Rigorous Precalculus and Calculus Textbooks + Intro to Linear Algebra

[PFuser1232]

I am currently studying A level Further Pure Mathematics, and I will be joining university next year. I want to brush up on precalculus (algebra, trigonometry, geometry) and calculus (differential, integral [single variable]), except that this time I need a more rigorous treatment of those topics. Most of what I covered in high school was very computational, with barely any focus on mathematical rigor. I am also looking for a good introduction to linear algebra textbook.


What textbooks would you recommend?

 

[MidgetDwarf]

Spivak or Apostol calculus? Serge Lang basic mathematics for pre cal.

 

[MidgetDwarf]

For linear algebra. There is a book mathwonk recommends (Math professor at a university who frequents this forum). The book is called Elementary Linear Algebra by Paul Shields. It is not rigorous, but it is by no means a trial book. USE this book first then Serge Lang Linear Algebra?

 

[PFuser1232]

Spivak or Apostol calculus? Serge Lang basic mathematics for pre cal.

Is Spivak Calculus better than Simmons/Stewart Calculus?

 

[MidgetDwarf]

Yes but it is more of the theory behind calculus book. In other words spivak is introductory analysis light. Spivak is a hard book but a good one. If you have the mathematical maturity go for it. Would advice strongly against spivak if you are not there yet. An older version of thomas with analytic geometry would be best. Are you still in highchool?

 

[PFuser1232]

Yes but it is more of the theory behind calculus book. In other words spivak is introductory analysis light. Spivak is a hard book but a good one. If you have the mathematical maturity go for it. Would advice strongly against spivak if you are not there yet. An older version of thomas with analytic geometry would be best. Are you still in highchool?

Yes, still in high school. I finished most of my subjects early though, so I know some basic differentiation and integration.

 

[MidgetDwarf]

Then spivak would be too hard if you don't have anyone to help you with. I would advice f getting an older edition of stewart with an older edition of thomas calculus with analytic geometry. I know that the 3rd ed is great but a little terse. I I've seen the 7th edition with lighthouse cover. I thought it was easier to read but it reminded me to much of stewart. Both books should be les than 20 dollars.

Both books compliment each other well. Avoid watching youtube videos as soon as you get. Getting stuck is the best process of the learning experience.

 

[PFuser1232]

Then spivak would be too hard if you don't have anyone to help you with. I would advice f getting an older edition of stewart with an older edition of thomas calculus with analytic geometry. I know that the 3rd ed is great but a little terse. I I've seen the 7th edition with lighthouse cover. I thought it was easier to read but it reminded me to much of stewart. Both books should be les than 20 dollars.

Both books compliment each other well. Avoid watching youtube videos as soon as you get. Getting stuck is the best process of the learning experience.

What about Simmons calculus?

 

[Stephen Tashi]

except that this time I need a more rigorous treatment of those topics.

Having read some of your posts on the forum, I'd say you have an high standard of rigor - a very high standard relative to the general population taking introductory courses. Textbooks that introduce those topics are not written in a rigorous fashion, so it may be impractical to find an introductory text that meets your standards.

(In the decade after Sputnik threw a scare in the USA education system, there were efforts to produce textbooks that were in harmony with respectable mathematics - versus being in harmony with convenient ways to teach students to work typical problems. An example of such a text is the 3rd edition of Calculus With Analytic Geometry by Johnson and Kiokemeister. The 4th edition takes an even more sophisticated approach but also has a slew of misprints. The last time I looked at Calculus books was in the 1980's and they were much less rigorous than J&K 3rd edition.)


Doing mathematics in a rigorous fashion can be laborious and slow, but mentally satisfying. You have to decide how to compromise the goal of "covering the material" with working in a way you enjoy. To cover material, you have to endure the fact that some aspects will not be presented rigorously. For example, the organization of mathematics education places the rigrorous approach to the properties of the real numbers in a graduate or advanced undergraduate course. Those who get to that course have been required to work with the real numbers in math courses for many years before they arrive.