[...]... counterexamples; • Learn to write mathematics, primarily formal proofs, but also intuitive explanations and conjectures A Survey Course More than just serving as a text for a transition course, A Transition to Advanced Mathematics is also designed to provide students with a broad survey of many fundamental areas of mathematics Students completing a calculus sequence may not realize that mathematicians are a. .. wide-ranging interests Indeed, many different areas of mathematics suit individual skills and insights as well as personal Preface xi interests and temperaments With the calculus sequence serving as the primary point of entry to the mathematics major, many students are unaware of the marvelous variety inherent in mathematics A Transition to Advanced Mathematics responds in a positive way to the need to. .. resulting creations as they use their “mathematical palette.” In summary, as a text for a transition course, A Transition to Advanced Mathematics encourages students to: • Develop careful reasoning skills as the student is transitioning from computationally oriented, algorithmic thinking to more sophisticated modes of reasoning; • Learn to read mathematics, specifically definitions, examples, proofs, and counterexamples;... with a broad survey of mathematical ideas and explorations, as it is intended to: • Provide students with a broad and comprehensive introduction to mathematics, including both continuous and discrete mathematics; • Introduce students to “upper-level” topics at an earlier stage in the mathematics major; • Create greater continuity and flow in the mathematics major, introducing various topics, mathematical... graphical representation, analytic functions, partial differentiation and the Cauchy–Riemann equations, power series representations of analytic functions, harmonic functions, and the Laplacian An application section explores the use of streamlines and equipotentials to understand and model fluid flow Key Elements of the Text We hope A Transition to Advanced Mathematics will be recognized as a clear and... mathematical logic as an essential tool that you can use in your attempts to determine the truth of mathematical statements In the study of more advanced mathematical ideas, one can go off on tangents that either have no basis in sound logic and are irrelevant, or that lead to incorrect conclusions and are counterproductive Mathematical logic can keep us on track, and this chapter is then essential as... see a variety of applications, to view purely abstract, theoretical ideas as not antithetical to using mathematics to benefit society The intent is for students to begin to perceive pure and applied mathematics as going hand in hand and strengthening one another in interplay: a search for applications often results in the development of new theoretical ideas, and theoretical mathematics often manifests... for a student to mature into a fully developed mathematician is to gain an ability to articulate precisely reasoned arguments that explain and justify the mathematical idea under scrutiny Unfortunately, as many in the mathematical community have recognized, a focus on the computational elements of calculus is not preparing students for this transition into theoretically oriented upper-level courses Many... proven in a suitable formal system) and are among the most significant mathematical and philosophical insights of the twentieth century Within another 3 4 A Transition to Advanced Mathematics 40 years, the use of formal languages began playing a key role in the design of the computer chips that are so essential to our technologically based society In this chapter, we develop a formal language in the... a transition course surveying mathematics It is designed to serve ideally in collaboration with mathematics professors helping students to explore new mathematical vistas, to grow into the perspectives of the mathematician, and to successfully practice mathematics The following elements of the text are intended to help facilitate this partnership between professor and text in the creation of a dynamic .