Learning Real Analysis: A High Schooler's Guide

[dissolver]

I know that AP Calculus BC is very watered down in comparison to a real analysis course at the college level. I'm taking high school Calc I, II this upcoming school year as a junior and would like to get a good book to learn real analysis from. I'm currently looking at getting Spivak's CALCULUS, but I have also heard good things about Courant and Apostle. I would like to hear your opinions, suggestions, etc.

 

[courtrigrad]

I think Courant and Apostol are very good. Apostol is very scholarly, and doesn't take shortcuts. Courant's Differential and Integral calculus provides a balance between theory and application. Spivak's book is probably the best for a one-variable calculus course. However, in my opinion I like Courant's book the best because of his style of writing. Its very conversational in tone, as is Spivak's. Apostol is more of a standard textbook.

 

[mathwonk]

a much more advanced book which is excellent and will last you a long time, is dieudonne, foundations of modern analysis, which assumes you already know classical advanced calculus.

it does not actually assume any knowledge, but does assume a lot of sophistication.

 

[leright]

Nobody seems to mentions Stewart's calculus book. IS there something wrong with it? That is what I used in calc 1-3. :/

 

[mathwonk]

stewart is a good non honors calc book, but he seems to be asking for a high level proof oriented honors calc book. i.e. he asked for an "analysis" book, not a calculus book.

the dieudonne book ir ecommended for instahnce starts with rigorous treatment of real numbers then metric spoaces, banach spaces, hilbert spaces, and caculus in infinite dimensions.

 

[leright]

Well, from what I have read, stewart focuses on intuition and less on rigor of proofs, and for engineering and science majors this is probably more important. This is probably why most engineering and science calculus classes use Stewart and not Courant of Spivak. I also think one who has a very strong intuition about mathematical ideas understands the subject more than one who is caught up in the "rigor". For instance, I clearly 'understand' the fundamental theorem of calculus (not just how to apply it), but I would have trouble writing a formal proof of it. Does that mean I understand it less than others who CAN write a formal proof...I personally don't think so. My explanation tends to be a more "hand waving" approach, but still logic and solid.

Almost all of the people at my uni that take calculus are engineers or science majors. The architecture students and non science majors take a watered down less rigorous calc sequence called math analysis.

 

[end3r7]

I'm using Stewart this fall for Calc III. In fact, Stewart teaches at my university (Texas A&M). I'm a science major

 

[fliptomato]

Calculus books are a commodity.

I suggest you go to a college/technical bookstore and you go look through books until you find one that you like. Just because one person found that a certain book resonates with them it doesn't mean that you'll like it as well.

It sounds like you want a book with more formalism, so you might consider looking into the Springer-Verlag textbooks (the ones with the yellow covers), which tend to be very well written.

Also, I tend to get confused about the subjects "real analysis" and "multivariable calculus" -- multivariable calculus refers primarily to what one would take after single variable calculus and deals with things like Stokes' theorem and integrals in 3-space. Real analysis overlaps with this, but can also be more formal dealing with definitions of integrals and such. Formal analysis is a slightly more specialized subject that isn't very prominent in physics (I do theoretical physics and don't know what a "Lebesgue integral" is... I suspect it's not important for anything I'll run into in my research).

After a good course in multivariable calculus and linear algebra, you have some flexibility about what kind of math you want to study, so don't feel like analysis is the 'next step'. (Differential geometry is also particularly interesting and relevant to physics.)

-F

 

[jbusc]

Stewart is a nice book, but he glosses over some important concepts (like the Jacobian)

 

[leright]

I'm using Stewart this fall for Calc III. In fact, Stewart teaches at my university (Texas A&M). I'm a science major

I thought he taught at McMaster.

 

[leright]

Calculus books are a commodity.

I suggest you go to a college/technical bookstore and you go look through books until you find one that you like. Just because one person found that a certain book resonates with them it doesn't mean that you'll like it as well.

It sounds like you want a book with more formalism, so you might consider looking into the Springer-Verlag textbooks (the ones with the yellow covers), which tend to be very well written.

Also, I tend to get confused about the subjects "real analysis" and "multivariable calculus" -- multivariable calculus refers primarily to what one would take after single variable calculus and deals with things like Stokes' theorem and integrals in 3-space. Real analysis overlaps with this, but can also be more formal dealing with definitions of integrals and such. Formal analysis is a slightly more specialized subject that isn't very prominent in physics (I do theoretical physics and don't know what a "Lebesgue integral" is... I suspect it's not important for anything I'll run into in my research).

After a good course in multivariable calculus and linear algebra, you have some flexibility about what kind of math you want to study, so don't feel like analysis is the 'next step'. (Differential geometry is also particularly interesting and relevant to physics.)

-F

Well, the terms analysis and caclulus are synonymous, but people tend to call more rigorous proof-based treatments of calculus "analysis".

 

[end3r7]

I thought he taught at McMaster.

You may be right. I might have just read too much inot the description here https://www.amazon.com/gp/product/0534493483/?tag=pfamazon01-20

 

[SpiffyKavu]

Has anyone used: Calculus by Ellis & Gulick, 6th Edition? I'll be using it in a couple of weeks, and I'd like to know if it is well writen.

 

[trinitron]

"My explanation tends to be a more "hand waving" approach, but still logic and solid."

Hand waving isn't logic. A proof is.

 

[leright]

"My explanation tends to be a more "hand waving" approach, but still logic and solid."

Hand waving isn't logic. A proof is.

ok, it may not be logical in the formal sense of the word, but I certainly have a strong intuitive grasp of it.

 

[trinitron]

I'm not saying intuition is not important,but there are things that intuition would lead you to believe are true that actually aren't. There's a book 'Counterexamples in Analysis' that's full of such things.

 

[MathematicalPhysicist]

btw, if you already mention courant's books i prefer his book which he wrote with john fritz.
it has even applications to physics which as far as i know the first book lacks of, and also it's written as a historical overview with the rigorous proofs.
i have also the two books by apostol's, and both books (courant's and apostol) start with the integral instead of the derivative, opposite of the way it is done in classes in universities, when you first encounter derivatives. (which could be akward for those who read the book roughly the same time when the topic is being covered in class).