Difference between Calc & Analysis?

[MathStudent]

Could someone explain to me the difference between Calculus courses and analysis courses?

Also, I am looking to get a book on the subject. I've heard Rudin's name tossed around quite a bit. Is this the best path to go? My main concern is properly learning the theory behind the material, as well as gain experience in applications. I especially love it when the author adds their personal interpretations and comments.

Thanks in Adavance!

 

[matt grime]

Approximately, and remember unlike the objects involved in a subject what constitutes a subject is subjective, calculus could be thought of as applied analysis. In analysis you would prove that the derivative of x^n is nx^{n-1}, in calculus you'd care about finding the local minumum of a function using analytically defined rules such as the derivative of x^n is nx^{n-1}

 

[MathStudent]

I see... so analysis is typically a more in-depth theoretical approach to calculus. Thanks!

 

[philosophking]

I think a good adjective to use for analysis is "rigorous".

 

[MathStudent]

how much of the material is brand new (ie: never seen in a calc class) and how much is just reworking some of the older proofs ( like mean value theorem )?

 

[HallsofIvy]

My understanding is that in Europe they tend to use "analysis" to mean what we (Americans or can I include the English) mean by "calculus". In the United States, at any rate, "Mathematical Analysis" is the theory behind calculus.

The basic theorems of analysis are often familiar from caculus but more general (abstract) and proved with more rigor.

For example, the intermediate value theorem, "If f is continuous on [a,b], f(a)< 0 and f(b)> 0, then f(c)= 0 for some value of c between a and b" becomes "A continuous image of a connected set is connected".
The theorem that "If f is continuous, then f takes on both maximum and minimum values on a closed and bounded interval" becomes "A continuous image of a compact set is compact."