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The right question is not what is a differential? but how do differentials behave? The differential equations class i took as a youth was disappointing, because it seemed like little more than a bag of tricks that would work for a few equations, leaving the vast majority of interesting problems insoluble Let me explain this by way of an analogy
Suppose i teach you all the rules for adding and multiplying rational numbers Which, as i understand it goes as follows: 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. 69 can someone please informally (but intuitively) explain what differential form mean See this answer in quora What is the difference between derivative and differential?
In simple words, the rate of change of function is called as a derivative and differential is the actual change of function We can also define a derivative in terms of differentials as the ratio of differentials of function by the differential of a variable. 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
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. Kernel of differential operator ask question asked 12 years, 7 months ago modified 12 years, 7 months ago It also leads to another point
The differential has a linear approximation meaning Basically, it denotes the change in the function If it's a scalar value function, the change would be scalar, and thus the differential (would map to a scalar) Is the above definition of the second differential used today in mathematics
This is the question for which i will accept an answer
Can the above definition be brought into consonance with the definition of the differential of a differential form
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