Formally, What is the difference between Calculus and Mathematical Analysis?
Calculus is a common abbreviation for "Differential and Integral Calculus" which in German is called "Differentialrechnung and Integralrechnung" and Mathematical Analysis is called generally simply Analysis.
The difference leads in the fact that the first is basically the Leibnizian and Newtonian Calculus, where the integral is the based on the Riemann approach. Its latest textbooks like "A Course of Pure Mathematics" by the English Godfrey Harold Hardy or the German "Differential and Integral Calculus" by Richard Courant, works with the so-called Elementary Functions.
Analysis instead in its normal approach looks for rigorous formalization of Calculus, as well of Real and Complex numbers, like did by Edmund Landau, in its "Grundlagen der Analysis", adopting the Dedekind cuts. However latest reset made by the Bourbaki pseudonym group includes in the formalization mathematical disciplines like Set theory (Théorie des Ensembles) and Topology (Topologie Générale) adopting also the concept of measure, originally introduced by Henri Lebesgue, in his book: "Leçons sur L'Integration", in 1928.
This is the formal approach today to (Mathematical) Analysis like offers the American "Real Analysis" by H. L. Royden, or the English E.T. Whittaker, G. N. Watson - A course of modern analysis
Further paths arise from this approach to offer a formal "Complex Analysis" offering proof of the so-called Fundamental Theorem of Algebra using Morera Theorem, as well to "Functional Analysis", originally implemented by the Polish Stephan Banach, in is "Stefan Banach - Théorie des opérations linéaires.pdf"
It is interesting and was developed by Prof. Jean Alexandre Eugène Dieudonné - Foundations Of Modern Analysis a formal introduction of Differential and Integral Calculus, with "Operators", and this approach is our adoption for analysis, after to verify students level.
The works of Jacobi and Niels Abel, are not generally included in "standard" courses.
Giovanni A. Orlando.