**Advanced Calculus** *By Avner Friedman*

The book by Friedman not only consists of functions of one variable but also several variables.

This is designed for students who have already studied elementary calculus. It shows the introduction of functions of several variables for as long as possible

It beholds clarity and simplicity and avoids a mixture of complicated and rigorous arguments.

The first part consists of functions of one variable, including numbers and sequences problems. It also shows continuous functions, differentiable functions, sequences, integration and series of functions.

The second part displays the functions of several variables. It consists of continuous functions, differentiation, line and surface integrals, and multiple integrals

At the end of the textbook, complete solutions to the problems have Been given.

The theorems are very precisely stated and proved in detail, but there is also a strong emphasis on computations for explicit functions.

Friedman makes a notational distinction between the Darboux and Riemann integrals and shows their equivalence.

Most of the mathematical authors gloss over this and confuse the two integrals, but showing their equivalence gives a deeper understanding of analysis. This is a very good book, in my view. It is very clear, rigorous and has good exercises.

Unlike other books on the same course that covers real analysis only through a single variable, this book takes it a step further. It has several chapters covering the n-dimensional calculus. Its writing is very clear with detailed proofs.

**Advanced Calculus (Theory and Practice)** By John Srdjan Petrovic

Advanced Calculus: Theory and Practice is the next step after elementary calculus. It organizes the course material in a structured manner.

The text tries to improve problem-solving skills and proof-writing skills. It makes them familiar with the historical development of calculus and its concepts. It helps them understand the connections among different topics.

Proper emphasis is given on historical perspectives, such that the notes give students an idea about the development of calculus.

The ideas from the age of Newton and Leibniz to the twentieth century are all provided in them.

Nearly 300 examples are listed and lead to important theorems. They also help students develop the necessary skills so that the students are able to closely examine the theorems and learn them easily.

The book makes an inspiring approach and makes theorems less unknown to students. It explains how various topics in calculus have common roots, even though they may seem unrelated.

Proofs are also presented in an organized and well-structured way to students. This textbook will help you master calculus with a subtle and unique approach.

It will help them succeed in their future whether in the field of mathematics or field of engineering studies.

I would say it perfectly caters to those higher on the learning curve of Calculus. It brushes over some important content and raises the bar of expectations of the reader. This can be great for those who want to be challenged, but rough for those who need a bit more of a foundation.

This is a very good textbook on advanced calculus and mathematical analysis. There are solutions and hints to selected problems at the end of the book which should be helpful to students studying the subject for the first time.

Little known facts and historical points sprinkled throughout the chapters enliven the text and may motivate the students to do some research on their own.

**Advanced Calculus** By Lynn Harold Loomis and Shlomo Sternberg

The book is divided roughly into two parts.

The first one mainly develops the differential calculus with the use of normed vector spaces, and the second half deals with the calculus of differentiable manifolds.

It is well written and presented.

The concepts are not simple but they are made easy to understand by the author.

This book cannot be said as simple maths, but it lays the foundations for getting into the deeper mathematics one needs to understand. Calculus is really important to appreciate the world of physics.

This book is probably the most mathematically rigorous and challenging. This book is based on the topic of abstract analysis in Banach spaces.

This topic is considered in differential calculus and for the Frechet derivative. It also lists the smooth manifolds for integral calculus, as the setting for the generalized Stokes theorem.

In addition to the former, the authors have also provided a large amount of abstract linear and multilinear algebra problems.

In order to develop these concepts, the text given serves as a general introduction to many important concepts. It also helps in learning the techniques of modern mathematics like duality and natural isomorphisms.

The writing is engaging and clear. The text is self-contained and contains only undergraduate calculus and linear algebra.

Although, only a learner with a strong ability to appreciate mathematical abstraction will benefit from this text. The level of mathematics in the textbook is much higher.

The text was written by Loomis and Sternberg for an “experimental” two-semester course. It was aimed at capable and motivated freshmen to allow them to quickly approach graduate-level courses after finishing the class.

This book is strongly recommended to test and strengthen your ability to grasp abstract mathematical concepts. However, refer to this only if you are good at basic calculus as this is a highly advanced level book.

**Calculus On Manifolds** By Michael Spivak

The approach to teach considered in this book uses elementary versions of modern methods of sophisticated mathematics.

The formal prerequisites include linear algebra, a reasonable amount of set theory and basic calculus course. You must be able to perform differentiation and integration up to one or two variables.

A little knowledge of abstract mathematics is also required.

Calculus on Manifolds is undoubtedly one of the more enticing, challenging and inspiring textbooks. This is not designed for an average student.

Spivak takes the one reading it on a unique journey which begins with simple topological notions of Euclidean n-dimensional space. It ends at the fundamentals of differential manifolds and differential geometry.

There are challenging exercises that contain original out of the box proofs. They are adorned with clever insights, which make this textbook stand a cut above the rest.

The first 3 chapters are understandable by anyone, but the real challenge commences from chapter 4. Then you need to fight the problems, learn dedicatedly, but it will be worth it.

**Advanced Calculus** By Patric M. Fitzpatrick

This book is self-contained and teaches us the creation of basic tools using the completeness axiom. This book is complemented by numerous exercises. Advanced Calculus is a perfect book for undergraduate students studying calculus thoroughly.

It presents the fundamental concepts of mathematical analysis in the clearest and simplest way. It uses illuminating examples and stimulating exercises.

Properties of functions of a single variable such as continuity, differentiability, integrality, and power series representation are provided. Topological and metric properties of Euclidean space are taught in the next few chapters. Special attention has been paid to the motivation for proof.

The author takes care to give intuitive examples and motivates the students with his proofs.

This book offers a far more efficient way to learn advanced calculus. The text shows that mathematical analysis is a coherent body of knowledge, rather than a collection of isolated facts.