However, here are some of the choices I have made: Level: “Calculus” and “analysis”-which for Newton et. viiĪfter three centuries of calculus texts, it is probably impossible to offer something truly new.
In the process, I have also become fascinated with the history of the subject, and I’m afraid my enthusiasm has infected the exposition. For myself as a mathematician, the intellectual exercise of working through all the theorems of single-variable calculus and constructing an elementary but coherent and self-contained presentation of this theory has been a fascinating experience. In contrast to “Calculus Lite”, the present book is “Calculus Tight”: a review of often familiar techniques is presented in the spirit of mathematical rigor (hopefully without the mortis ), in a context of ideas, and with some sense of their history. Keeping in mind these competing demands, I opted for a course which approaches the tools of calculus through the eyes of a mathematician. On the other hand, these bright students would be bored out of their minds by a straight repetition of familiar topics. On one hand, the great variability among high school calculus courses required us to cover all the standard topics again, to make sure students end up “on the same page” as their counterparts in the mainstream course. The challenge of the Honors Calculus course was twofold. It could, however, prove equally useful for a course using analysis as the topic for a transition to higher mathematics. This book was initially developed for the first semester of the Honors Calculus sequence at Tufts, which gives entering students with a strong background in high school calculus the opportunity to cover the topics of the three-semester mainstream calculus sequence in one year. The present volume is an introduction to the elementary parts of this structure, for the reader with some exposure to the techniques of calculus who would like to revisit the subject from a more conceptual point of view. Underpinning these methods is an intricate structure of ideas and arguments in its fullest form this structure goes by the name of real analysis. Johann Wolfgang von Goethe Maximen und ReflexionenĬalculus is a collection of incredibly efficient and effective tools for handling a wide variety of mathematical problems. Mathematicians are a kind of Frenchmen: whatever you say to them they translate into their own language and it is then something quite different. Leonhard Euler Introductio in Analysin Infinitorum (1755) From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use. Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. Same effects can also take place for other known stellar bounds on dark sector particles.Calculus Deconstructed A Second Course in First-Year Calculus Zbigniew H. This also implies that nonstandard strongly self-interacting neutrino is not consistent with the SN1987a observation. Our findings thus imply that the existing supernova bounds on light dark particles can be generally eluded by a similar self-trapping mechanism. For 3 m A ′ = m χ, bounds in regions where α D ≳ 10 − 7 for m χ ≲ 20 MeV can be evaded similarly. In particular, for the mass range m χ ≲ 20 MeV, supernova bounds can only be applied to weakly self-interacting dark sector with α D ≲ 10 − 15. For m A ′ = 3 m χ, we show that this effect can completely evade the supernova bounds on widely examined dark photon parameter space for a dark sector with α D ≳ 10 − 7. We consider specifically two mass ratios m A ′ = 3 m χ and 3 m A ′ = m χ which represent scenarios where the decay of A ′ to χ χ ¯ is allowed or not. This effect strongly limits the energy luminosity carried away by dark sector particles from the supernova core and thus drastically affects the parameter space that can be constrained by SN1987a. We find that even with a small dark sector fine structure constant α D ≪ 1, dark sector self-interactions can easily lead to their own self-trapping. Considering explicitly a dark photon portal dark sector model, we compute the relevant interaction rates of dark photon ( A ′) and dark fermion ( χ) with the Standard Model particles as well as their self-interaction inside the dark sector. We examine the constraints on sub-GeV dark sector particles set by the proto-neutron star cooling associated with the core-collapse supernova event SN1987a.