Are calculus and real analysis the same thing? 
*

*I guess this may seem stupid, but
how calculus and real analysis are
different from and related to each
other?
I tend to think they are the same
because all I know is that the
objects of both are real-valued
functions defined on $\mathbb{R}^n$,
and their topics are continuity,
differentiation and integration of
such functions. Isn't it?

*But there is also
$\lambda$-calculus, about which I
honestly don't quite know. Does it
belong to calculus? If not, why is
it called *-calculus?

*I have heard at the undergraduate course level, some people mentioned the
topics in linear algebra as
calculus. Is that correct?


Thanks and regards!
 A: *

*A first approximation is that real analysis is the rigorous version of calculus. You might think about the distinction as follows: engineers use calculus, but pure mathematicians use real analysis. The term "real analysis" also includes topics not of interest to engineers but of interest to pure mathematicians. 

*As is mentioned in the comments, this refers to a different meaning of the word "calculus," which simply means "a method of calculation."

*This is imprecise. Linear algebra is essential to the study of multivariable calculus, but I wouldn't call it a calculus topic in and of itself. People who say this probably mean that it is a calculus-level topic. 
A: My take on this: One would use the word 'calculus' when one is applying the mathematical tools - chain rule, integration- by-parts, etc - to solve problems in science, engineering, and so on; whereas one would use the word 'analysis' when one is developing/justifying the same tools - proving the chain rule, inventing integration-by-parts, etc.  I.e, analysis is what the pure mathematicians do, calculus is the product of analysis which engineers use.
A: In Eastern Europe (Poland, Russia) there is no difference between calculus and analysis (there is mathematical analysis of function of real/complex variable/s).
In my opinion this distinction is typical for Western countries to make the following difference:


*

*calculus relies mainly on conducting "calculations" (algebraic transformations applied to function, derivation of theorems/concepts by methods of elementary mathematics, computations applied to specific problem)

*analysis relies mainly on conducting "analysis" of properties of functions (derivation of theorems, proving theorems)
However, still this distinction is unnecessary: 


*

*(the most important) issues of "calculus" and "analysis" are very often linked together so that distinction is impossible (e.g. consideration of concept of limit in calculus due to Cauchy or Heine is actually the same as in analysis)

*it makes artificial ambiguity in perception of mathematical analysis

*it isolates common sense approach obtained from elementary mathematics and disables straightforward transition from elementary mathematics to higher mathematics

*issues of "calculus" and "analysis" treated together enables acquisition of deeper understanding of subject by making extension from methods gained from elementary mathematics.
A: This is a purely Anglo-Saxon distinction. In most countries, however, there is no distinction between "rigorous" analysis and "non-rigorous" calculus. There are just different levels of analysis courses, e.g. "real analysis for engineers".
The term "calculus" itself just means "method of calculation". Even simple arithmetic is a kind of "calculus". What people in Anglo-Saxon countries refer to as "calculus" is actually just a short version of "infinitesimal calculus", the original ideas and concepts introduced by Leibniz and Newton. Nonetheless, even the lowest-level "calculus" courses usually refer to concepts that were introduced much later after Newton and Leibniz, e.g. Riemann sums (Riemann lived about 200 years after these two).
A: As I understand the terms, calculus is just differentiation and integration, whereas real analysis also includes such topics as the definition of a real number, infinite series, and continuity. But perhaps I am out of date.
A: Calculus is about integration and differentiation. In real analysis we talk about Measure theory and lebesgue integral, proving theorems etc .And that introduces Topology , Functional analysis , Complex analysis , PDE and ODE etc .
A: The classes in the U.S. are definitely separated by name, but the content of the 4  "Calculus" courses I took for electrical engineering wasn't quite as distinct as some are portraying it.   We spent a substantial amount of time on sequences/series, limits, convergence, Taylor series, differentiability and integrability, etc., and related theorems.  It was quite a bit more than "how to compute";  however, it wasn't nearly as rigorous as the proof-based real analysis class my son described from Caltech.
I'm sure that the engineering disciplines are a big reason for the distinction. The distinction may have other historical roots, but engineering schools sustain it.  In the U.S., engineering (especially electrical) has changed in the last 50 years because the world is changing; instead of Fortran, it's incredibly helpful to know C++, Python, AND MATLAB;  Newer high-speed transistors has shifted emphasis to RF circuit design classes;  Increases in computer frequencies is pushing signal/power integrity to the forefront because-- in industry-- it's a dominant issue that barely existed in the 1980s.  One EE prof. I spoke with said that electromagnetics, which formerly included distinct classes in electrostatics & magnetics, was condensed into 1 course that is barely adequate.  There is an ever-widening list of courses/requirements and many students can barely finish a degree in 4 years;  Math is just one of several areas receiving pressure to shed "unnecessary" requirements.  My point is that real analysis would be good for engineers, but it's competing with many other math/non-math topics.  
One can argue whether engineering should be a 5 or 6 year degree;  but unless someone forces that upon everyone, which university will be the first to step forward and inform students that they must now pay for 5-6 years of school?  In my opinion, economics is the sustained pressure to streamline courses and focus on practical "calculus" topics.
A: The issue here is that Calculus has several different meanings, some of them overlapping in context.
In mathematics, calculus can mean:


*

*a calculation or method of calculation (archaic, but motivates the other usages)

*a formal system in which symbolic expressions are manipulated according to fixed rules (and/or the rules themselves)
e.g propositional calculus, predicate calculus, lambda calculus

*infinitesimal calculus,
i.e. differential calculus & integral calculus

*By extension of 3, differential and integral calculus considered as a single subject in mathematical education (in addition to other pre-requisites taught as part of such courses)

*By extension of 4, a synonym of analysis (or some vague subset thereof) as a subject and/or branch of mathematics, since introductory analysis courses are generally based on 4.



- https://mathoverflow.net/questions/36758/difference-between-a-calculus-and-an-algebra
- What is the difference between a calculus and an algebra?
- What exactly is calculus?
- What do Algebra and Calculus mean?
- When do you call something "a calculus" vs. "a logic"?
- terminology: what is meant if someone writes "calculus of .."?
- What does "calculus" mean?
- Why "calculus"?
