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.