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.
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)?
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.
I have a problem understanding how to define a linear or non-linear Differential equation. These are my answers to the questions, however, my teacher's answers are not the same as mine. Questions ...
The main difference between a differential such as dx d x, dy d y, or more generally df d f (for a function f f of several variables) and a differential form is generality: a differential is a 1-form whereas in differential geometry one works with more general k k -forms where k k can be as large as the dimension of the manifold one is in ...
Next semester (fall 2021) I am planning on taking a grad-student level differential topology course but I have never studied differential geometry which is a pre-requisite for the course. My plan i...
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. Simmons' book fixed that.
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 ...
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.
