Should I Become a Mathematician?

[mathwonk]

i didn't know where he got his BA, I meant he was on the Brandeis faculty before he went to the faculty at Harvard. As I recall, he went from the faculty at Brandeis to that at Columbia and then to Harvard. I was trying to illustrate the fact that many less famous schools have outstanding faculties. Hironaka also lived in Kyoto when the ICM was held there in about 1992, and I believe he was also on the faculty there at RIMS.

 

[AcidRainLiTE]

Hi Everyone,

I want to apply to graduate school this year for a PhD (or possibly masters, but with the eventual goal of a PhD) in math. As an undergrad, I did a double major in math and physics with physics being my primary major.

My application is strong, with the exception of my Math Subject GRE scores. I did really poorly on them (43rd percentile or 620 out of 900), mainly because I didn't study correctly. Other than that, I have straight A's or A- in all of my math and physics courses and have good scores on the general GRE (94 percentile in verbal, 91 percentile in quantitative). I also am sure my professors will give me strong recommendations.

My questions is, will my bad Subject GRE scores destroy my chances at getting into grad school this year? I know I could take them again, study properly, and do much better, but that would mean I would have to wait to apply until next year. I would really like to go this year. I graduated in 2010 with my BS, have been working since then, but am ready to go back to school and would like to do so without working another year in the corporate world. What do you think? What are my chances of getting in?

 

[homeomorphic]

Hi Everyone,

I want to apply to graduate school this year for a PhD (or possibly masters, but with the eventual goal of a PhD) in math. As an undergrad, I did a double major in math and physics with physics being my primary major.

My application is strong, with the exception of my Math Subject GRE scores. I did really poorly on them (43rd percentile or 620 out of 900), mainly because I didn't study correctly. Other than that, I have straight A's or A- in all of my math and physics courses and have good scores on the general GRE (94 percentile in verbal, 91 percentile in quantitative). I also am sure my professors will give me strong recommendations.

My questions is, will my bad Subject GRE scores destroy my chances at getting into grad school this year? I know I could take them again, study properly, and do much better, but that would mean I would have to wait to apply until next year. I would really like to go this year. I graduated in 2010 with my BS, have been working since then, but am ready to go back to school and would like to do so without working another year in the corporate world. What do you think? What are my chances of getting in?

Get in where? I don't think it's that important that you have to get into the top schools. You might not get into Princeton or Harvard, but that's not the end of the world. Assuming you aren't broke, you should just apply and see what happens. You won't lose anything except a modest amount of time and money. You can always try again later. If you don't insist on going to one of the very top places, I don't think you'll have any trouble. Just apply to a whole bunch of places. Like 8 places, let's say.

 

[mathwonk]

the grades (and letters) are more impressive if you went to a strong undergraduate school. is that the case?

 

[bpatrick]

You said you have solid grades and also focused a lot in physics, so assuming you went to a good undergraduate institution, you'll probably be fine almost anywhere other than the top math programs in the US. I'd imagine that an admissions committee would see your 43rd % on the GREs along with your solid undergrad physics and math course grades and reason that you did fine on the calculus / differential equations / linear algebra end of it but didn't know much when it came to the algebra/topology/geometry/discrete areas that were tested by the math GRE.

So if that all is the case, I'd imagine you have a good chance at getting into many programs if you're personal statement reflects eventually getting into applied math / mathematical physics / PDE's / etc...

If you're trying to go for ivy league programs or any that focus mainly in pure math, I'd imagine they wouldn't take the risk of investing in you with those test scores, especially since many of the top programs are more "pure math" oriented and wouldn't want to bring in a student who possibly needed remedial work or didn't stand a great chance at passing their qualifiers after the first year.

 

[homeomorphic]

You said you have solid grades and also focused a lot in physics, so assuming you went to a good undergraduate institution, you'll probably be fine almost anywhere other than the top math programs in the US. I'd imagine that an admissions committee would see your 43rd % on the GREs along with your solid undergrad physics and math course grades and reason that you did fine on the calculus / differential equations / linear algebra end of it but didn't know much when it came to the algebra/topology/geometry/discrete areas that were tested by the math GRE.

So if that all is the case, I'd imagine you have a good chance at getting into many programs if you're personal statement reflects eventually getting into applied math / mathematical physics / PDE's / etc...

If you're trying to go for ivy league programs or any that focus mainly in pure math, I'd imagine they wouldn't take the risk of investing in you with those test scores, especially since many of the top programs are more "pure math" oriented and wouldn't want to bring in a student who possibly needed remedial work or didn't stand a great chance at passing their qualifiers after the first year.

Not really. The math GRE barely tests that stuff. It's 50% calculus. If he's like me, he just made a lot of calculation errors or wasn't fast enough. And that's probably what the admissions committees will think. It's a silly test. It's basically all about having lightning calculation reflexes (since it is long enough that one must be some kind of demon in order to get through the whole thing, in terms of pure manual dexterity in writing (sarcasm)) and not being prone to trivial oversights, which, conveniently, are exactly my weaknesses (I think I got 52nd percentile). Not much to do with pure math. That's only a small portion of it. It barely has any topology in it and no geometry. Tiny bit of algebra. Of course, it was 6 years ago when I took it, but I doubt it's much different.

But yes, they will see it as suspect because they are looking for an overall strong application. Sort of like one more consistency check because it's an outside source that isn't coming from the particular institution.

By the way, I know a grad student at a, let's say top 20 school, who did pretty badly on the math GRE. Maybe even in the 30s, definitely no higher than 40s. I don't remember. She didn't get many offers, but just one is enough.

 

[tyler_T]

Not really. The math GRE barely tests that stuff. It's 50% calculus. If he's like me, he just made a lot of calculation errors or wasn't fast enough. And that's probably what the admissions committees will think. It's a silly test. It's basically all about having lightning calculation reflexes (since it is long enough that one must be some kind of demon in order to get through the whole thing, in terms of pure manual dexterity in writing (sarcasm)) and not being prone to trivial oversights, which, conveniently, are exactly my weaknesses (I think I got 52nd percentile). Not much to do with pure math. That's only a small portion of it. It barely has any topology in it and no geometry. Tiny bit of algebra. Of course, it was 6 years ago when I took it, but I doubt it's much different.

But yes, they will see it as suspect because they are looking for an overall strong application. Sort of like one more consistency check because it's an outside source that isn't coming from the particular institution.

By the way, I know a grad student at a, let's say top 20 school, who did pretty badly on the math GRE. Maybe even in the 30s, definitely no higher than 40s. I don't remember. She didn't get many offers, but just one is enough.

Is it even worth reporting a score around 50th percentile to schools that recommend, but don't require, the subject GRE, like applied math programs?

 

[eliya]

Is it even worth reporting a score around 50th percentile to schools that recommend, but don't require, the subject GRE, like applied math programs?

I think you can email the program and ask if they look at the subject GRE scores or not. If they don't then there's no need to report them.
I can't see how reporting a low score will help you in any way. On the other hand, if you don't report it, they can immediately assume that you just did horribly on the exam. However, I think I wouldn't report a low score to a program that doesn't require it.
I should say, I haven't applied to grad school yet, so take it with a grain of salt.

To the "original" poster, there are a few good schools that don't require the subject GRE scores. I think Stony Brook is one of them.

 

[homeomorphic]

Is it even worth reporting a score around 50th percentile to schools that recommend, but don't require, the subject GRE, like applied math programs?

Most places require it.

Actually, someone I e-mailed when applying to grad schools said something to the effect that he considered 50th percentile to be sort of a minimum. Around 50th percentile. But that's just one opinion. People on admissions committees all have their own philsophies and that's why you have to apply to many places.

 

[tyler_T]

Most places require it.

Actually, someone I e-mailed when applying to grad schools said something to the effect that he considered 50th percentile to be sort of a minimum. Around 50th percentile. But that's just one opinion. People on admissions committees all have their own philsophies and that's why you have to apply to many places.

How strong was the program at that schools? (Top 20, top 50, etc)

 

[AcidRainLiTE]

the grades (and letters) are more impressive if you went to a strong undergraduate school. is that the case?

I went to a small private school, Grove City College (http://colleges.usnews.rankingsandreviews.com/best-colleges/grove-city-college-3269). It is pretty strong academically, but I am doubtful that it is very well known.

I am looking to apply to programs in pure math, not applied math. From some of the replies, it seems that may be a bit harder with the low GRE score.

To the "original" poster, there are a few good schools that don't require the subject GRE scores. I think Stony Brook is one of them.

Stony Brook is on my list. The only thing that bothers me a bit about the program is that it has very little in mathematical logic/set theory/foundations of math. Some of the other programs I have looked at (Maryland, Penn State, Illinoise, UCLA, Michigan, Notre Dame) seem to have a bit more in that area. And that is one of the areas I am potentially interested in. Either that or Topology or Geometry. The good thing about stony brook, though, is they would be a good place for me to pursue my dual interest in math&theoretical physics (if I do want to continue in physics), since they seem to have a good theoretical physics department.

 

[homeomorphic]

How strong was the program at that schools? (Top 20, top 50, etc)

Maybe 20-40 range. Actually, I think even he would probably say it's just a general rule that could have exceptions if the rest of the application was particularly strong.

I am looking to apply to programs in pure math, not applied math. From some of the replies, it seems that may be a bit harder with the low GRE score.

I don't think it makes any difference whether it's pure or applied. Typically, you get to choose AFTER you get accepted, whether you want to do pure or applied. They don't even ask, except you might mention it in your application.

Stony Brook is on my list. The only thing that bothers me a bit about the program is that it has very little in mathematical logic/set theory/foundations of math. Some of the other programs I have looked at (Maryland, Penn State, Illinoise, UCLA, Michigan, Notre Dame) seem to have a bit more in that area. And that is one of the areas I am potentially interested in. Either that or Topology or Geometry.

Most of those sound at least somewhat realistic. If you don't mind spending the money, apply to Michigan and UCLA, but be aware that places like that turn down fantastic people. I've known people with seemingly unbeatable applications that apply mostly in the top 15 and they typically only get into 2 or 3 out of 8. No one should count on getting into a place like Harvard unless they already have 2-3 years of graduate level classes under their belt, more or less straight As in their subject, preferably already published, etc, plus fit the department. I think sometimes they just decide to take a chance with someone who doesn't have that kind of record, though. Ed Witten was a history major and somehow got into Princeton. I'm not sure exactly how that happened, except, evidently, his dad was a physicist, so he wasn't just starting from scratch.

 

[mathwonk]

you should choose your school based on whether they have your interests, keeping in mind that your interests can easily change once you join a strong department.

to get a look at your school compared to others, you might take a look at their departmental website and check out where your professors got their own phd's. you will note that they themselves went to good but not super famous schools. if you ask one of your own teachers who got a phd at say notre dame, she can tell you whether you would find that school a reasonable fit.

admissions committees do not look just at one factor, but try to discover from all aspects of it, which candidates have the most potential. i am virtually sure you will gain admission to a school where you can fit in well. those gre scores are not as low as you seem to think.

i do not recommend applying to harvard, unless your letters are incredibly strong, but i do suggest applying to schools comparable to my institution, university of georgia.

i must observe however that set theory is not widely considered a serious research topic, and may not be represented at many places. geometry on the other hand is an extremely serious subject in pure math. ( of course i am a geometer)having looked at your list, i suggest speaking with some professors at those places to see what they think. in this context however, i recall my own story again - i was interviewed and rejected by eilenberg for columbia in 1965. then in 1980 when i was at harvard i met eilenberg again and he apologized for not recognizing my ability. so the story is never over until it's over.

 

[eliya]

Stony Brook is on my list. The only thing that bothers me a bit about the program is that it has very little in mathematical logic/set theory/foundations of math. Some of the other programs I have looked at (Maryland, Penn State, Illinoise, UCLA, Michigan, Notre Dame) seem to have a bit more in that area.

UIC have a few logicians, and they seem to hold a few logic and set theory seminars. I don't know how that compares to Stony Brook or the other schools on your list that in terms of logic and set theory.

 

[gniedom]

Calc midterm tomorrow. Anyone want to help with this suggested problem?
Limits of the form limxgoesto inf xe^-x.
(i) Evaluate limxgoesto inf xe^-x.
(ii) Fix a positive intger n and assume that limxgoesto inf xe^-x.= 0. Show that
it must be the case that limgoesto inf x^n+1e^-x = 0 as well.
(iii) From (i) and (ii), what can you say about the value of limxgoesto inf xe^-x.
for any positive integer s? Explain your reasoning.

 

[mathwonk]

try one of the math or homework threads. (hint: l'hopital)

 

[DrummingAtom]

So this semester is winding down to an end and a couple months ago I was intrigued by Differential Equations but my whole interest has shifted to Linear Algebra. There is a slight disclaimer though when I say I'm interested in Linear Algebra... The class I'm currently taking is a combined DE/LA class and actually it kinda stinks. We learned a whole bunch of LA concepts in about 3 weeks then used them to solve DE's. I went out and self studied some things on my own and the geometric properties/concepts in LA are fascinating.

I'm probably going to spend all winter break studying more LA but I don't really know where to pick up. In class I have learned vector spaces, eigenstuff, matrix operations, linear independence, and a little about basis. This class has been a crash course and although I am picking up somethings I want to know the theory of LA much more because I can feel there is something very deep going on. What are usually the first topics studied in LA from a pure math perspective?

Also, I was pretty disappointed with the presentation of some of the topics in LA because it seemed like they were making things much more complicated than need be. The book I have for class made it seem like the determinant was just pulled out of thin air and never once mentioned it's geometry! After browsing through a bunch of books in my school's library I noticed almost no LA books talk about the geometry of determinants. Is anyone else disappointed by this?

 

[matqkks]

Yes it has been frustrating to see lack of geometric interpretation of LA concepts. Have a look at Gilbert Strang website at MIT to liven this subject.
I have also set up topics on this issue through this forum.
Determinants of 2 by2 gives the area scale factor, 3by 3 gives the volume scale factor.
Negative determinant changes the orientation of the area, volume.
Determinant of 1 preserves the lengths, angles etc.
I have some notes on this topic, let me know via email if you would like a copy.

 

[Nano-Passion]

Yes it has been frustrating to see lack of geometric interpretation of LA concepts. Have a look at Gilbert Strang website at MIT to liven this subject.
I have also set up topics on this issue through this forum.
Determinants of 2 by2 gives the area scale factor, 3by 3 gives the volume scale factor.
Negative determinant changes the orientation of the area, volume.
Determinant of 1 preserves the lengths, angles etc.

At some point I kind of accepted that linear algebra is devoid of geometric interpretations. If others have links that supply the geometric interpretations then that would be appreciated.

I have some notes on this topic, let me know via email if you would like a copy.

Hmm, not to be nosy or anything but I would also like an email of this. That is if you don't have to go out of your way typing it out and stuff.

 

[homeomorphic]

At some point I kind of accepted that linear algebra is devoid of geometric interpretations. If others have links that supply the geometric interpretations then that would be appreciated.

I think it's a pretty geometric subject. However, when you pass to complex vector spaces or vectors spaces with other fields, it gets more abstract and it's not exactly so geometric, but you can pretty much think of it the same way by analogy. The bulk of what I understand about linear algebra was learned after I took the class in the course of learning other things.

I didn't really learn the subject from books, but you might try Linear Algebra Through Geometry. Never read it. I just like the sound of it. For more advanced stuff, I think Linear Algebra Done Right is good.

What are usually the first topics studied in LA from a pure math perspective?

Vector spaces, bases, linear independence, linear transformations, matrices that represent linear transformations. That's the core of it.

Also, I was pretty disappointed with the presentation of some of the topics in LA because it seemed like they were making things much more complicated than need be. The book I have for class made it seem like the determinant was just pulled out of thin air and never once mentioned it's geometry! After browsing through a bunch of books in my school's library I noticed almost no LA books talk about the geometry of determinants. Is anyone else disappointed by this?

Yes. I find it extremely annoying. Why do they insist on being so rigid, formal, and boring? It's a shame.

 

[DrummingAtom]

Yes. I find it extremely annoying. Why do they insist on being so rigid, formal, and boring? It's a shame.

Yeah, I don't know anything about higher level math because I'm only up to Diffy Q's right now. But it seems the biggest mystery to me is why most math professors teach abstractly first then proceed to go back and do examples and conceptual things. The lectures and most of the books I've seen follow this method and I can't understand why. To me, definition, theorem, proof, then concepts is backwards in every aspect; heck sometimes the concepts don't even follow it just ends with the proof.

I really like math and want to know more about it but honestly my "discovering" math is when I sift through the abstract stuff and find out the concepts really aren't as hard as they are made it out to be.

By the way, homeomorphic, I took your advice and checked out V.I. Arnold and he seems exactly like my type of math guy. Visual and intuitive. His article on teaching math was inspiring for someone like me. Once I get Linear Algebra out of my system I'm going to buy his ODE book. Thanks for the recommendation.

 

[homeomorphic]

Yeah, I don't know anything about higher level math because I'm only up to Diffy Q's right now. But it seems the biggest mystery to me is why most math professors teach abstractly first then proceed to go back and do examples and conceptual things. The lectures and most of the books I've seen follow this method and I can't understand why. To me, definition, theorem, proof, then concepts is backwards in every aspect; heck sometimes the concepts don't even follow it just ends with the proof.

There are books that are like definition, theorem, proof that are pretty good. It depends on how it's done.

I really like math and want to know more about it but honestly my "discovering" math is when I sift through the abstract stuff and find out the concepts really aren't as hard as they are made it out to be.

Yeah, pretty much. I'm usually aware from the start that a textbook/prof is not giving nearly as much intuition as they should, but it can be even more striking when you actually understand the subject and see how badly they butchered it. Math is difficult. The fact that it is difficult is part of what makes it so outrageous for them to complicate matters and make it 20 times more difficult than it ought to be.

By the way, homeomorphic, I took your advice and checked out V.I. Arnold and he seems exactly like my type of math guy. Visual and intuitive. His article on teaching math was inspiring for someone like me. Once I get Linear Algebra out of my system I'm going to buy his ODE book. Thanks for the recommendation.

The ODE book is good, but it's a little difficult. I think you'll definitely see his visual, intuitive thinking show through, but it's not an easy book. I think probably my favorite book of Arnold might be the one on classical mechanics, although I don't consider to be perfect. The best place to start would be Visual Complex Analysis. Pretty elementary. Very visual, as the title suggests. Very entertaining.

 

[sponsoredwalk]

You guys should never look at Bourbaki's or Hoffman/Kunze's definition of the determinant so

 

[Deveno]

one can look at mathematics as a type of game people play. as with any game, there are differing strategies:

a) one can play by "gut feeling"
b) one can make an exhaustive analysis of the rules
c) one can devise a "toolkit" which covers most common situations
d) one can adapt strategies from some other game, and hope they work
e) something else entirely

there are certain attractive, and unattractive features of every strategy, including: how much information has to be internalized, how efficient the application of the strategy is, how successful it is.

the "theoretical" approach aligns most closely with (b). this requires a long "learning curve" and a good deal of retained information. it is highly successful and efficient in application. most people in point of practice go with (c), which represents a compromise between (b) and (d). it should be noted that people who stick with (d) usually resort to (a) if their approach doesn't work. (d) doesn't require a great deal of retained information, because adaptation is certainly easier than assimilation.

(a) is arguably the worst explicit strategy listed, because it relies extensively on internal inductive reasoning (unconscious pattern recognition). some people use it reasonably well, arguably because they are better at recognizing relevant information without first "translating" it into some other area.

in any case, there's an inherent tension between abstract/instance. how deeply does one examine the particular example of an interesting case with "nice" properties (like, for example, the real numbers, instead of an arbitrary field), versus examining the shared characteristics of a wide variety of disparate examples (like abstract linear operators in a hilbert space, rather than complex matrices)?

in one sense, topics like category theory, and differential equations lie at "opposite ends" of this spectrum. one studies structures so general they seem removed from anything remotely "real" at all, while the other studies things so rooted in reality, and particular in nature, that the methods are tailored to the distinct case in hand (this type of function, with these types of numbers, subjected to this constraint, under these sets of assumptions). what it means to get a "result" and what is meaningful, is very different for these two areas.

i argue that a well-rounded individual needs both: an orderly set of cupboards to organize the ideas (abstract), and plenty of food in them (particular and interesting examples). for example: the abstract properties of a determinant aren't needed if you never actually calculate any determinants, and for a particular determinant calculation, knowing the abstract properties can make the calculation easier (short-cuts). the abstract gives mathematics shape, and the concrete gives mathematics flavor and texture.

 

[homeomorphic]

You guys should never look at Bourbaki's or Hoffman/Kunze's definition of the determinant so

Not sure what you're getting at, but if it's the exterior-algebra approach, I don't see it as being fundamentally different from the visual definition as the signed volume of the parallelepiped spanned by the column vectors. It's basically the same thing, except it's more general. But, then, I don't know their definition. I don't think there is a definition of determinant that I can't interpret geometrically or at least intuitively in one way or another.

 

[homeomorphic]

i argue that a well-rounded individual needs both: an orderly set of cupboards to organize the ideas (abstract), and plenty of food in them (particular and interesting examples).

I agree, but your cupboard analogy makes it seem like the cupboards should come before the food, which is backwards. Design the cupboards with the food in mind. The cupboards are not an end in themselves.

It's not so much abstraction that I object to. It's unmotivated definitions (i.e. not laying the groundwork for the abstractions), and gruesome calculations that obscure the concepts. I'm not against all gruesome calculations, but I see it as extremely unfortunate if they should impinge on the theory. Ideally, they ought to be more like the end result of a theory, not part of the theory itself. For example, you can understand curvature very well theoretically, and then compute some curvature in some example and it will be horrific, and you won't understand it conceptually. But you just want the answer, so it's fine. Also, maybe you can compute the curvature of a sphere because it's good practice, even though it can be understood conceptually. So, as practice or as a way to get answers, I have no problem with calculation. But it irks me when it REPLACES conceptual understanding. That is the real shame.

 

[sponsoredwalk]

Not sure what you're getting at, but if it's the exterior-algebra approach, I don't see it as being fundamentally different from the visual definition as the signed volume of the parallelepiped spanned by the column vectors. It's basically the same thing, except it's more general. But, then, I don't know their definition. I don't think there is a definition of determinant that I can't interpret geometrically or at least intuitively in one way or another.

All I meant by that comment was that these books give the most "rigid, formal" definitions
of the determinant that I've seen (well, except for one in a book called Linear Algebra &
Group Representations) with such a lack of motivation that it caused me to stop reading
both despite the fact I was pulling my hair out trying to understand the concept at that
abstract a level

 

[mathwonk]

here are free class notes on linear algebra (intended as a second course), with determinants discussed starting on page 62.

https://www.math.uga.edu/sites/default/files/inline-files/4050sum08.pdf

 

[mathwonk]

happy thanksgiving! (sorry, off topic)

 

[kramer733]

What kind of textbook do you guys recommend with a course description of the following
:

http://www4.carleton.ca/calendars//ugrad/current/courses/MATH/2000.html

Calculus and Introductory Analysis II (Honours)
Higher dimensional calculus, chain rule, gradient, line and multiple integrals with applications. Use of implicit and inverse function theorems. Real number axioms, limits, continuous functions, differentiability, infinite series, uniform convergence, the Riemann integral.

Would i still use spivak calculus?

I'm using spivak now for the following course:

http://www4.carleton.ca/calendars//ugrad/current/courses/MATH/1002.html

Elementary functions. Limits. Continuity. Differentiation. L'Hôpital's rules. Indefinite and definite integrals. Improper integrals. Sequences and series, Taylor's formulae. Introduction to differential equations.

I'm supposed to use stewart's calculus book but I've heard bad things about it and instead bought spivak for $50.00. Compared to what they were selling stewart's book for, spivak was 3-4x cheaper.

 

[deluks917]

Spivak "Calculus" doesn't have any multi-variable stuff in it. Spivaks calculus on Manifolds is quite good and covers Higher Dim Stuff. If you haven't already taken it I really suggest you learn some linear algebra before taking that multi-variable class.

 

[bpatrick]

Granted, it has been a while, but I used the 5th edition of Stewart's single variable and multivariable calculus books when I learned the majority of my calculus, and for what it's worth, I thought they were wonderful books ... not sure why they'd have a reputation of being bad.

 

[Nano-Passion]

Granted, it has been a while, but I used the 5th edition of Stewart's single variable and multivariable calculus books when I learned the majority of my calculus, and for what it's worth, I thought they were wonderful books ... not sure why they'd have a reputation of being bad.

Who said they were bad? I saw them recommended a few times. I use stewart's 4th edition as a supplement with Larson's Calculus.

 

[homeomorphic]

Granted, it has been a while, but I used the 5th edition of Stewart's single variable and multivariable calculus books when I learned the majority of my calculus, and for what it's worth, I thought they were wonderful books ... not sure why they'd have a reputation of being bad.

They are okay, but they could be better. I can't recall specific examples, but there are many cases in which there are better explanations of things out there. They aren't terrible. You can learn calculus from it and you'll get the main concepts, but you won't have the deepest possible understanding of everything. Which, maybe you can't expect the first time you learn it, anyway, but it could be improved upon.

The last chapter about Stokes theorem and that stuff has some particularly bad sections in it.

 

[Sina]

What kind of textbook do you guys recommend with a course description of the following
:

http://www4.carleton.ca/calendars//ugrad/current/courses/MATH/2000.html

Calculus and Introductory Analysis II (Honours)
Higher dimensional calculus, chain rule, gradient, line and multiple integrals with applications. Use of implicit and inverse function theorems. Real number axioms, limits, continuous functions, differentiability, infinite series, uniform convergence, the Riemann integral.

Would i still use spivak calculus?

I'm using spivak now for the following course:

http://www4.carleton.ca/calendars//ugrad/current/courses/MATH/1002.html

Elementary functions. Limits. Continuity. Differentiation. L'Hôpital's rules. Indefinite and definite integrals. Improper integrals. Sequences and series, Taylor's formulae. Introduction to differential equations.

I'm supposed to use stewart's calculus book but I've heard bad things about it and instead bought spivak for $50.00. Compared to what they were selling stewart's book for, spivak was 3-4x cheaper.

Marsden, vector analysis. I think it has Newton on its cover.