebraic geometry and model theory occupies a pivotal position at the interface of algebra, geometry, and logic. Differential algebraic geometry investigates solution sets ...
Gauge theories form the backbone of modern theoretical physics, underpinning the Standard Model and many approaches to quantum gravity. Traditionally, these theories utilise continuous symmetries and ...
The right question is not "What is a differential?" but "How do differentials behave?". Let me explain this by way of an analogy. Suppose I teach you all the rules for adding and multiplying rational numbers. Then you ask me "But what are the rational numbers?" The answer is: They are anything that obeys those rules. Now in order for that to make sense, we have to know that there's at least ...
Then one thinks of differential operators as a linear maps between such spaces. Often the space of all linear maps between two spaces is itself a vector space and so one can indeed start to manipulate differential operators as if they are ‘objects’ in their own right eg add them together.
Why can we treat differential operators as if they behave like ...
Anyone who sees calculus in application is likely to encounter both derivatives and differentials. The two concepts have confusingly similar notation. For that reason, this post is a very important contribution.