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i am an engineering student , i had little interest in mathematics but my interest in mathematics inadvertently( exactly because i am loving that) rises to extreme level, although i have little knowledge (subject-syllabus oriented) on mathematics, i am eagerly trying to learn the mathematics as art ( can be apllied anywhere .....). so please suggest books on topics geometry and calculus to like beginner with interests to learn the contents as art(passion).

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  • $\begingroup$ What is Mathematics? by Courant and Robbins. $\endgroup$ – user49640 Feb 24 '17 at 21:10
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Introduction To Geometry by Donald Coxeter.... You may find Mathematical Models by Cundy and Rollet to be fun....Also : The Fractal Geometry of Nature by Mandelbrot, which, as he says, is not a textbook but an essay, exploring various ideas.... Engineering will require calculus, and you must have a clear understanding of the logical foundations of the real number system, and of what limits are, to understand calculus.

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  • $\begingroup$ I dk why you got a down-vote. I think a written justification for a down-vote should be required, and subject to review by the moderators. $\endgroup$ – DanielWainfleet Feb 24 '17 at 18:42
  • $\begingroup$ bro, can you suggest books on calculus also $\endgroup$ – Imran Kabir Feb 25 '17 at 13:40
  • $\begingroup$ I learned the basics of calculus by reading on my own, decades ago. I really cannot recall the titles. BTW I also recommend What Is Mathematics? by Courant & Robbins. Also Infinite Sequences and Series by Bromwich, as a supplement to analysis. $\endgroup$ – DanielWainfleet Feb 26 '17 at 0:34
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Numbers and Functions: Steps into Analysis and Groups: A Path to Geometry by R. P. Burn. They are both problem books, and I find one learns, and more importantly, understands much better from books of this sort. They come with solutions to all the problems as well, so they are helpful if you get stuck for a very long time and need a little bit of a hint.

Prerequisites are covered in the introductory chapters of both books. For example, the analysis book covers induction and inequalities in the first chapters and the group theory book covers functions in the first chapter which builds into a chapter on permutations of a finite set.

The group theory book really surprised me. I'm currently reading about groups of permutations of $\mathbb{R}$ and $\mathbb{C}$ and isometries of the plane. The very first problem has given me more insight about linear algebra compared to all the matrix-heavy representations of the same topic I have seen before. Why all linear algebra books don't start this way, I do not know.

I heartily recommend both books as they seem to coincide with both aspects of what you seek to learn.

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O'Neil's differential geometry is probably a great place to learn both calculus and some elementary differential geometry at the same time! It has been posted to the internet archive. It isn't so easy to find a physical copy of this book, unless you're at a large university. https://archive.org/details/ElementaryDifferentialGeometry

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  • $\begingroup$ is differential geometry is suitable for the beginner $\endgroup$ – Imran Kabir Feb 25 '17 at 13:39
  • $\begingroup$ This book is. You could easily read this book in concert with a book on vector calculus, $\endgroup$ – RougeSegwayUser Feb 26 '17 at 0:37
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Check out Spivak's "Calculus on Manifolds." It is free online and one of the classic great textbooks on calculus.

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  • $\begingroup$ Thanks bro, I will go through that $\endgroup$ – Imran Kabir Feb 26 '17 at 14:18

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