Higher Mathematics Books [upd] ✮ (Deluxe)

Undergraduate Level

1. Calculus:
   • "Calculus" by Michael Spivak (7th edition)
   • "Calculus: Early Transcendentals" by James Stewart (8th edition)
   • "Calculus: A Modern Approach" by Karl Menger (2nd edition)
2. Linear Algebra:
   • "Linear Algebra and Its Applications" by Gilbert Strang (4th edition)
   • "Introduction to Linear Algebra" by James DeFranza (5th edition)
   • "Linear Algebra: A Geometric Introduction" by David C. Lay (2nd edition)
3. Real Analysis:
   • "Real and Complex Analysis" by Walter Rudin (3rd edition)
   • "Real Analysis: A First Course" by Frank Jones (2nd edition)
   • "Introduction to Real Analysis" by Bartle and Sherbert (4th edition)
4. Abstract Algebra:
   • "Abstract Algebra" by David S. Dummit and Richard M. Foote (3rd edition)
   • "A First Course in Abstract Algebra" by John A. Carter (2nd edition)
   • "Introduction to Abstract Algebra" by W. Keith Nicholson (2nd edition)

Graduate Level

1. Advanced Calculus:
   • "Advanced Calculus" by Robert W. Weirstrass (2nd edition)
   • "Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus" by Michael Spivak (1st edition)
   • "Advanced Mathematical Analysis" by Walter A. Strauss (1st edition)
2. Measure Theory:
   • "Real Analysis: Measure Theory, Integration, and Hilbert Spaces" by Elias M. Stein and Rami Shakarchi (1st edition)
   • "Measure Theory" by Frank Jones (1st edition)
   • "Introduction to Measure Theory" by Yehuda Pinchover and Jacob Rubinstein (1st edition)
3. Functional Analysis:
   • "Functional Analysis: Theory and Applications" by Gerald Teschl (2nd edition)
   • "Introduction to Functional Analysis" by James R. Kirkwood (1st edition)
   • "Functional Analysis" by Peter D. Lax (1st edition)
4. Differential Equations:
   • "Partial Differential Equations" by L.C. Evans (2nd edition)
   • "Ordinary Differential Equations" by Vladimir I. Arnold (3rd edition)
   • "Differential Equations and Dynamical Systems" by Lawrence Perko (3rd edition)

Specialized Topics

1. Number Theory:
   • "An Introduction to the Theory of Numbers" by G.H. Hardy and E.M. Wright (6th edition)
   • "Number Theory: An Introduction" by George E. Andrews (2nd edition)
   • "A Classical Introduction to Modern Number Theory" by Ireland and Rosen (2nd edition)
2. Algebraic Geometry:
   • "Algebraic Geometry: A First Course" by Joe Harris (1st edition)
   • "Introduction to Algebraic Geometry" by Richard Hartshorne (1st edition)
   • "Algebraic Geometry" by Robin Hartshorne (1st edition)
3. Topology:
   • "Topology" by James R. Munkres (2nd edition)
   • "Introduction to Topology" by Bert Mendelson (3rd edition)
   • "Algebraic Topology" by Edwin H. Spanier (1st edition)

Online Resources

1. arXiv: A repository of electronic preprints in physics, mathematics, computer science, and related disciplines.
2. MathWorld: A comprehensive online encyclopedia of mathematics.
3. n-Category Café: A blog on mathematics, physics, and philosophy.
4. Mathematics Stack Exchange: A Q&A platform for mathematics.

Authors and Series

1. Springer-Verlag: A renowned publisher of mathematics books, with many series, including Graduate Texts in Mathematics and Universitext.
2. American Mathematical Society: Publisher of many mathematics books, including the AMS Graduate Studies in Mathematics series.
3. Cambridge University Press: A leading publisher of mathematics books, with series like Cambridge Mathematical Texts and Cambridge Studies in Advanced Mathematics.

Tips

1. Start with the basics: Make sure you have a solid understanding of undergraduate-level mathematics before moving to more advanced topics.
2. Read actively: Engage with the material by working through exercises and proofs.
3. Seek help when needed: Consult online resources, textbooks, or experts if you're struggling with a particular concept.

Navigating the Abstract: A Guide to the Best Higher Mathematics Books

Stepping into the world of higher mathematics is like learning a new language. You move away from the "plug-and-play" arithmetic of high school and into a realm of proofs, structures, and infinite abstractions. Whether you are a student, a self-taught enthusiast, or a professional looking to sharpen your logic, the right book is the difference between clarity and total confusion.

Here is a curated roadmap of the foundational and advanced texts that have shaped the minds of mathematicians for decades. 1. The Gateway: Transition to Higher Math

Before diving into Calculus or Algebra, you need to learn how to speak the language: Mathematical Proofs.

"How to Prove It: A Structured Approach" by Daniel J. VellemanThis is widely considered the "gold standard" for anyone transitioning from computational math to theoretical math. It teaches you how to think logically and how to construct a rigorous argument from scratch.

"The Foundations of Mathematics" by Ian Stewart and David TallA great companion for understanding set theory, relations, and the logic that underpins every other branch of math. 2. Analysis: The Rigor Behind Calculus

If Calculus is about "how" things change, Analysis is about "why." It’s where you deal with the "epsilon-delta" definitions that make calculus work.

"Principles of Mathematical Analysis" by Walter Rudin (The "Baby Rudin")Infamous for its brevity and elegance, this book is a rite of passage. It’s dense and difficult, but mastering it gives you a level of mathematical maturity that few other books can provide.

"Understanding Analysis" by Stephen AbbottIf Rudin feels like a brick wall, Abbott is the ladder. It is exceptionally well-written, focusing on the intuition behind the proofs without sacrificing rigor. 3. Algebra: Beyond Solving for X

In higher math, "Algebra" means Abstract Algebra—the study of groups, rings, and fields.

"Abstract Algebra" by David S. Dummit and Richard M. FooteAn encyclopedic text. It’s heavy, but it covers almost everything an undergraduate or beginning graduate student needs to know. It’s a fantastic reference book to keep on your shelf for life.

"Contemporary Abstract Algebra" by Joseph GallianA more accessible entry point. Gallian uses plenty of examples and historical notes to make the abstract concepts feel more "real." 4. Geometry and Topology: The Shape of Space

Topology is often described as "rubber-sheet geometry," where you study properties that remain unchanged even if you stretch or twist an object.

"Topology" by James MunkresThe definitive introductory text. Munkres is incredibly clear, making a notoriously difficult subject feel manageable.

"Visual Complex Analysis" by Tristan NeedhamFor those who prefer a geometric approach to complex numbers and functions, this book is a masterpiece. It uses diagrams to explain concepts that are usually buried in equations. 5. Linear Algebra: The Workhorse of Modern Math

Linear algebra is the backbone of data science, physics, and engineering.

"Linear Algebra Done Right" by Sheldon AxlerA favorite among theorists. Axler avoids using determinants until the very end, focusing instead on linear maps and operators to provide a deeper understanding of the structure of vector spaces.

"Introduction to Linear Algebra" by Gilbert StrangIf you want a more practical, application-heavy approach, Strang’s book (and his famous MIT lectures) is the way to go. How to Choose Your Next Book

When picking a book in higher mathematics, consider your learning style:

The Minimalist: Go for Rudin or Axler. They provide the bare essentials and expect you to do the heavy lifting.

The Visualist: Look for Needham or Gallian. They use intuition and imagery to bridge the gap.

The Practicalist: Strang or Dummit & Foote provide the exhaustive examples you need to see the math in action.

Higher mathematics is a marathon, not a sprint. The best book isn't necessarily the hardest one—it’s the one that keeps you turning the page until the "aha!" moment finally hits.

The journey through higher mathematics is less about memorizing formulas and more about learning a new language—one that describes the hidden architecture of the universe. Whether you are an aspiring physicist, a data scientist, or a pure math enthusiast, the right books act as mentors, bridging the gap between mechanical calculation and abstract reasoning.

Here is a curated guide to the essential pillars of higher mathematics and the definitive texts that define them. 1. The Gateway: Transitioning to Proofs

Before diving into specialized fields, you must master the art of the mathematical proof. Higher math is not about "solving for x"; it is about proving why "x" must exist.

"How to Prove It: A Structured Approach" by Daniel J. Velleman: This is widely considered the gold standard for students moving from calculus to abstract math. It deconstructs the logical syntax of proofs, teaching you how to communicate complex ideas with rigor.

"Book of Proof" by Richard Hammack: A fantastic, often free alternative that covers set theory, logic, and various proof techniques with clear, approachable examples. 2. Analysis: The Rigor Behind Calculus higher mathematics books

Calculus in high school is often taught as a set of rules. Analysis is the branch of math that proves those rules. It deals with limits, continuity, and the nature of real numbers.

"Principles of Mathematical Analysis" by Walter Rudin: Known affectionately (and sometimes infamously) as "Baby Rudin," this book is dense and demanding. However, it is the most influential text in the field, providing a level of mathematical maturity that few other books can match.

"Understanding Analysis" by Stephen Abbott: If Rudin feels like a brick wall, Abbott is the ladder. It is exceptionally well-written, focusing on the "why" and the beauty of the subject rather than just the technical grind. 3. Abstract Algebra: The Study of Structure

Abstract algebra moves away from numbers and looks at structures like groups, rings, and fields. It is the foundation of modern cryptography and theoretical physics.

"Abstract Algebra" by David S. Dummit and Richard M. Foote: A massive, comprehensive tome that serves as both a textbook and a reference. It is famous for its vast array of exercises that help solidify abstract concepts.

"Contemporary Abstract Algebra" by Joseph Gallian: A more accessible entry point that uses modern examples and visual aids to explain how algebraic structures work in the real world. 4. Linear Algebra: The Engine of Modern Science

From Google’s search algorithms to quantum mechanics, linear algebra is the workhorse of modern mathematics.

"Linear Algebra Done Right" by Sheldon Axler: This book is a favorite among purists. Axler avoids using determinants until the very end, forcing the reader to understand the underlying geometry and operators of vector spaces.

"Introduction to Linear Algebra" by Gilbert Strang: If you prefer a more computational and intuitive approach, Strang’s book (and his famous MIT OpenCourseWare lectures) is the gold standard for practical application. 5. Topology: "Rubber Sheet" Geometry

Topology studies the properties of spaces that remain unchanged under continuous deformation (stretching or twisting, but not tearing).

"Topology" by James Munkres: The definitive introductory text. It is rigorous yet clear, taking the reader from basic set theory into the complexities of topological spaces and fundamental groups. How to Choose Your Next Book

When selecting a higher mathematics book, consider your mathematical maturity. If you are self-studying, look for books with "Introduction" in the title or those praised for their "exposition" (the quality of the explanations).

Higher mathematics is a marathon, not a sprint. The best book isn't necessarily the most advanced one—it’s the one that challenges you enough to grow without discouraging you from the beauty of the pursuit.

8. Resources & formats

• Use printed books for deep reading, PDFs for searching, and lecture videos for intuition.
• Consult solution manuals, community forums (for hints, not full answers), and university course pages for curated problem sets.

If you want, I can:

• generate a tailored 3- or 6-month reading schedule for a specific background and goal, or
• provide a short annotated bibliography for one chosen topic.

(Next: related search suggestions.)

Higher mathematics moves away from rote calculation and toward abstract reasoning, rigorous proofs, and structural patterns. If you're looking to build a solid library or start a self-study journey, here are the essential pillars and the "gold standard" books for each. 1. The Foundation: Set Theory & Logic Before the "math" starts, you need the language of proofs. "Book of Proof" by Richard Hammack:

The best entry point for learning how to write direct proofs, inductions, and contradictions. It’s accessible and free online. "Naive Set Theory" by Paul Halmos:

A classic that explains the building blocks of mathematics without getting bogged down in overly formal logic. 2. Analysis: The "Why" of Calculus

Analysis takes the "how-to" of calculus and proves it using limits and epsilon-delta definitions. "Principles of Mathematical Analysis" by Walter Rudin:

Known affectionately (and infamously) as "Baby Rudin." It is dense and difficult, but mastering it is a rite of passage for every mathematician. "Understanding Analysis" by Stephen Abbott:

A much friendlier alternative to Rudin. It focuses on the intuition behind the proofs. 3. Abstract Algebra: Symmetry & Structure

This field looks at "operations" (like addition) on "sets" (like integers) and abstracts them into Groups, Rings, and Fields. "Abstract Algebra" by Dummit & Foote:

The comprehensive "encyclopedia" for the subject. It’s heavy but contains everything you’ll need from undergraduate to early graduate level. "Contemporary Abstract Algebra" by Joseph Gallian:

Highly recommended for beginners due to its clear examples and applications to things like DNA and cryptography. 4. Linear Algebra: Beyond the Matrix

While intro linear algebra is about solving equations, higher linear algebra is about vector spaces and linear transformations. "Linear Algebra Done Right" by Sheldon Axler:

This book is famous for avoiding determinants until the very end, focusing instead on the geometry and structure of linear maps. 5. Topology: "Rubber Sheet" Geometry

Topology studies properties that stay the same even when you stretch or bend a shape. "Topology" by James Munkres:

The undisputed standard. It’s clear, well-organised, and covers both point-set and fundamental concepts. 6. Number Theory: The Secrets of Integers

"An Introduction to the Theory of Numbers" by Hardy & Wright:

A legendary text. It’s more of a reference than a textbook, but it contains beautiful insights into primes and divisibility. Tips for Success Don't just read; do.

Higher math is a "contact sport." You can't understand a proof until you try to recreate it on paper. Expect to be stuck.

It is normal to spend three hours on a single page of a Rudin or Munkres book. Use Visuals.

For subjects like Topology and Complex Analysis, look for "Visual Complex Analysis" by Tristan Needham to see the geometry behind the symbols. Are you looking to dive into a specific branch first, or are you looking for a structured curriculum for self-study? Undergraduate Level

Higher mathematics is a vast landscape of abstract structures, logical proofs, and complex problem-solving

. Whether you are a student transitioning from secondary school or a researcher looking for specialized references, the right books serve as the essential map for this journey. The Bridge to Higher Mathematics

For many, the jump from "calculation-based" math to "proof-based" math is the most challenging transition. These books focus on the logic and rigor required for undergraduate studies: Towards Higher Mathematics: A Companion Cambridge University Press

: Designed to bridge the gap between school and university, covering extension material with over 1500 exercises. Learning Higher Mathematics

by L.S. Pontrjagin: A foundational text that introduces analytic geometry, limits, and calculus in a conversational yet rigorous style. A Course of Pure Mathematics

by G.H. Hardy: A classic resource that has introduced generations to the rigorous study of analysis. Core Subjects and Specialised Texts

As you progress, the curriculum branches into specific domains like abstract algebra, topology, and advanced analysis. This Will Make You Better at Mathematics FOREVER 9 Dec 2024 —

Whether you're a self-taught learner or a graduate student looking for deeper intuition, finding the right "higher math" book can be daunting. Many advanced texts can feel like a "wall of definitions", but some masterpieces bridge the gap between abstract rigor and genuine understanding. Accessible "Entry Points" into Higher Mathematics

Before diving into formal graduate coursework, these "popular" math books help build excitement and intuition for advanced concepts: The Joy of X

: A guided tour by Steven Strogatz that starts from basic counting and builds up to infinity with an entertaining, informative style. How Not to Be Wrong

: Jordan Ellenberg uses a playful writing style and real-world connections to show how mathematical thinking prevents mental pitfalls. Fermat's Enigma

: A beautifully written narrative by Simon Singh detailing the epic quest to solve one of the world's most famous mathematical problems. Foundational Textbooks for Self-Study

If your goal is to move from undergraduate basics to professional-level proficiency, these texts are highly recommended for their clarity: A Concise Introduction to Pure Mathematics

: Written by Martin Liebeck, this is an ideal "intro to proofs" that prepares you for the way pure mathematicians think. Linear Algebra Done Right

: Sheldon Axler focuses on building genuine understanding of the structure of linear algebra rather than just mechanical computation. Analysis I & II

: Terence Tao's series on real analysis is praised for including essential discrete math and set theory often skipped in other texts.

Unlocking the Secrets of Higher Mathematics: A Comprehensive Guide to Advanced Math Books

For students and professionals looking to dive deeper into the world of mathematics, higher mathematics books are an essential resource. These advanced texts provide a gateway to understanding complex mathematical concepts, theories, and applications that underlie many fields, including physics, engineering, economics, and computer science. In this article, we'll explore the realm of higher mathematics books, highlighting key topics, notable authors, and influential texts that can help you master advanced mathematical concepts.

What are Higher Mathematics Books?

Higher mathematics books, also known as advanced mathematics books, are texts that cover mathematical topics beyond the introductory level. These books typically assume a strong foundation in basic mathematics, including calculus, linear algebra, and differential equations. They delve into more specialized areas, such as abstract algebra, real analysis, topology, and differential geometry, providing a deeper understanding of mathematical structures, theorems, and proofs.

Key Topics in Higher Mathematics Books

Some of the key topics covered in higher mathematics books include:

1. Abstract Algebra: This branch of mathematics deals with algebraic structures, such as groups, rings, and fields. Books on abstract algebra explore the properties and behavior of these structures, including group theory, ring theory, and Galois theory.
2. Real Analysis: Real analysis is concerned with the study of real numbers and real-valued functions. Advanced texts on real analysis cover topics like measure theory, functional analysis, and differential equations.
3. Topology: Topology is the study of shapes and spaces, focusing on properties that are preserved under continuous deformations. Higher mathematics books on topology cover point-set topology, algebraic topology, and differential topology.
4. Differential Geometry: This field combines differential equations and geometry to study curves, surfaces, and higher-dimensional manifolds. Books on differential geometry explore topics like curvature, geodesics, and Riemannian geometry.
5. Number Theory: Number theory is concerned with the properties and behavior of integers and other whole numbers. Advanced texts on number theory cover topics like prime numbers, modular forms, and elliptic curves.

Notable Authors and Influential Texts

Some notable authors and influential texts in the realm of higher mathematics books include:

1. Michael Artin's "Algebra": This classic text provides a comprehensive introduction to abstract algebra, covering group theory, ring theory, and Galois theory.
2. Walter Rudin's "Principles of Mathematical Analysis": This influential book provides a rigorous introduction to real analysis, covering topics like measure theory, functional analysis, and differential equations.
3. James Munkres' "Topology": This text provides a thorough introduction to point-set topology, covering topics like compactness, connectedness, and separation axioms.
4. Serge Lang's "Differential Geometry": This book provides a comprehensive introduction to differential geometry, covering topics like curvature, geodesics, and Riemannian geometry.
5. Andrew Wiles' "Modular Forms and Fermat's Last Theorem": This text provides an introduction to modular forms and their application to number theory, including the proof of Fermat's Last Theorem.

Benefits of Reading Higher Mathematics Books

Reading higher mathematics books can have numerous benefits, including:

1. Deepening your understanding of mathematical concepts: Advanced texts help you develop a deeper understanding of mathematical structures, theorems, and proofs.
2. Developing problem-solving skills: Higher mathematics books provide numerous exercises and problems, helping you develop your problem-solving skills and mathematical intuition.
3. Preparing for advanced study or research: Reading higher mathematics books can prepare you for advanced study or research in mathematics, physics, engineering, or other fields.
4. Enhancing your critical thinking and analytical skills: Advanced mathematical texts help you develop critical thinking and analytical skills, valuable in a wide range of fields.

Challenges and Tips for Reading Higher Mathematics Books

Reading higher mathematics books can be challenging, but with the right approach, you can overcome these challenges. Here are some tips:

1. Start with a strong foundation: Make sure you have a solid understanding of basic mathematics, including calculus, linear algebra, and differential equations.
2. Work through exercises and problems: Exercises and problems are an essential part of learning advanced mathematics. Work through them carefully, and don't be afraid to ask for help.
3. Join a study group or find a study partner: Joining a study group or finding a study partner can provide motivation, support, and a chance to discuss challenging topics.
4. Take your time: Advanced mathematics books require patience and persistence. Take your time, and don't be discouraged if you encounter difficulties.

Conclusion

Higher mathematics books are a valuable resource for students and professionals looking to deepen their understanding of advanced mathematical concepts. With their rigorous treatment of abstract algebra, real analysis, topology, differential geometry, and number theory, these texts provide a gateway to understanding complex mathematical structures, theorems, and proofs. By reading higher mathematics books, you can develop problem-solving skills, critical thinking, and analytical skills, preparing you for advanced study or research in mathematics, physics, engineering, or other fields. Whether you're a student, researcher, or simply a math enthusiast, higher mathematics books offer a rich and rewarding journey into the world of advanced mathematics.

Conclusion: The Infinite Ascent

The search for the right higher mathematics books is a personal journey. One reader’s "elegant and terse" is another’s "inscrutable and arrogant." Steven Strogatz (Cornell) loves Abbott; Terence Tao (Fields Medalist) recommends a blend of Rudin and counterexamples.

Start with a bridge book like How to Prove It. Move to Understanding Analysis and Linear Algebra Done Right. Then, and only then, gaze upon the summit of Munkres or Dummitt & Foote.

These books are not just textbooks. They are keys to a secret society. Once you have worked through them, you will never look at a graph, a pattern, or a piece of data the same way. You will see the scaffolding of logic beneath the chaos. Calculus :

Pick one. Buy a pencil. Turn to page one. And begin.

Which higher mathematics book changed your life? Do you prefer Rudin’s iron fist or Abbott’s gentle hand? Share your recommendations in the comments below.

The transition from calculus to higher mathematics involves a shift from computation to rigorous proof and abstraction. The "best" book often depends on your specific goal—whether you are preparing for a professional exam, self-studying for a math major, or looking for a historical perspective on great theorems. 📘 Essential Foundations (The Transition) These books bridge the gap between "solving for " and "proving Foundations of Analysis by Joseph L. Taylor

: This text is highly regarded for introducing the depth and rigor of higher mathematics. It covers single and several variable calculus with a focus on real number properties and linear algebra integration.

A Review of Algebra: Enriched Edition by Romeyn Henry Rivenburg

: A comprehensive look at algebraic principles that balances theoretical foundations with practical applications. Linear Algebra by Friedberg, Insel, and Spence

: Often recommended for its clear conceptual development and rigor, making it a staple for college-level study. 🎓 Self-Teaching & Mastery

For those navigating higher mathematics independently, these titles are frequently cited by students and experts alike. Calculus by Thomas

: A classic, rigorous text for mastering single-variable and multivariable calculus concepts. Introduction to Linear Algebra

by Gilbert Strang: Known for its intuitive approach and the accompanying MIT OpenCourseWare videos. A First Course in Differential Equations by Dennis Zill

: A standard introductory text that provides a solid foundation for more advanced analysis. Complex Variables and Applications

by Ruel V. Churchill: Widely recommended for its clear treatment of complex analysis. 🛠️ Professional & Competitive Prep

If your interest in higher math is driven by engineering certifications or competitive exams. UK Civil/Mechanical Engineering Mastery

by Chandresh Agrawal: These compendiums offer thousands of advanced MCQs focused on higher-level calculus and structural analysis for professional reviews like the ICE or IMechE Advanced Problems in Mathematics for JEE

💡 Pro Tip: If you are just starting your journey into higher math, check out Khan Academy for intuitive video explanations before diving into dense, proof-heavy textbooks.

To help me recommend the perfect book for you, could you tell me:

What is your current math level (e.g., high school, college student, professional)?

Is there a specific branch you are interested in (e.g., Number Theory, Topology, Real Analysis)? Best Books About Mathematics (451 books) - Goodreads

To build a curated list of exceptional higher mathematics books, the selection must balance rigorous theory with intuitive explanation. The books below are widely regarded by the mathematical community as some of the best texts for mastering advanced concepts. 📐 Foundations & Transition to Proofs

Before diving into advanced fields, a student must master mathematical rigor, logic, and proof construction. How to Prove It: A Structured Approach by Daniel J. Velleman

Teaches the transition from computational math to proof-based math.

Explains how to dissect and construct complex logical arguments. Proofs: A Long-form Mathematics Textbook by Jay Cummings

Provides a famously conversational and accessible introduction to proofs. 📈 Analysis & Calculus

Analysis provides the rigorous, proof-based foundation for the calculus concepts you already know.

Calling All Math Enthusiasts: What Are Your Favorite Math Books?

Higher mathematics encompasses the rigorous study of abstract structures, ranging from the fundamental logic of proofs to the complex behaviors of manifold surfaces. For students and enthusiasts alike, selecting the right literature is a "rite of passage" that often defines their intellectual trajectory. Abstract algebra

This is an excellent goal. "Higher mathematics" typically means moving beyond calculus (analysis) and linear algebra into proof-based, abstract reasoning. The right book depends entirely on your current level and goal (pure math, physics, engineering, self-study).

Here is a helpful, tiered guide to higher mathematics books, from foundations to advanced topics.

6. Differential Geometry

• Differential Geometry of Curves and Surfaces – Manfredo do Carmo
  Standard for first course.
• Riemannian Geometry – Manfredo do Carmo
  Graduate-level.
• Introduction to Smooth Manifolds – John M. Lee
  Very thorough and readable (part of his "Manifolds" trilogy).