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The right question is not what is a differential? but how do differentials behave? Please explain with example both solutions (implicit and explicit)of same initial value problem Let me explain this by way of an analogy
Suppose i teach you all the rules for adding and multiplying rational numbers What is difference between implicit and explicit solution of an initial value problem 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. 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) An introduction by walter strauss an introduction to partial differential equations by michael renardy partial differential equations by fritz john partial differential equations by lawrence c evans my background is having read a first course in differential equations with modelling applications by dennis zill. 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 Which, as i understand it goes as follows: 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
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