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I am applying for a PhD in pure maths. Now a lot of places that I am applying to are asking for detailed lists of courses and books that I have followed for anything more advanced than calculus.

Now I am not quite sure what they mean here, because in our institute the course that was named calculus was an introduction to differential forms and theorems regarding those. And from there it went on to give us an introduction to manifolds. (This was our third semester undergrad; prior to this we had taken linear algebra, analysis 1, multivariable analysis, topology, and group theory.) I am from India.

Now if this is considered to be a basic calculus course then I am not quite sure how to judge exactly what courses are supposed to be more advanced than this.

Since some folks have asked regarding the syllabus:

Axioms of the real number system without construction, applications of the least-upper-bound- property, Archimedean principle, existence of nth roots of positive real numbers, ax for a > 0 and x > 0.

Convergence of sequences, monotonic sequences, subsequences, Heine-Borel theorem, lim sup and lim inf Cauchy sequences, completeness of R. Infinite series, absolute convergence, comparison test, root test, ratio test, conditional convergence, complex numbers, power series, radius of convergence of power series.

Continuous functions on intervals of R, intermediate value theorem, boundedness of continuous functions on closed and bounded intervals.

Differentiation, mean value theorem, Taylor's theorem, application of Taylor's theorem to maxima and minima, L'Hôpital rules to calculate limits.

Construction of ez using power series, proof of the periodicity of sin and cos.

Riemann Integration: Riemann integrals, Riemann integrablity of continuous functions, fundamental theorem of calculus.

This is the analysis 1 syllabus. I suppose this is going to be equivalent to calculus.

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    For U.S. universities a reference to "calculus" means an introductory 3 semester sequence (sometimes done in 2 semesters, sometimes done in 4 semesters) covering the topics listed here. See also the descriptions of Math 115 (1st semester), Math 116 (2nd semester), Math 215 (3rd semester) here. The topics you mentioned are well beyond this. – Dave L Renfro Nov 14 '20 at 8:44
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    @DaveLRenfro: so Integration and Derivation are not taught in high school? Or just not with rigourous proofs? – user111388 Nov 14 '20 at 8:53
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    not taught in high school --- It is often the case that students (at least those intending to major in a math, engineering, or physical sciences area) will have studied these in high school (usually it's just the 1st semester calculus material), but they still have to prove competence in some way (varies with the college/university) to obtain credit or to obtain permission to take the next higher level course (these are two different things, the former requiring a higher "standard of proof" of the student's competence). – Dave L Renfro Nov 14 '20 at 9:11
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    Incidentally, "rigorous proofs" are usually not done in a typical university calculus sequence. For example, epsilon-delta (or sequence-based) proofs are usually not done, although some instructors will include simple ones and textbooks usually include them somewhere (and if offered, a "honors level" sequence has them). On the other hand, the courses cover algebraic limit calculations of the derivative of simple polynomials and simple rational functions (e.g. x^(-2) and (2x - 1)/(3x + 2)) and probably sqrt(x) before the usual power rule, product rule, etc. short cuts are introduced. – Dave L Renfro Nov 14 '20 at 9:19
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    @jamesqf I think this depends somewhat on the degree of specialization in your university system. In some countries, such as in the US, education at the university level is relatively unspecialized and programs usually do not start with specialized topics such as proof-based math. In other countries (such as in the Netherlands or Germany), you decide to do math the moment you enter the university level, and it is common to have an introduction to proofs in the first seminar, with most courses following being proof-based. – Discrete lizard Nov 15 '20 at 13:14
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This probably means anything beyond the semi-standarized three introductory Calculus courses.

Examples include:

  • Differential Equations
  • Linear Algebra
  • Discrete Mathematics
  • Probability
  • Statistics
  • Ring Theory

Or basically topics that might consider Calculus as a prerequisite to performing well in the class.

Classes that build a mathematical foundation to take calculus won't apply, like:

  • Algebra
  • College Algebra
  • Pre-Algebra
  • Any math topic "for some other non-math major" (Statistics for Business majors)

To clarify I've attached the University of Houston's Math department class offerings. Note that Calculus I, II, and III are 1000 (Freshman) and 2000 (Sophmore) level courses. I would assume any 3000 or 4000 level course would satisfy the requirement, and possibly some of the 2000 level courses.

This should help clarify some of the comments about "Algebra" courses. Higher level Algebras that would be post-Calculus include:

  • Elements of Algebra and Number Theory
  • Abstract Algebra

I hope this provides a little more clarity.

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    ok so basically. this pretty much means every math courses that I have ever taken. Thanks.. – Pablo Bhowmik Nov 14 '20 at 18:53
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    @PabloBhowmik Glad to help. Sounds like you have your prerequisite covered. – Edwin Buck Nov 14 '20 at 20:59
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    There is the unfortunate aspect of mathematics that the "Algebra" that studies group theory would count as more advanced than calculus, while the "Algebra" that culminates in the quadratic formula would be less advanced than calculus. – Teepeemm Nov 14 '20 at 22:22
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    @Acccumulation "College Algebra" is a remedial course for those who cannot place into the introductory Calculus courses. It is sometimes termed "College Algebra" instead of just "Algebra" in undergraduate class listings. I'm not 100% sure why they chose this terminology; but, in the USA it may have to do with the variance of learning in High School Algebra offerings. It covers algebra including basic trigonometry, basic matrices, etc. – Edwin Buck Nov 15 '20 at 3:52
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    Incidentally, "remedial" is a technical term at many colleges/universities, and College Algebra rarely qualifies, as one can obtain college credit at most places. Remedial usually ends with Intermediate Algebra (typical content: properties of numbers; fundamental operations with algebraic expressions; polynomials; systems of equations; ratio and proportion; factoring; functions; graphs; solutions of linear inequalities; and linear and quadratic equations). Also, College Algebra was a lot more standard as a first math course before roughly the 1950s, and often calculus started in 2nd year. – Dave L Renfro Nov 15 '20 at 9:42
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I think your question is: What does calculus mean?

Calculus would usually include learning to compute derivatives and integrals.

If you are learning to prove the theorems used to compute derivatives and integrals, that would be more advanced than calculus.

linear algebra analysis 1, multivariable analysis, topology, group theory

All of those would usually be considered more advanced than calculus.

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    Just out of interest: could you say which courses are not more advanced than calculus? Are there any? – user111388 Nov 14 '20 at 10:29
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    Nothing is completely standardized in the US, but precalculus, "college algebra", and introductory statistics might be common examples. – Anonymous Physicist Nov 14 '20 at 10:52
  • Ok sry about dragging this further but what exactly is precalculus college algebra ? ( I tried googling it but the varied response from different places seems sorta overwhelming ) – Pablo Bhowmik Nov 14 '20 at 15:42
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    @Pablo: Precalculus generally covers the behavior of certain common functions (polynomials and trigonometric functions) for students as a preparation before their first exposure to calculus. College algebra is basic equation manipulation ("solve for x"). If you're majoring in mathematics you probably took them before college. – Nick Matteo Nov 14 '20 at 19:09
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As far as the US is concerned, "Calculus" is the first introduction to the material. It typically is light on proofs and often geared to the Engineering Curriculum. In Germany (where I grew up) this material was partially high school, partially (in College) classes called ``Higher Mathematics for Engineers''.

To get a more detailed idea, http://www.cds.caltech.edu/~marsden/volume/Calculus/ are (by now somewhat old, but the material has not changed) Calculus textbooks that would be at the upper (more ambitious) level -- many books are weaker.

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    In the USA, we had already covered most of the Calculus I (limits and derivatives, and very basic integration (fitting quadrilaterals under curves) topics in High School; however, I was very aware that others had not taken these courses as they were electives. The USA High School elective system permits various classes to be optionally taken, and "higher" math classes compete with music, drama, speech, art, auto mechanics, higher level science, foreign languages, computer classes, wood shop, school sponsored early work entry, and any other non-core class. – Edwin Buck Nov 15 '20 at 4:01
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In light of various clarifications, here's the bottom line for your current situation: You should definitely include this course on a list of courses you have taken "more advanced than calculus" for the purposes of US universities. And you should call it "Analysis" (or maybe "Real Analysis"), not "Calculus".

More generally, the other answers and Dave L Renfro's comments have explained well the difference in how those two labels are used in naming undergraduate university courses in the US. It's not always widely appreciated (either in US academia or elsewhere) that the distinction between those words in US course titles may differ from how they are used either to refer to fields of mathematics (independently of naming classes) or to name courses in other countries.

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    Thanks for your input but given in my transcript it's mentioned as calculus also given other than this there are 3 other courses which are called analysis (I ,II, III) I don't suppose I should be liberal about this abrupt name changes. Also given the calculus course is sorta computational ( computing tangent spaces , working with differential form etc ) I suppose there is valid reason why folks in our department chose to call it calculus. – Pablo Bhowmik Nov 14 '20 at 15:41
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    Some schools literally call their real analysis undergrad course "advanced calculus". It's essentially calc one but with rigorous proofs for everything. – eps Nov 15 '20 at 17:41
  • @eps That's a good point. Although some other schools use "advanced calculus" to mean a somewhat more advanced real analysis course, which focuses on rigorous multivariable calculus and the beginnings of manifold theory. There's really no way to know exactly what's in a given course just from the title. (But I strongly doubt that it's really important for the OP to provide that much detail in this context anyway.) – Mark Meckes Nov 16 '20 at 11:37
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    @NoahSnyder: I gave the answer that appeared to me to be correct at the time I wrote it, based on what had been written in the post and various comments up to that point. Since then the situation has gotten murkier (at least from my point of view), including the very basic point of just how many different courses are being discussed. – Mark Meckes Nov 16 '20 at 15:18
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    But in the hope of giving some more clarity for the sake of anyone reading this thread in the future, let's point out explicitly (as you said in your answer) that courses named "Advanced Calculus" and "Calculus on Manifolds" do not count as "calculus" classes. They are definitely "above calculus". – Mark Meckes Nov 16 '20 at 15:21
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You can get an idea of what constitutes "Calculus" by looking at the Calculus AP test.

You can get an idea of "pre-calculus" with the UC admission requirements:

Three years of college-preparatory mathematics that include the topics covered in elementary and advanced algebra and two- and three-dimensional geometry. A geometry course or an integrated math course with a sufficient amount of geometry content must be completed. Approved integrated math courses may be used to fulfill part or all of this requirement, as may math courses taken in the seventh and eighth grades if the high school accepts them as equivalent to its own courses; also acceptable are courses that address the previously mentioned content areas and include or integrate probability, statistics or trigonometry. Courses intended for 11th and/or 12th grade levels may satisfy the required third year or recommended fourth year of the subject requirement if approved as an advanced math course.

There are a few courses that are "parallel" to Calculus , neither pre-requisites of Calculus, nor having Calculus as a pre-requisite. In the case of Statistics, the subject does rely on Calculus, but there are basic courses that don't require students to actually do Statistics, so they may be considered "before Calculus". Most other parallel courses would probably be considered "after": Abstract Algebra, Number Theory, Complex Algebra, Linear Algebra, Logic, Set Theory, Graph Theory, Topology. My interpretation is that if it's not taught in American high schools, it should be included.

If in doubt, it's probably better to put something in that you shouldn't than to leave something else that you should have. They probably put this qualification in to save you time, and assure you that you don't need to put in every single math course you've ever taken.

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I might call the class you called “Calculus” instead “Calculus on Manifolds,” which is the name of a famous text by Spivak covering that material. Even though it has “calculus” in the title everyone would consider it “more advanced than calculus.”

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    Funnily enough that's the book that we followed. :) – Pablo Bhowmik Nov 16 '20 at 6:00
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    Great! People know that book and you’re listing the textbook so it’ll be easy for them to understand. Maybe the honest but clear thing is to write the title as “Calculus [on Manifolds].” Square brackets means words you’ve added for clarity that aren’t in the original. – Noah Snyder Nov 16 '20 at 12:40
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Other courses besides those indicated in earlier answers could be: Numerical Analysis, Complex Variables, Mathematical Statistics, and Theory of Functions.

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