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 ...
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.
Now we define differential of f(x) as follows: df(x): = f ′ (x)dx Where df(x) is the differential of f(x) and dx is the differential of x. 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.
70 can someone please informally (but intuitively) explain what "differential form" mean? I know that there is (of course) some formalism behind it - definition and possible operations with differential forms, but what is the motivation of introducing and using this object (differential form)?
I am a bit confused about differentials, and this is probably partly due to what I find to be a rather confusing teaching approach. (I know there are a bunch of similar questions around, but none o...
Partial Differential Equations: 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.
Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian ...
The answer mentions that this is an exercise. Is that in fact correct? If so, it would be helpful to know which section / chapter of the textbook this exercise comes from, since that would give a hint as to how to approach the problem.
For differential equations (ordinary, partial, with deviating argument, etc.), as well as for some integral equations, a rule of thumb is the following: the equation is called autonomous when for any of its solutions any (admissible) time translate of that solution is its solution, too.
I was attempting to solve the following integro-differential equation using convolutions. My answer also had a convolution which did not seem right and was wondering if someone would check my proce...
