Advanced calculus

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1. sixbrxNov 2022
   Harold Edwards: Advanced Calculus: A Differential Forms Approach (https://www.maa.org/press/maa-reviews/advanced-calculus-a-di....)

   Tristan Needham: Visual Complex Analysis (https://www.maa.org/press/maa-reviews/visual-complex-analysi...)

   Tristan Needham: Visual Differential Geometry and Forms (https://press.princeton.edu/books/paperback/9780691203706/vi...)

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2. bmitcJul 2022
   What is the meaning then?

   dy = f'(x)dx is just a definition for notional convenience, primarily employed when doing u or u-v substitution. My point is that dx in single variable calculus is notation. It is not an intrinsic object. dx is an intrinsic object as a differential form on a smooth manifold. Of course, the real line R is a 1-manifold, so dx does have that meaning, but you need to understand what a differential form is to know that.

   One doesn't necessarily need the full generality of smooth manifolds though. Harold Edwards' Advanced Calculus: A Differential Forms Approach and Advanced Calculus: A Geometric View teach differential forms for Euclidean manifolds.

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3. bmitcFeb 2021
   You may like the books Conceptual Mathematics: A First Introduction to Categories, Creative Mathematics, Advanced Calculus: A Differential Forms Approach, and Geomtrical Vectors.

   https://www.amazon.com/dp/052171916X/

   https://www.amazon.com/dp/0883857502/

   https://www.amazon.com/dp/0817637079/

   https://www.amazon.com/dp/0226890481/

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4. bmitcNov 2020
   The problem exists in the educational methods, in my opinion. It seems most, not just in mathematics, feel that learning is the same as if the student can simply be played a tape that cannot be paused or rewound but can sometimes be sped up. Nearly nothing works like this, and it’s just that the penalties in mathematics are greater.

   I think of the ideal learning path as a sort of circular, recursive path that bends back and goes back over itself from time to time. Imagine a human painting a canvas. One does not simply do a raster scan, painting pixel by pixel in a linear fashion. By the time the painting is done, the piece has been gone over multiple times with multiple layers of finer and finer detail. A painting is a mishmash of broad and finer strokes, with some layers simply forming foundational layers for the later portions.

   A pyramid is the wrong metaphor. In my opinion, learning mathematics should be like creating a painting.

   One of the best examples of this I know of is the book Advanced Calculus: A Differential Forms Approach by Harold Edwards. The first three chapters introduce the material heuristically. The next three chapters circle back on the material, thoroughly proving everything introduced in the first three chapters. The remaining chapters greet the student with applications and extensions of the material. In the preface, Edwards states that he wanted the book to be able to be opened to any page and be read and make sense. It’s a wonderful goal, and he achieves it.

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5. bmitcJul 2020
   Since you mention analysis, I recommend Yet Another Introduction to Analysis and Metric Spaces by Victor Bryant. They should be at the right level and a lot of fun.

   Another analysis suggestion is Creative Mathematics by H.S. Wall. It is a book that walks a high school level student through creating the proofs themselves. The topic covered is a stripped down version of analysis, calculus, and later even differential geometry. It's really brilliant. Going slow and having fun when you're a teen would be much more productive than going fast and burning out.

   The Spivak book is a good suggestion and might be a little difficult depending on their actual background. The books by Gelfand mentioned by someone else (there's actually a series of them that cover algebra, functions, coordinates, trigonometry, etc.) would help provide the needed background.

   The book Conceptual Mathematics is claimed to be aimed at high school students. Maybe give it a whirl and see what happens. If they know calculus, then Advanced Calculus: A Differential Forms Approach by Harold Edwards is a gem. The first three chapters should be readable, as they give heuristic discussions of the topic.

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6. bmitcMay 2020
   For anyone wanting to learn more about this, I highly recommend Advanced Calculus: A Differential Forms Approach by Harold Edwards and An Introduction to Manifolds by Loring Tu. The former reads almost like a novel and is a real treat of mathematical exposition. It's also a little quirky which is always nice. Tu's book is simply the gold standard of an introduction to the mathematics of manifolds and differential forms. It is the most concise and straightforward introduction to the full theory. It is also a wonderful book.
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