The book provides an introduction to Differential Geometry of Curves and Surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet theorem, both in its local and in its global form, using basic properties (fully proved in the complementary material) of triangulations of surfaces. As an application, we shall prove the Poincaré-Hopf theorem on zeroes of vector fields. Finally, the last chapter will be devoted to several important results on the global theory of surfaces, like for instance the characterization of surfaces with constant Gaussian curvature, and the orientability of compact surfaces in R3.
Advocacy Excellence: The Jury Trial, Second Edition, teaches students the art and science of 21st Century trial advocacy through the eyes of two seasoned, tenacious, and successful female trial attorneys who bring over 60 years of combined experience to the text. With a sharp and practical focus on how the digital age has changed trial practice, students will gain the ability to successfully advocate in today’s smart courtrooms using electronically stored information, social media, and technology in all phases of trial. This text teaches classic courtroom skills with a modern and spirited tone, using examples from real trials and step-by-step practice guides along with insider tips about the strategy and execution techniques that win trials. This clear, concise, and easy-to-understand text is organized into three distinct sections: Part I:Preparation — investigation, preliminary case analysis, developing a case theory, and merging the case theory into the actual trial Part II:Practice — techniques and advice that provide simple steps to successful jury selection, openings, direct and cross examination, impeachment, cross of special witnesses, and summation Part III:Strategy — navigating the courtroom, how to admit or oppose evidence at trial, objections, and the end game of jury deliberation. Learn the law, ethics, and strategy of trial advocacy with step-by-step instructions and useful chapter ending process guides and infographics to reinforce skills. Professors and students will benefit from: Question-and-answer examples in every chapter teach students how to ask strategic and purposeful questions during jury selection, depositions, pretrial hearings, direct examination, cross examination, impeachment, and the admitting or opposing of evidence. Illustrations and charts demonstrate how to create various proof matrices, timelines, witness statement charts, transcript keys, and how to structure opening, direct, and cross examination. Sidebars highlight practice tips, legal ethics, and cautions Why This Works sidebars explain why skills or methods are used in practice Timely coverage of the role of social media includes emojis as evidence, plus how to authenticate social media and other electronic or digital evidence at trial. Reference sheets designed for use in both an academic, experiential setting and the first years of practice as a trial lawyer.
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