اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدFun with Analysis I: basic theorems in calculus revisited
385
0
0.0
(
0
)
تأليف
Daniel Reem
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the extreme, namely we present two proofs of the Extreme Value Theorem: the programmer proof that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values of the given function in a metric space setting, and an abstract space proof (the level-set proof) for semicontinuous functions defined on compact topological spaces. Next, in the intermediate part, we consider the Intermediate Value Theorem, generalize it to a wide class of discontinuous functions, and re-examine the meaning of the intermediate value property. The trek reaches the final frontier when we discuss the Uniform Continuity Theorem, generalize it, re-examine the meaning of uniform continuity, and find the optimal delta of the given epsilon. Have fun!
قيم البحث
اقرأ أيضاً
Do you want to know what an anti-chiece Latin square is? Or what a non-consecutive toroidal modular Latin square is? We invented a ton of new types of Latin squares, some inspired by existing Sudoku variations. We cant wait to introduce them to you a
nd answer important questions, such as: do they even exist? If so, under what conditions? What are some of their interesting properties? And how do we generate them?
What initial trajectory angle maximizes the arc length of an ideal projectile? We show the optimal angle, which depends neither on the initial speed nor on the acceleration of gravity, is the solution x to a surprising transcendental equation: csc(x)
= coth(csc(x)), i.e., x = arccsc(y) where y is the unique positive fixed point of coth. Numerically, $x approx 0.9855 approx 56.47^circ$. The derivation involves a nice application of differentiation under the integral sign.
We provide a Rademacher theorem for intrinsically Lipschitz functions $phi:Usubseteq mathbb Wto mathbb L$, where $U$ is a Borel set, $mathbb W$ and $mathbb L$ are complementary subgroups of a Carnot group, where we require that $mathbb L$ is a normal
subgroup. Our hypotheses are satisfied for example when $mathbb W$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.
The Monty Hal problem is an attractive puzzle. It combines simple statement with answers that seem surprising to most audiences. The problem was thoroughly solved over two decades ago. Yet, more recent discussions indicate that the solution is incomp
letely understood. Here, we review the solution and discuss pitfalls and other aspects that make the problem interesting.
E661 in the Enestrom index. This was originally published as Variae considerationes circa series hypergeometricas (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He look
s at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers think they understand please I would appreciate it if you could email me. I am presently examining Eulers work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
