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This exploration of a selection of fundamental topics and general purpose tools provides a roadmap to undergraduate students who yearn for a deeper dive into many of the concepts and ideas they have been encountering in their classes whether their motivation is pure curiosity or preparation for graduate studies. The topics intersect a wide range of areas encompassing both pure and applied mathematics. The emphasis and style of the book are motivated by the goal of developing self-reliance and independent mathematical thought. Mathematics requires both intuition and common sense as well as rigorous, formal argumentation.
This book attempts to showcase both, simultaneously encouraging readers to develop their own insights and understanding and the adoption of proof writing skills. The most satisfying proofs/arguments are fully rigorous and completely intuitive at the same time.
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"The author succeeds at giving the reader a good notion of the content and feel of these basic areas in a fairly short book. Will prospective students find it of value? For some students I have known - ones who expect to see only more things like the calculus they really like - it offers a sort of warning. For students desiring a more realistic picture of the mathematics they might see in advanced courses, this offers a solid and appealing treatment." Bill Satzer in: https://maa.org/press/maa-reviews/advanced-mathematics-an-invitation-in-preparation-for-graduate-school (06..11.2023)
