Advice on preparation for Cambridge math undergraduate courses
I'm a 25fall Cam math offer-holder who had just achieved an unconditional offer, so i was wondering whether there is anything i can do before i enter uni, for instance, reading relevant books or self-studying some notes. I learnt that for the Michaelmas Term, there would be four courses: differential equations, groups, numbers and sets, and vectors and matrices. Hence, looking for advice on books or videos to prepare for these four courses!
Reply 1
I'm a 25fall Cam math offer-holder who had just achieved an unconditional offer, so i was wondering whether there is anything i can do before i enter uni, for instance, reading relevant books or self-studying some notes. I learnt that for the Michaelmas Term, there would be four courses: differential equations, groups, numbers and sets, and vectors and matrices. Hence, looking for advice on books or videos to prepare for these four courses!
Reply 2
Thx, I got 11 in STEP and went through about 20 past papers
Reply 3
Thank you for your reply! I'll pay back your kindness in good spirit and show you this doc I found which gives more insight into the Maths Course, and using this guide and chatgpt I think you can find some resources to prep adequately for the course. I wish you well! https://www.maths.cam.ac.uk/undergrad/files/coursesIA.pdf
I also asked ChatGPT on what to recommend you lol:
"1. Vectors and Matrices
•
Textbooks:
•
“Linear Algebra Done Right” – Sheldon Axler (deep, conceptual)
•
“Introduction to Linear Algebra” – Gilbert Strang (more applied, with insight)
•
Videos:
•
3Blue1Brown – Essence of Linear Algebra
•
MIT OCW – Gilbert Strang’s lectures (entire MIT course)
•
Proof Practice: Learn to prove linear independence, rank-nullity, etc.
•
Challenge Problems:
•
MIT OpenCourseWare problem sets
•
Art of Problem Solving Linear Algebra problems
•
Textbooks:
•
“A Book of Abstract Algebra” – Charles Pinter (approachable with proofs)
•
“Visual Group Theory” – Nathan Carter (builds strong intuition)
•
Videos:
•
VisualGroupTheory (YouTube)
•
Imperial College’s algebra lectures (on YouTube)
•
Enrichment:
•
Learn Cayley’s theorem, group actions, and symmetries beyond the syllabus.
•
Practice: Do proof-heavy questions — many undergrads crumble on abstraction.
•
Textbooks:
•
“How to Prove It” – Daniel Velleman (gold standard for beginners)
•
“Naive Set Theory” – Paul Halmos (foundational, rigorous)
•
Key Topics to Master:
•
Proofs (induction, contradiction, contraposition)
•
Modular arithmetic, Euclidean algorithm, countability
•
Bonus Training:
•
Do TMUA and MAT Q1 problems (pure logic)
•
Spivak’s Calculus (the intro chapters) to sharpen mathematical reasoning
•
Textbooks:
•
“Ordinary Differential Equations” – Tenenbaum and Pollard (massive but intuitive)
•
Boyce & DiPrima – standard intro text
•
Videos:
•
Professor Leonard’s ODE playlist (YouTube)
•
MIT OCW ODE lectures
•
Stretch Yourself:
•
Understand existence and uniqueness theorems
•
Solve ODEs from physics (e.g. damped harmonic motion, Newtonian gravity)
•
Textbooks:
•
“Introduction to Probability” – Blitzstein (used at Harvard, very strong examples)
•
“A First Course in Probability” – Sheldon Ross
•
Videos:
•
Harvard Stat 110 by Blitzstein (YouTube)
•
Khan Academy – if you need to brush the basics
•
Key Concepts:
•
Random walks, Bayes, Central Limit Theorem
•
Generate your own simulations with Python/R if you want to stand out
•
Textbooks:
•
“Div, Grad, Curl and All That” – H.M. Schey (must-read)
•
“Vector Calculus” – Marsden & Tromba (clear & comprehensive)
•
Videos:
•
MIT OCW – Multivariable Calculus (Prof. Denis Auroux)
•
3Blue1Brown – relevant visualizations
•
Projects:
•
Try applying divergence/Stokes’ theorem to real physical fields (e.g. electromagnetism)
•
Textbooks:
•
“Understanding Analysis” – Stephen Abbott (intuitive but rigorous)
•
“Principles of Mathematical Analysis” – Walter Rudin (this is hard, but legendary)
•
Videos:
•
Dr. Tadashi Tokieda’s intuition-driven analysis lectures (where available)
•
Essence of Calculus – 3Blue1Brown (for visual reinforcement)
•
Train like a wrangler:
•
Go deep into ε-δ proofs until it becomes second nature.
•
Prove convergence theorems yourself without looking.
•
Textbooks:
•
“Classical Mechanics” – John Taylor (clear, physics-style derivations)
•
“Spacetime Physics” – Taylor and Wheeler (intro to special relativity)
•
Videos:
•
MIT OCW – Walter Lewin’s mechanics lectures
•
PBS Spacetime or MinutePhysics for visual relativity
•
Top tip: Do the derivations yourself (e.g. time dilation, Lorentz transforms)."
Reply 4
I also asked ChatGPT on what to recommend you lol:
"1. Vectors and Matrices
•
Textbooks:
•
“Linear Algebra Done Right” – Sheldon Axler (deep, conceptual)
•
“Introduction to Linear Algebra” – Gilbert Strang (more applied, with insight)
•
Videos:
•
3Blue1Brown – Essence of Linear Algebra
•
MIT OCW – Gilbert Strang’s lectures (entire MIT course)
•
Proof Practice: Learn to prove linear independence, rank-nullity, etc.
•
Challenge Problems:
•
MIT OpenCourseWare problem sets
•
Art of Problem Solving Linear Algebra problems
•
Textbooks:
•
“A Book of Abstract Algebra” – Charles Pinter (approachable with proofs)
•
“Visual Group Theory” – Nathan Carter (builds strong intuition)
•
Videos:
•
VisualGroupTheory (YouTube)
•
Imperial College’s algebra lectures (on YouTube)
•
Enrichment:
•
Learn Cayley’s theorem, group actions, and symmetries beyond the syllabus.
•
Practice: Do proof-heavy questions — many undergrads crumble on abstraction.
•
Textbooks:
•
“How to Prove It” – Daniel Velleman (gold standard for beginners)
•
“Naive Set Theory” – Paul Halmos (foundational, rigorous)
•
Key Topics to Master:
•
Proofs (induction, contradiction, contraposition)
•
Modular arithmetic, Euclidean algorithm, countability
•
Bonus Training:
•
Do TMUA and MAT Q1 problems (pure logic)
•
Spivak’s Calculus (the intro chapters) to sharpen mathematical reasoning
•
Textbooks:
•
“Ordinary Differential Equations” – Tenenbaum and Pollard (massive but intuitive)
•
Boyce & DiPrima – standard intro text
•
Videos:
•
Professor Leonard’s ODE playlist (YouTube)
•
MIT OCW ODE lectures
•
Stretch Yourself:
•
Understand existence and uniqueness theorems
•
Solve ODEs from physics (e.g. damped harmonic motion, Newtonian gravity)
•
Textbooks:
•
“Introduction to Probability” – Blitzstein (used at Harvard, very strong examples)
•
“A First Course in Probability” – Sheldon Ross
•
Videos:
•
Harvard Stat 110 by Blitzstein (YouTube)
•
Khan Academy – if you need to brush the basics
•
Key Concepts:
•
Random walks, Bayes, Central Limit Theorem
•
Generate your own simulations with Python/R if you want to stand out
•
Textbooks:
•
“Div, Grad, Curl and All That” – H.M. Schey (must-read)
•
“Vector Calculus” – Marsden & Tromba (clear & comprehensive)
•
Videos:
•
MIT OCW – Multivariable Calculus (Prof. Denis Auroux)
•
3Blue1Brown – relevant visualizations
•
Projects:
•
Try applying divergence/Stokes’ theorem to real physical fields (e.g. electromagnetism)
•
Textbooks:
•
“Understanding Analysis” – Stephen Abbott (intuitive but rigorous)
•
“Principles of Mathematical Analysis” – Walter Rudin (this is hard, but legendary)
•
Videos:
•
Dr. Tadashi Tokieda’s intuition-driven analysis lectures (where available)
•
Essence of Calculus – 3Blue1Brown (for visual reinforcement)
•
Train like a wrangler:
•
Go deep into ε-δ proofs until it becomes second nature.
•
Prove convergence theorems yourself without looking.
•
Textbooks:
•
“Classical Mechanics” – John Taylor (clear, physics-style derivations)
•
“Spacetime Physics” – Taylor and Wheeler (intro to special relativity)
•
Videos:
•
MIT OCW – Walter Lewin’s mechanics lectures
•
PBS Spacetime or MinutePhysics for visual relativity
•
Top tip: Do the derivations yourself (e.g. time dilation, Lorentz transforms)."
Thank you so much! Wish you well too!
I'm a 25fall Cam math offer-holder who had just achieved an unconditional offer, so i was wondering whether there is anything i can do before i enter uni, for instance, reading relevant books or self-studying some notes. I learnt that for the Michaelmas Term, there would be four courses: differential equations, groups, numbers and sets, and vectors and matrices. Hence, looking for advice on books or videos to prepare for these four courses!
Honestly I wouldn't try to prepare too much by going through notes in advance. Learning how to learn through lectures is part of the process of first year, and if you don't do it then, you will have to when everyone else has already learned how.
I would instead advise really perfecting the knowledge you have already, going through STEP questions and ensuring you have a really great foundation.
There is also an introductory sheet in the style of the example sheets we do (see https://www.damtp.cam.ac.uk/user/examples/A0.pdf ). I found this really enjoyable and useful to getting used to the course. Note that it is much more difficult than A-Level and even perhaps moreso than STEP as it is not meant to be done in exam conditions. Feel free to use notes or books, come back to it after a break and don't feel obliged to finish every question.
If you still have all lot of extra time, I would recommend doing learning / improving some coding. As well as a useful skill in general, it is useful for the Cambridge Maths course as there is a coding coursework option in second and third year. There is support for the programming language MATLAB but you may do it in any. I do mine in Python as I already knew Python fairly well and there is a lot of online support for it. I would recommend trying various Project Euler projects. It's a free website with a bunch of Maths challenges designed to be solved with coding.
If you really want to do the lectures in advance, which, again, I would strongly advise against, some notes are here: https://dec41.user.srcf.net/notes/
Reply 6
I would instead advise really perfecting the knowledge you have already, going through STEP questions and ensuring you have a really great foundation.
There is also an introductory sheet in the style of the example sheets we do (see https://www.damtp.cam.ac.uk/user/examples/A0.pdf ). I found this really enjoyable and useful to getting used to the course. Note that it is much more difficult than A-Level and even perhaps moreso than STEP as it is not meant to be done in exam conditions. Feel free to use notes or books, come back to it after a break and don't feel obliged to finish every question.
If you still have all lot of extra time, I would recommend doing learning / improving some coding. As well as a useful skill in general, it is useful for the Cambridge Maths course as there is a coding coursework option in second and third year. There is support for the programming language MATLAB but you may do it in any. I do mine in Python as I already knew Python fairly well and there is a lot of online support for it. I would recommend trying various Project Euler projects. It's a free website with a bunch of Maths challenges designed to be solved with coding.
If you really want to do the lectures in advance, which, again, I would strongly advise against, some notes are here: https://dec41.user.srcf.net/notes/
Thank you so much! I would definitely try to do the introductory sheet! But where can I find the answer to check whether I'm correct or not?
I'm not sure if the answers are publicly available but many of the questions are self-checking. You could try asking the DoS at your college if they would be willing to send over the answers but honestly, just trying the questions is a good use of your time. Remember it's not designed to be done under exam conditions so you can always look things up to help you.
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