Theoretical Calculus worth the time?

[Mohamad Ternanni]

Hello, I already took multivariable calculus at my school but I did not take it rigorously. I am unsatisfied with my performance(I only plug numbers and have a small sense of what is happening behind the scenes, and I am concerned that I may need to grasp the underlying theories to prepare me for studying stochastic calculus). Can someone please advise me if it is worthwhile to study Calculus with theory(18.014 and 18.024 ) on mit ocw during the summer. (The tradeoff would be giving up a course on computational inference)

Links to courses:
Calculus:
Calculus with Theory | Mathematics | MIT OpenCourseWare
Multivariable Calculus with Theory | Mathematics | MIT OpenCourseWare
Inference:
Computational Probability and Inference

 

[Pavlos Sakoglou]

Hello, I already took multivariable calculus at my school but I did not take it rigorously. I am unsatisfied with my performance(I only plug numbers and have a small sense of what is happening behind the scenes, and I am concerned that I may need to grasp the underlying theories to prepare me for studying stochastic calculus). Can someone please advise me if it is worthwhile to study Calculus with theory(18.014 and 18.024 ) on mit ocw during the summer. (The tradeoff would be giving up a course on computational inference)

Links to courses:
Calculus:
Calculus with Theory | Mathematics | MIT OpenCourseWare
Multivariable Calculus with Theory | Mathematics | MIT OpenCourseWare
Inference:
Computational Probability and Inference

Although computational probability (machine learning in this case) is quite useful, you still can't do much with it if you don't understand the models and the math they are using. Using calculus is one thing, and understanding calculus is another.
I would recommend to invest your time in gaining intuition on math that are used (almost) everywhere -- even though they are theoretical, and once you finally get the concepts then to move forward to more applied stuff, which will seem easier to master; having no gaps on their background.

 

[IntoDarkness]

do it

 

[Daniel Duffy]

Calculus is basically learning methods and useful tricks. Nothing wrong with that. But indeed using calculus and understanding real and complex analysis are miles apart.

 

[Mohamad Ternanni]

Although computational probability (machine learning in this case) is quite useful, you still can't do much with it if you don't understand the models and the math they are using. Using calculus is one thing, and understanding calculus is another.
I would recommend to invest your time in gaining intuition on math that are used (almost) everywhere -- even though they are theoretical, and once you finally get the concepts then to move forward to more applied stuff, which will seem easier to master; having no gaps on their background.

Thank you. Also I will enroll in a differential equations course(not theoretical) in the upcoming fall. Does the same argument apply? (I can supplement my regular course with theoretical material also on ocw)

Although computational probability (machine learning in this case) is quite useful, you still can't do much with it if you don't understand the models and the math they are using. Using calculus is one thing, and understanding calculus is another.
I would recommend to invest your time in gaining intuition on math that are used (almost) everywhere -- even though they are theoretical, and once you finally get the concepts then to move forward to more applied stuff, which will seem easier to master; having no gaps on their background.

Calculus is basically learning methods and useful tricks. Nothing wrong with that. But indeed using calculus and understanding real and complex analysis are miles apart.

May you please give me some insight on how can complex analysis help me succeed in quantitative finance? I understood that real analysis is essential for quantitative finance but in my limited experience I didn't come across complex analysis uses yet. Thank you.

 

[Daniel Duffy]

Complex variables pop up here and there:

Characteristic function of a random variable
Fourier and Laplace transforms (e.g. option pricing)
PDEs
Complex variants of functions like N(z), erfc(z), Bessel etc.
Time series
Signal processing
Compute option greeks using Complex Step Method

 

[Mohamad Ternanni]

Complex variables pop up here and there:

Characteristic function of a random variable
Fourier and Laplace transforms (e.g. option pricing)
PDEs
Complex variants of functions like N(z), erfc(z), Bessel etc.
Time series
Signal processing
Compute option greeks using Complex Step Method

Okay Thank you for the much appreciated insight.

Complex variables pop up here and there:

Characteristic function of a random variable
Fourier and Laplace transforms (e.g. option pricing)
PDEs
Complex variants of functions like N(z), erfc(z), Bessel etc.
Time series
Signal processing
Compute option greeks using Complex Step Method

Okay Thank you for the much appreciated insight.

 

[Daniel Duffy]

You're welcome. Here's a readable book + lots of exercises


https://www.amazon.com/Schaums-Outline-Complex-Variables-Outlines/dp/0071615695#customerReviews

 

[bigbadwolf]

Can someone please advise me if it is worthwhile to study Calculus with theory(18.014 and 18.024 ) on mit ocw during the summer.

The Apostol book on calculus is still a calculus book. There seems to be a spectrum running from plug-and-chug calculus to real analysis. Apostol's calc book is still on the plug-and-chug side except with some proofs and some structure. Apostol's "Mathematical Analysis" is oriented more towards real analysis (but won't be used in the MIT courses whose links you gave).

 

[Quasar Chunawala]

Hello, I already took multivariable calculus at my school but I did not take it rigorously. I am unsatisfied with my performance(I only plug numbers and have a small sense of what is happening behind the scenes, and I am concerned that I may need to grasp the underlying theories to prepare me for studying stochastic calculus). Can someone please advise me if it is worthwhile to study Calculus with theory(18.014 and 18.024 ) on mit ocw during the summer. (The tradeoff would be giving up a course on computational inference)

Links to courses:
Calculus:
Calculus with Theory | Mathematics | MIT OpenCourseWare
Multivariable Calculus with Theory | Mathematics | MIT OpenCourseWare
Inference:
Computational Probability and Inference

I used these videos uploaded by Harvey Mudd College as a gentle, first introduction to Real Analysis.

Analysis Yawp!

The video lectures broadly follow Baby Rudin, and the prof covers all proofs including Dedekind's construction.

Another gem is Understanding Analysis by Stephen Abott. It builds an intuitive feel for the subject - something you'd miss, if you were to self-study using Rudin's PMA.

 

[Daniel Duffy]

I think Dedekind cuts is only relevant for maths degrees. It has no added value in the current context. There is a good chance that they will scare some people off.

 

[Quasar Chunawala]

I think Dedekind cuts is only relevant for maths degrees. It has no added value in the current context. There is a good chance that they will scare some people off.

True that.

 

[bigbadwolf]

I think Dedekind cuts is only relevant for maths degrees. It has no added value in the current context. There is a good chance that they will scare some people off.

There is an equivalent way of introducing real numbers and that is Cauchy sequences. This does have practical value, but the applications -- other than proofs of various convergence tests -- can be found only in classical texts or modern texts with "classical analysis" in the title. These applications are the raison d'etre for analysis but unfortunately they tend not to be found in modern real analysis books and they are too advanced for the usual US calculus book. I would argue that the typical great American calculus text -- at anywhere between 800 and 1200 pages -- actually makes one stupider. As well as causing slipped discs.

 

[Mikey V]

I used these videos uploaded by Harvey Mudd College as a gentle, first introduction to Real Analysis.
Another gem is Understanding Analysis by Stephen Abott. It builds an intuitive feel for the subject - something you'd miss, if you were to self-study using Rudin's PMA.

Couldn't agree more, I used this book for my analysis class and it is fantastic.

 

[Mohamad Ternanni]

Couldn't agree more, I used this book for my analysis class and it is fantastic.

I will be taking real analysis this spring. Do you recommend me to read the material before my course or use it as means of studying the course itself?

 

[Mikey V]

I will be taking real analysis this spring. Do you recommend me to read the material before my course or use it as means of studying the course itself?

It definitely can't hurt to read some in advance to begin thinking about some of the problems. It can certainly serve as great supplemental text depending on what your instructor ends up using.

 

[Mohamad Ternanni]

It definitely can't hurt to read some in advance to begin thinking about some of the problems. It can certainly serve as great supplemental text depending on what your instructor ends up using.

Okay thank you.

 

[Daniel Duffy]

There is an equivalent way of introducing real numbers and that is Cauchy sequences. This does have practical value, but the applications -- other than proofs of various convergence tests -- can be found only in classical texts or modern texts with "classical analysis" in the title. These applications are the raison d'etre for analysis but unfortunately they tend not to be found in modern real analysis books and they are too advanced for the usual US calculus book. I would argue that the typical great American calculus text -- at anywhere between 800 and 1200 pages -- actually makes one stupider. As well as causing slipped discs.

Speak of the devil!

There are two kinds of people: those who see Cauchy and them that don't want to see.

Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians? - wilmott.com

I can't imagine doing analysis without them, especially numerical analysis/methods.

The well-kept secret in mathematics is Functional Analysis.

 

[Daniel Duffy]

When students say "calculus" a question could be to explain (construction of) Riemann, Lebesque and {Riemann, Lebesgue} Stieltjes integration and compute them by hand on let's say the function f(x) = 4x(1-x) using pencil and paper.
These are very concrete questions.
Seeing how there are used later in probability and stochastics.


Lebesgue summarized his approach to integration in a letter to Paul Montel:

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.