Math analysis and ODE

[TGV320]

Hello,

I am currently planning on self studying math analysis with MIT ocw courses, but I cannot find the analysis course. I have found on the web that analysis and ODE are the same thing. If I want to get a good introduction to analysis, is the MIT ODE course fit for such a use?

Thanks

 

[fresh_42]

Hello,

I am currently planning on self studying math analysis with MIT ocw courses, but I cannot find the analysis course. I have found on the web that analysis and ODE are the same thing.

They are not.

If I want to get a good introduction to analysis, is the MIT ODE course fit for such a use?

Thanks

No. They are related but ODE is normally not part of analysis. If you search "analysis + pdf" and "ODE + pdf" on Google, you will almost certainly find lecture notes which tell you the difference.

Or look at
https://en.wikipedia.org/wiki/Mathematical_analysis
https://en.wikipedia.org/wiki/Ordinary_differential_equation
for a brief overview what the two areas cover.

 

[TGV320]

Hi,
Thanks for the reply.
Sorry for having my questions sound so silly, I am really quite lost here.

I am currently preparing to self study the same classes as MIT physics majors, at least during the first two or three years of college. Will do the best I can. Is only studying ODE at first without analysis for physics majors something that is usually done?

Thanks

 

[fresh_42]

Hi,
Thanks for the reply.
Sorry for having my questions sound so silly, I am really quite lost here.

I am currently preparing to self study the same classes as MIT physics majors, at least during the first two or three years of college. Will do the best I can. Is only studying ODE at first without analysis for physics majors something that is usually done?

Thanks

Analysis usually covers 3 parts: real functions, real multivariate functions, complex functions.
ODE deal with relations between derivatives of functions. Normally taken after Analysis 1 and 2.
(You need integration techniques for ODE, usually taught at school and in Analysis I. Depending on the course, you will also need basics in linear algebra for ODE systems.)

However, all are relevant for physics. Analysis is about functions and functions are everywhere in physics. ODE is (roughly, for details see the Wikipedia link) about the behavior of functions in time. But behavior in time is physics!

In analysis, you learn that $\displaystyle{\int \dfrac{1}{x}}\,dx = \log |x| +C$ and in ODE you learn to solve $y'(x)=\dfrac{1}{x}$ and what it needs to determine $C.$

It cannot be said any shorter, and such a description lacks plenty of content. You really should at least read the first section on the Wikipedia pages, possibly in your own language.

 

[jtbell]

Different countries or languages may use different terminology here.

In my experience in the US, physics and math students usually learn calculus at the beginning of university, from courses labeled "Calculus". There are are usually three semesters: Calculus 1 & 2 covering single-variable calculus, and Calculus 3 covering multi-variable calculus. These courses are oriented towards practical use and techniques. They may include a simple introduction to differential equations.

Then come courses focusing on differential equations: Ordinary Differential Equations for single variables, and Partial Differential Equations for multiple variables.

A course labeled "Analysis" is usually intended only for math majors who have already taken the Calculus 1-3 sequence. It basically covers calculus again from a more rigorous point of view, appropriate for a mathematician, focusing on theorems and proofs. Physics majors do not normally take this course, unless they are double-majoring in physics and math, as I did when I was an undergraduate.

 

[fresh_42]

We do not use the word Calculus in Germany. It is called Analysis.

 

[malawi_glenn]

We do not use the word Calculus in Germany. It is called Analysis.

Same in sweden. Same word for analysis and calculus.

 

[fresh_42]

Same in sweden. Same word for analysis and calculus.

We also use calculus completely differently, and I miss the English correspondence.

A calculus is a certain framework that defines a logical environment to perform deductions (e.g. Fitch calculus) or a system to perform calculations (e.g. residue calculus). It describes a framework, not analysis.

 

[malawi_glenn]

Earlier it used to be called "kalkyl"

 

[fresh_42]

Earlier it used to be called "kalkyl"

It is Kalkül here. And even Kalkül has yet a double meaning: the framework, and someone's mindset which explains his actions. Putin's Kalkül that the West would silently stand by didn't work out.

So giving it a third meaning for Analysis would definitely be confusing.

 

[malawi_glenn]

It is Kalkül here. And even Kalkül has yet a double meaning: the framework, and someone's mindset which explains his actions. Putin's Kalkül that the West would silently stand by didn't work out.

So giving it a third meaning for Analysis would definitely be confusing.

I have an analysis book showing the resemblence between Cauchy and young Putin. I read that chapter the 23rd of February this year... pretty spooky

 

[fresh_42]

I have an analysis book showing the resemblence between Cauchy and young Putin. I read that chapter the 23rd of February this year... pretty spooky

Tell me when you start Tolstoi's doorstop.

 

[malawi_glenn]

Tell me when you start Tolstoi's doorstop.

I don't read books that does not have equations and diagrams

 

[jtbell]

I am currently planning on self studying math analysis with MIT ocw courses, but I cannot find the analysis course.

Searching for “analysis” on the MIT OCW website gives me

https://ocw.mit.edu/search/?q=analysis

which includes e.g. 18.100A Introduction to Analysis.

Searching for “calculus” instead gives me

https://ocw.mit.edu/search/?q=calculus

which includes e.g. 18.01 Single Variable Calculus and 18.02 Multivariable Calculus. I strongly suspect that these are what you want to study, to go along with intro physics.

 

[CrysPhys]

Hi,
Thanks for the reply.
Sorry for having my questions sound so silly, I am really quite lost here.

I am currently preparing to self study the same classes as MIT physics majors, at least during the first two or three years of college. Will do the best I can. Is only studying ODE at first without analysis for physics majors something that is usually done?

Thanks

If you want to know what math courses undergrad physics majors at MIT take, look here: https://physics.mit.edu/academic-programs/undergrads/requirements/. All freshmen are required to take 18.01 (Calculus I) and 18.02 (Calculus II). MIT now offers different physics tracks. Physics majors planning to continue on to grad physics take "The Focused Track", which requires 18.03 (Differential Equations) and "one subject given by the Mathematics Department beyond 18.03". 18.03 is taken directly after 18.02.

Many, many moons ago when I was there, after 18.03, most physics majors who were more interested in physics than math per se then took a two-semester sequence called Advanced Calculus for Scientists and Engineers (or some variation of the theme). I see that the current offering is only a one-semester course called 18.075 Methods for Scientists and Engineers. There is an OCW version of 18.075 called Advanced Calculus for Engineers (https://ocw.mit.edu/courses/18-075-advanced-calculus-for-engineers-fall-2004/pages/syllabus/). I see that the textbook is one version by F.B. Hildebrand (Advanced Calculus for Applications). Hildebrand was what I used way back when.

So a course specifically on analysis is not required for physics undergrads at MIT: not decades ago, and not now.

 

[jtbell]

All freshmen are required to take 18.01 (Calculus I) and 18.02 (Calculus II)

Aha, that explains why I didn’t see calculus listed among the physics major requirements.

 

[WWGD]

ODE's and PDE's are more towards the applied side, though you can do some more purely Mathematical work on them. Function Spaces and Functional Analysis in general. You may want to
think of which aspect you prefer to focus on.

 

[Delta2]

If anybody cares what's happening here in Greece, we use both words.

"Analysis" is an ancient greek word which is used in modern greek as well.

The Greek word for calculus is "logismos" ("Diaforikos logismos" for differential calculus for example) and it's general meaning is as in German, it means thinking or more precisely the way of thinking. And yes "logismos" relates to logic, both are ancient Greek words.

 

[TGV320]

Hello
Thanks, I shall study ODE then.

 

[pbuk]

As you have been advised elsewhere, if you are selecting courses from MIT OCW you need to check the prerequisites listed under the "Syllabus" section of the course home page. For instance for the ODE course https://ocw.mit.edu/courses/18-03-differential-equations-spring-2010/pages/syllabus/ you need https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/, for which in turn you need only "high school algebra and trigonometry. Prior experience with calculus is helpful but not essential". This sounds like you so I would start there.

In general (and specifically at MIT), courses labelled analysis are concerned with proofs (which is what mathematicians do) rather than learning methods for solving problems (which is what physicists do).

 

[TGV320]

Hello，
I shall study in that order then. Loads of work ahead of me.
Thanks

 

[CrysPhys]

Hello，
I shall study in that order then. Loads of work ahead of me.
Thanks

Even though 18.02 is listed as a corequisite for 18.03, I would recommend that you follow the sequential order 18.01, 18.02, 18.03 (rather than completing 18.01 and taking 18.02 and 18.03 at the same time). Besides, since you are primarily interested in math needed for the physics curriculum, you should note that you will need 18.02 before you need 18.03 for your physics courses.

 

[pbuk]

Even though 18.02 is listed as a corequisite for 18.03, I would recommend that you follow the sequential order 18.01, 18.02, 18.03 (rather than completing 18.01 and taking 18.02 and 18.03 at the same time). Besides, since you are primarily interested in math needed for the physics curriculum, you should note that you will need 18.02 before you need 18.03 for your physics courses.

Yes I think that's probably a good idea: remember these courses were designed for an American University year so three courses in series means you would take a year and a half to get to the end of ODEs. When studying independently you don't have this problem.

 

[CrysPhys]

Yes I think that's probably a good idea: remember these courses were designed for an American University year so three courses in series means you would take a year and a half to get to the end of ODEs. When studying independently you don't have this problem.

Yes, but that's a mixed blessing. When attending a university, there is a calendar that regulates the pace. When studying independently unguided, it's too easy to take on too much.

At any rate, the typical physics and math schedule for physics majors at MIT would be:

First semester: 8.01 and 18.01
Second semester: 8.02 and 18.02
Third semester: 8.03 and 18.03

 

[TGV320]

Thanks, I've heard that MIT students had usually 4 main classes per term, quite a heavy load for sure.

 

[vanhees71]

We do not use the word Calculus in Germany. It is called Analysis.

I'd say in the usual German university-physics curriculum the equivalent of "calculus" are lectures usually called "Mathematische Methoden der Physik". At my university (Goethe University Frankfurt) we have "Mathematische Methoden" accompanying the early theoretical-physics course lectures (mechanics, electrodynamics, quantum mechanics 1). They provide a more applied approach to mathematics, usually not giving mathematically rigorous proofs but rather calculational techniques needing to solve problems in physics.

Analysis is usually taught by mathematicians and give rigorous proofs and all that. In some universities the physicists take the same lectures as the pure mathematics students in some there are extra lecture for physicists and other STEM students.

 

[fresh_42]

I'd say in the usual German university-physics curriculum the equivalent of "calculus" are lectures usually called "Mathematische Methoden der Physik". At my university (Goethe University Frankfurt) we have "Mathematische Methoden" accompanying the early theoretical-physics course lectures (mechanics, electrodynamics, quantum mechanics 1). They provide a more applied approach to mathematics, usually not giving mathematically rigorous proofs but rather calculational techniques needing to solve problems in physics.

Analysis is usually taught by mathematicians and give rigorous proofs and all that. In some universities the physicists take the same lectures as the pure mathematics students in some there are extra lecture for physicists and other STEM students.

I don't remember how it was when I studied. Physicists and mathematicians could easily study the same first-year content. I remember the one who taught experimental physics and some of us (mathematicians) attended the course just for the show. The professor was a real entertainer.

Nevertheless, Calculus is an American word and we do not use it in German. Whether you call it analysis or methods for physicists doesn't matter, it is never 'Kalkül' which has a completely different meaning.

 

[mathwonk]

This is quite interesting. Here is apparently the original German title of Courant's famous book, (which is titled Differential and Integral Calculus, in English, omitting "Lectures on"):

Vorlesungen über Differential- und Integralrechnung.

Apparently, perhaps the German speakers will tell us, "rechnung" or "Rechnung" has a meaning like computation, reckoning, or calculation.

 

[fresh_42]

This is quite interesting. Here is apparently the original German title of Courant's famous book, (which is titled Differential and Integral Calculus, in English, omitting "Lectures on"):

Vorlesungen über Differential- und Integralrechnung.

Apparently, perhaps the German speakers will tell us, "rechnung" has a meaning like computation, reckoning, or calculation.

I don't really know the difference between computation and calculation.
Vorlesung = lecture
-rechnung = calculus
Rechnung = calculation, or bill, it depends

 

[mathwonk]

Here is an interesting paragraph on the historical use of the term calculus and its connection with the German word rechnen, e.g. in the use of the term "reckoners" for skilled calculators at one time, in English. The answer of interest is the one by Alan Kennington.

https://hsm.stackexchange.com/quest...ed-the-word-calculus-and-what-did-it-describe

 

[WWGD]

Doesn't the term ' Calculus' refer to a pebble/stone?

 

[pbuk]

Doesn't the term ' Calculus' refer to a pebble/stone?

Yes; what can you do with a collection of pebbles (apart from throw them)?