### Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1

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Your options are the latter part of Keisler's book above for an infinitesimals approach; Lang's "Calculus of Several Variables" or the latter part of Simmons' book to continue with their approach for weaker and less prepared students; and "Calculus, Vol. II" by Richard Courant and Fritz John the paperback is split into 2 parts to continue on with the standard rigorous approach. The following texts take a slightly more rigorous approach than Apostol or Courant and go a bit deeper into the subject by covering differential forms and manifolds.

Most single semester courses on vector calculus do not have time to reasonably cover this material, and consequently is usually skipped until later, but this advanced perspective can greatly aid one's understanding of the subject. You could study this material either when you first learn multivariable calculus or when you want a second pass on the subject, after just learning the basic methods, to improve your understanding while deepening your knowledge by generalizing what you've seen before.

They can also be used as supplements or stepping stone to an advanced multivariable analysis course.

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This subject is the study of Geometry using the tools that you learned in Vector Calculus and serves as a preparation to more abstract approaches to Differential Geometry you'll see in the future. Most schools only quickly pass through the subject during multivariable calculus but it will help in the long run if you study the material early on. The standard text used in college courses is "Elementary Differential Equations" by Boyce and DiPrima, which many people do seem to like not me however.

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The necessary prerequisite knowledge is just precalculus but some calculus knowledge is useful and may appear in a few examples. For a first exposure to the subject there really isn't that much to learn. You typically cover systems of equations, matrix operations, Gaussian elimination also known as row reduction , LU decomposition, determinants, eigenvectors and eigenvalues, and diagonalization possibly with a few additional fluff subjects to round out a whole course. Many times you can pick up this material while studying calculus or ODEs like with Apostol or Hubbard 2 's book so you can just skip to more advanced material.

Also, the introductory material in first few chapters of advanced textbooks are often good enough to learn matrix algebra from if you're in a rush. But while some students seem to inhale these topics and quickly move on, others will need to take their time before operating with matrices becomes natural to them. Learners with slightly better math abilities can benefit more from "Matrices and Linear Transformations" by Cullen which is aimed at STEM students and contains extra material at the end on advanced material.

A free book for students seeking a honors introduction to linear algebra and basic proofs is "Linear Algebra Done Wrong" by Treil Don't worry, the title is a pun on Axler's "done right" book below. Another popular free book is Hefferon's Linear Algebra. There's also a whole host of vulgarly over expensive textbooks used by college courses at this level like Strang's Introduction to Linear Algebra, Lay's Linear Algebra and Its Applications, Friedberg's Elementary Linear Algebra, etc but most of them aren't very good and even if they were, the first 2 aforementioned books above are far cheaper thanks to them being published by Dover and the last 2 are free.

Bonus is they are all free. For a first book in applied linear algebra, "Linear Algebra and Its Applications" by Strang is the standard text used but it is one of those love it or hate it texts. After reading one of them, you'll be more than ready to move onto advanced Numerical Linear Algebra and Matrix Analysis textbooks.

To get started on the theoretical side of linear algebra you obviously should be familiar with the basics of proofs. Once you are, theory side has a lot of classic and well loved textbook to choose from:.

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Of course there's also "Linear Algebra done Right" by Axler and on the one hand, the stuff he does is great Because of that you shouldn't use his book alone to learn from and you really should read Shilov alongside of it. But Axler certainly gives an unique development of the subject. It is good for learning linear algebra for the first time if you're a hot shot freshman, using it as a second book on linear algebra, or as a 3rd refresher book for those who are entering graduate school.

Another good 3rd book for deeper linear algebra study, and if you have the abstract algebra background for it, is Roman's "Advanced Linear Algebra". The term Advanced Calculus has come to mean different things over the course of the past century. During the first half of the 20th century, Advanced Calculus courses consisted of what's now commonly found in Multivariable and Vector Calculus possibly with some Differential Equations topics thrown in. Lately, it has been fashionable to call very watered down "Real Analysis" courses Advanced Calculus even though it's not advanced nor calculus and goes no deeper into analysis than a good rigorous calculus book does.

Here Advanced Calculus means what the name implies, advanced topics in calculus and tools from analysis typically not found in the usual calculus sequence but still very useful for solving difficult problems in science, engineering, and mathematics. A good supplement to any of the above is Visual Complex Analysis by Needham. Special Functions used to be the subject of a second semester complex variables course until it was sucked into Mathematical Physics, Advanced Engineering Mathematics and other similar courses.

The problem with such courses is that they spend far too little time developing subject as they try to cover complex variables, PDEs, differential geometry, topology, variations, algebra, and numerical methods among other subjects at the same time. The following books give a more focused and fuller development of special functions:. Whittaker and Watson has been the bible for special functions for over a century now. Part 1 contains a review of the essential real and complex analysis needed for Part 2 which details the major special functions.

The Fourier transform and related transforms are powerful techniques used throughout STEM that convert a function into its frequency components. Tragically, many science and engineering programs can't find room for such a course in their curricula and try to get away with throwing in brief discussion of how to use them into the courses that require them. This in the end fails to create any conceptual understanding of what's going on beyond the mindless crank turning. These books will help you see the Fourier transform beyond just a 'trick' and be better equipped to apply them:.

For more mathematical detailed books see the Fourier Analysis books below. Calculus of Variations is the subject of finding functions that maximize or minimize some equation. For example, finding a path that minimizes the distance traveled from point a to b. If you have read about Feynman, you may have heard his story of coming across Advanced Calculus by Woods and discovering the differentiating parameters under the integral sign [1] [2] [3] trick and using it to his advantage over and over again. Just as you can iterate to get second derivatives and triple integrals, it's possible to extend the order of these operators from integers to fractions or to any real or complex number.

For example, you can define a half derivative operator where if you apply it twice to a function, you get the usual derivative of that function. This is the domain of Fractional Calculus which has a wide variety of applications in many branches of physics and engineering. The idea of fractional calculus is an old one dating back to Leibniz in and its applications were examined by the electrical engineer Oliver Heaviside in the s but the first textbook on the subject was only published in by Oldham and Spanier.

Since then fractional calculus has steadily been gaining more attention but it still remains relatively unknown to many in the STEM field. Historically, the study of PDEs was a major impetus for the development of many results of analysis. General Theorems regarding the Probability of Compound Events Evaluation of Probabilities involving Factorials of Large Numbers. Life Contingencies Bemarks General and Bibliographical. Algebra, an Elementary Text-book for the Higher Classes of Exercises V.

Exercises XXI. Exercises XXII. Exercises XXIV. Exercises XXV. Exercises VII. Exercises IX. Exercises X. Exercises XL.

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