Preface to 100+1 Problems in Advanced Calculus PDF
We are about to embark on a journey. Imagine, if you will, that we are on a silent, smooth-running boat, and that we are preparing to enter some of the most beautiful “fjords” in Mathematical Analysis. A fjord, you will recall, is a long, narrow, and deep inlet of the sea, between high cliffs. In this book, since we are exploring mathematical “terrain,” the cliffs are immense mathematical structures, stretching higher than we can imagine. As we cruise between and beneath them in our boat, deeply entering these mathematical “fjords,” we will be able to see the underlying foundations of the massive mathematical theories and structures above us.
The key foundations, or “ports of call,” of Mathematical Analysis are not as numerous as the unsuspecting tourist might think. On our journey, we will have the opportunity to pass near to, and thus to closely examine, the fundamental concepts of function, of infinity, of limit, and of differentiability. In order to “visit” these “destinations” in a complete way, however, we will have to look at them from many different vantage points; we will have to explore them in some really unexpected ways. It is necessary to allow them the possibility of expressing their full potentials, without making excessively restrictive assumptions. Keep the image of an unexplored coastal shoreline in mind. If we look at the shore from our vessel, while we are still some significant distance away from the land, it often resembles a straight line. But as we approach more and more closely, we may discover that it has many irregularities and inlets. Similarly, in order to appreciate the fine details of the whole mathematical picture, it is important that we set our imaginations loose as we simultaneously explore with more and more precision.
To successfully achieve such exploration, mathematically speaking, it will be necessary to remain faithful to the definition of function as a relation between sets. In fact, a function is not always a mere combination of symbols and operations that transform one set into another. Formulas or equations do not exhaust all possibilities; functions can be much more general and may exhibit fantastic and incredible properties. By committing to this approach, the notion of set returns to the forefront of our exploration; the study of sets and their topology becomes crucial. We will see that the “Pythagorean-Euclidean drama” concerning the notion of point naturally makes its appearance, with startling consequences related to the idea of infinity. For, if we agree to drop the basic classical notions discussed in textbooks, it turns out that the idea of mathematical infinity furnishes an unfathomable wealth, and can play remarkable—indeed, infinitely many—tricks on our intuition. Another aspect of our journey will revolve around the limits of functions defined on arbitrary sets and not merely intervals.
This will provide us with a “window” through which to view inaccessible “terrain,” and scenarios playing out on that terrain, which it would otherwise be impossible for us to encounter on our voyage. This part of our journey will give the notion of limit the central role it deserves in the program of Mathematical Analysis. The notion of continuity also yields many surprises and is another fertile plain that we will have the chance to survey on our journey. Indeed, while the definition of continuity appears very simple when we associate it with the idea of an uninterrupted wire, it is very suggestive and can even prove quite tricky when it is more fully revealed by the Squeeze Theorem. (Not to mention the problem of tangency…a “caress” produced by the derivative.) In order to give readers the greatest chance of navigating through the mathematical landscape in a relaxed, enjoyable manner, the book includes some additional material on the order properties of real numbers, the topology of the real line, the definition of limit, and a few graphic features of algebraic curves, as well as the notion of integral. The book proposes a series of problems which require a bit of imagination and critical thinking. Readers will have the opportunity to compare their answers with our solutions, and to then proceed in journeying down these fjords on their own— possibly forging new paths and discovering uncharted inlets worthy of admiration and further exploration.
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