Differential-algebraic equations are important for mathematical modeling and scientific computation. If you write down the mathematical laws for some chemical, electrical, or physical system, you often will just end up with a system of equations involving parameters, various partial derivatives and purely algebraic quantities. Maybe you also get some equations involving integrals. Now ...
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.
calculus - What is the practical difference between a differential and ...
What bothers me is this definition is completely circular. I mean we are defining differential by differential itself. Can we define differential more precisely and rigorously? P.S. Is it possible to define differential simply as the limit of a difference as the difference approaches zero?: $$\mathrm {d}x= \lim_ {\Delta x \to 0}\Delta x$$ Thank you in advance.