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The right question is not what is a differential? but how do differentials behave? If it's a scalar value function, the change would be scalar, and thus the differential (would map to a scalar) Let me explain this by way of an analogy

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Suppose i teach you all the rules for adding and multiplying rational numbers Basically, it denotes the change in the function 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. 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. 69 can someone please informally (but intuitively) explain what differential form mean I am quite new to differential equations and derivatives I want to derive an differential form for equation of an ellipse

If i start with an ordinary ellipse equation \\begin{equation} \\frac{x^2}.

The meaning of the notation is indeed a second order differential, i.e A difference of difference, not a squared difference Then about any function will show you that the square of the first derivative isn't the second derivative. What is difference between implicit and explicit solution of an initial value problem

Please explain with example both solutions (implicit and explicit)of same initial value problem 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. Differential forms are things that live on manifolds So, to learn about differential forms, you should really also learn about manifolds To this end, the best recommendation i can give is loring tu's an introduction to manifolds

Tu develops the basic theory of manifolds and differential forms and closes with a exposition of de rham cohomology, which allows one to extract topological. 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 It also leads to another point The differential has a linear approximation meaning