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Books for the first two undergraduate years

J. Oumamar
 
Problem-Solving Strategies (Problem Books in Mathematics)
Problem-Solving Strategies (Problem Books in Mathematics)
"You should really already have a copy of this, but I include it anyway because it is an incredible book. Has some use for the Numbers and sets first year course."
Introduction to Metric and Topological Spaces
Introduction to Metric and Topological Spaces
"Decent problem sets and essential to take in Easter term of the first year. Make sure to grasp the inverse function theorem."
Algebra and Geometry
Algebra and Geometry
"Transition book from pre-university mathematics; covers half of first year group theory (Orbit stabilizer and isomorphism theorem), spherical/hyperbolic geometry, elementary results in quadratic forms"
Vector Calculus (Springer Undergraduate Mathematics Series)
Vector Calculus (Springer Undergraduate Mathematics Series)
"Easy guide to doing basic computations in vector calculus and intuitive proofs of the results. A book that is quickly outgrown, but still serves some purpose."
A First Course in Mathematical Analysis
A First Course in Mathematical Analysis
"After 45 years this is still the best first year analysis book on the market, with more stimulating problems that Rudin. Also written in a "transition to university maths" style."
Probability: An Introduction (Oxford Science Publications)
Probability: An Introduction (Oxford Science Publications)
"Covers most of the first year probability course succintly; problem sets a bit lacking though."
Rings, Modules and Linear Algebra (Mathematics Series)
Rings, Modules and Linear Algebra (Mathematics Series)
"Covers the material for the latter half of the rings and modules optional first year course (which you should take!)"
Problem-Solving Through Problems (Problem Books in Mathematics)
Problem-Solving Through Problems (Problem Books in Mathematics)
"An alternative/complement to Engel with more focus on calculus problems."
A First Course in Mechanics (Oxford Science Publications)
A First Course in Mechanics (Oxford Science Publications)
"First year mechanics book. Perhaps a bit lacking in tripos-difficulty and length problems..."
Introduction to Complex Analysis
Introduction to Complex Analysis
"Superb book; you will likely cover the first half in a session at the end of the first year and then complete in the second. Make sure to pick up open mapping, maximum modulus and Rouche's theorem."
An Introduction to Quantum Theory (Oxford Graduate Texts in Mathematics)
An Introduction to Quantum Theory (Oxford Graduate Texts in Mathematics)
"Straightforwardly readable coverage of second and part of third year quantum mechanics."
Elementary Fluid Dynamics (Oxford Applied Mathematics and Computing Science Series)
Elementary Fluid Dynamics (Oxford Applied Mathematics and Computing Science Series)
"Reasonable book for second and a lot of third year fluid mechanics. Very lacking in challenging problems for the earlier material though."
A Second Course in Mathematical Analysis (Cambridge Mathematical Library)
A Second Course in Mathematical Analysis (Cambridge Mathematical Library)
"Many interesting problems and coverage of Analysis II and an alternative view of elementary complex analysis."
Probability and Statistics by Example: Volume 1, Basic Probability and Statistics
Probability and Statistics by Example: Volume 1, Basic Probability and Statistics
"Provides good coverage of first year probability problem solving and second year statistics. Be careful not to abuse this book!"
Elementary Differential Geometry (Springer Undergraduate Mathematics Series)
Elementary Differential Geometry (Springer Undergraduate Mathematics Series)
"Covers the latter half of the second year geometry course. Provides quality solutions to exercises if you are having difficulty."
Algebraic Topology
Algebraic Topology
"The standard text for undergrad algebraic topology. The amount you will cover varies; the first course finishes at homology whilst previous courses had included de Rham cohomology and homotopy theory."
Representations and Characters of Groups
Representations and Characters of Groups
"Despite its size it only covers the first half of the course using slightly annoying conventions. Provides extensive experience with finite groups (up to order 1000 in fact!)"
Principles of Mathematical Analysis (Int'l Ed) (International Series in Pure & Applied Mathematics)
Principles of Mathematical Analysis (Int'l Ed) (International Series in Pure & Applied Mathematics)
"The book most great modern mathematicians learned elementary real analysis from."
Complex Analysis
Complex Analysis
"Covers more elementary topics in the Riemann surfaces course. Absolute classic book ."
An Introduction to the Theory of Numbers (Oxford Science Publications)
An Introduction to the Theory of Numbers (Oxford Science Publications)
"Covers number theory from first year material to the prime number theorem."
Galois Theory, Third Edition (Chapman Hall/CRC Mathematics Series)
Galois Theory, Third Edition (Chapman Hall/CRC Mathematics Series)
"Slow paced and readable introduction to Galois theory. Covers most of the third year course."
Linear Analysis (Cambridge Mathematical Textbooks)
Linear Analysis (Cambridge Mathematical Textbooks)
"Functional Analysis"
Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics)
Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics)
"2nd year Markov chains and more!"
General Relativity
General Relativity
"A decent choice of text for third year General relativity. Exercises are perhaps a bit lacking."
Algebra [1st Edition]
Algebra [1st Edition]
"Comprehensive coverage of Algebra for the first two years, but a bit light on Representation and Galois theory."