I believe that most of us are unaware of a certain no of books at the undergraduate level . Although most of the times it happens because of lack of exposure , sometimes we are too inept to make an attempt on knowing. Eitherway the unawareness remains. Also the books listed below were or still are preferred by us ( I say “us” because I studied in a group which consisted of, prayagdeep (firstname.lastname@example.org) who is now in CMI, pratik(Pratikmisra007@gmail.com) who is in IIT Madras and pratyush (Mishrapratyushkumar@gmail.com ) who is in IISER Mohali). These are our personal choices. There are a lot other very good books other than those listed below. These books find a mention because we like them( which makes them good too).
- Robert G, sherbert: Introduction to real analysis
- Introduction to mathematical analysis,walter rudin (most of the times we all think of it as difficult. But trust me difficulty lies in our mind, no where else.)
- A basic course in real analysis by kumaresan and ajith kumar(this is a very new book. It finds a mention because before theorems are proved some time has been given to “how they are going to prove it” and “why they are doing so?”)
- Elementary analysis: The theory of calculus by Ross, springer
- A companion to analysis by korner, ams ( this is termed as a graduate level book but I believe that this is accessible to undergraduates and they will gain a lot from this wonderful book)
- Linear algebra done right by AXLER.(This is one of my favorite books. The solution manual can be found here http://math-test.googlecode.com/files/Linear%20Algebra%20Done%20Right%202e%20Solutions%20Manual.pdf.
- Linear algebra by friedberg ( sometimes called as insel linear algebra too)
- Matrix algebra by fuzhen zhang
- Algebra by michael artin
- Algebra by artin( A must read book. Has insights)
- Contemporary abstract algebra by Gallian( Has a lot of problems. Probably a good book to start with)
- Abstract algebra by dummit and foote
- A Course in Algebra by vinberg( Another AMS publications master class, has a lot of examples)
- Topics in algebra by I N Herstein( Everyone knows about it. Some parts of the solution can be found here:www.math.iisc.ernet.in/~rakesh13/group_theory.pdf
- Ordinary differential equations with historical notes and applications by George F Simmons
- Ordinary differential equations by Coddington
- Ordinary differential equations with boundary value problems by boyce and diprima
Metric Space and topology
- Topology of metric spaces by kumaresan
- Basic topology by Armstrong
- Topology by munkres
- Calculus by Thomas and Finney
- Basic Multivariable Calculus by Marsden, Tromba and Weinstein ( the drawback with this book is that it has only answers to odd numbered questions)
- Caculus, VOL-1&2, by Tom. M. Apostol
- Complex Analysis , J Bak and D.Newman
We learn mathematics by solving problems. The following are some very good problem books:
- Problems in Mathematical Analysis, VOL-1,2,3, Kraczor and Nowak, AMS Publications
- Berkely Problems in Mathematics, De Souza and Silva
- Problems in Linear Algebra, Fuzhen Zhang
Happy problem solving. I will keep on updating these as I go along.
All these books are best for selfstudy and can be bought online from sites like amazon.in, flipkart.com, infibeam.com etc at cheap prices.The answers to the exercises are either found on the back or online after a little bit of google search.
- stackexchange.com : You can post your own questions here and also answer others. More like a facebook of mathematics.
- org,edx,mit courseware: A lot of videos available online for free,
- iitm.ac.in: a lot of math video lectures
- libgen : a lot of ebooks available online for free (for download)
- bookos.org is again the same
- indiabookstore.net ( Here you can check the availability of various books online for buying at the lowest price in India)
Note: It is best and more than enough to pick one book and solve it completely for each topic.