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Best mathematical book that include all of the following:

  • Algebra and linear algebra: (Numbers, fractions, exponents, log, factoring, equations, matrices and determinants, etc...)

  • Trigonometry: (Trig. ratios; sin,cos,tan, csc,sec,cot, sine rule, cosine rule, Hero's formula, sum to product, half and double angles, etc...)

  • Calculus I: (Limits, ordinary differentiation, partial differentiation, implicit differentiation, directional derivatives, etc...)

  • Calculus II: (integration, tables of integration, IBP, trig sub, trig integrals, arc length, surface of revolution, volume of revolution, ets...)

  • Calculus III: (Vectors, double integral, triple integral, volume integral, optimisation, ...)

  • Statistics: (Probability, central tendency: mean, median, mode, variance, SD, counting, sum and product rules, permutation, combination, etc...)

  • Differential Equations: (Variable separable DE, homogeneous DE, growth and decay such as Newton's low of cooling/heating, variation of parameters, Wronskain, Rung-kuta method, Newton method, Euler method, Laplace transform, ets...)

  • Engineering mathematics: (Line integrals, Stoke's theorem, Green theorem, Contor integrals, ets...)

It is not necessary to have deep/details of all of the above, but it is necessary to have all of the categories (Algebra, Trigonometry, Calculus I,II,III, Statistics, Differential Equations, Engineering Mathematics) even with brief and basic ideas.

In other words, I want a comprehensive book even not detailed one.

Are there such books? Which one do you prefer? Is it hard copy or soft copy (eg. PDF file)? Is it cheap or expensive?

No matter if it is old, like 1980 or older, it is ok if it is handful.

AGAIN, the book should not be detailed, but at least contains all of the above briefly.

Any help/suggestion would be really appreciated. THANKS!

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There are books called some variation on "Mathematical methods in $X$" which cover almost all, if not all, of the topics you mention. Three examples I am aware of:

  • Mathematical Methods in Physics and Engineering by Riley, Hobson and Bence (Cambridge University Press, currently in its third edition, 2006).

  • Mathematical Methods in the Physical Sciences by M. L. Boas (Wiley, third edition 2005).

  • Mathematical Methods for Physicists: A Comprehensive Guide by Arfken, Weber and Harris (Academic Press, seventh edition 2012).

You can check the tables of contents of all three on Amazon.

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  • $\begingroup$ I check the table of contents of the first book. It seems so useful. THANKS DEAR. $\endgroup$ – Hussain-Alqatari Sep 17 at 20:02
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You probably won't find a single book that meets all of your requirements so you have to settle for one book that more or less covers what you think you need and in enough detail. I suggest the 1999 Dover Publications one volume book Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrent'ev. It has over 1000 pages and much more than you wanted to know. You can read more about this book in its entry at goodreads.

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One I advice is "The Princeton Companion to Mathematics": comprehensive, complete and so well written! Good luck!

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