In this post I started out presenting possibilities of learning physics online with the only prerequisite being to understand english. Now physics is notoriously formulated in mathematical language. So in order to follow the classroom presentations and common textbooks you need to have some college level math, but not too much. More precisely: you need, at every corner and really in every science: calculus. This, on top of high school math is really everything you need to work your way up all the way to the special theory of relativity, that is the beginning of modern physics, just before quantum mechanics comes into play.

Last time I presented Walter Lewin as an extraordinary teaching personality. For your mathematical needs, there is an equally appealing instructor online: Herb Gross.

Herb Gross provides us with a very clear explication of all the basic mathematical fields: calculus, linear algebra, complex numbers, vectors, series, differential equations. When I say, very clear, I mean this. In 30-45 minutes, you fill your head with nothing but relevant information to reconstruct the logic behind mathematical methods.

I said that all you needed for physics is calculus. Those other fields really only enhance your comprehension of what you are doing mathematically. Linear Algebra gives you insight into the structure of mathematical reasoning and you will learn, while one formula can explain really everything that you can formulate as a solution to it. Vectors of course are your basic building blocks for physical models, but their rules you learn in almost one lesson. Series and complex numbers again, make the mathematical thinking easier, because they give you synonyms, which enrich your way to express yourself mathematically. They really add new ways to the solution of problems, broader in scope and sometimes easier to handle. This is especially important for everything periodic, waves for example.

Now with Herb Gross you will get an effective overview about all these fields and with some exercises and more importantly with basic knowledge of a mathematical programming package like Mathematica or mathlab you really can start solving physical problems.

Where his courses are maybe a bit short, is multivariable calculus. This subject needs a little more attention, as you will learn to imagine problems in a new way. In general, calculus is a cookbook part of maths (citing Mr Gross here), which means that you learn some formulas, basic justification and than you apply it in certain cases. You learn of situations, where they don’t work and you learn of special formulas and so on. This corresponds to what MIT on their open courseware present as 18.01, 02 and 03.

In 18.01 you learn what differentials and integration are and you get to know methods to approximate and calculate numerical integrals.

18.02 gives you the mentioned multivariable view. You learn to represent up to 3D problems in different coordinates and how to integrate and differentiate accordingly.

18.03 goes deeper into solving differential equations. Now compared to Herb Gross, the course presented by Prof. Arthur Mattuck is slow paced. It is also less well explained to my taste. Therefore I’d advise you to start with Herb Gross. But the end of the course introduces what might be the most important formulas of this level of mathematical sophistication: Fourrier Series, Laplace Transforms, solving systems of differential equations with matrices. Those you have to know.

This is really all you need to this point. From here on, things get a bit more complicated. If you want to immerse yourselves into technical things, like construction, dynamics, complex systems, computersimulation: introductions to all those things are out there and mostly in fine quality on edX, coursera, MIT OpenCourseware and Stanford Online Courses. To follow physics through the 20th century to our times, quantum mechanics, particle physics, nuclear physics, statistical physics and the beautiful general theory of relativity and cosmology, there is really only one coherent online source. String theorest Leonard Susskind hosts adult education courses at Stanford about these topics, which are available on his website: The theoretical minimum. From my point of view though, these courses confuse you more than they help, because Prof. Susskind talks from the perspective of someone, who has already seen all this. You will lack explications. There are also tons of youtube videos from all kinds of personalities, which I find hard to follow with some exceptions to which I will turn in a second. There is however an excellent book (judging by the german version) by Randy Harris, entitled: Modern Physics, which is a really gentle introduction to all of these fields. It is comfortable, comprehensive and with a very pleasant way to present mathematical reasoning.

Mathematically the next steps are above all statistics, differential geometry and topology. The first is indispensible for all particle physics, quantum and statistical mechanics, the last two for cosmology and the general theory of relativity. This is really my current level of sophistication so I can’t present much further. However after some research I can already point in the right directions. Statistics courses are available on EdX (look at the archives maybe). The course there is really as non-mathematical as can be, but the concepts are there and the mathematical way to express things is presented in most textbooks anyhow. For differential geometry, I’m using the youtube channel mathview, which gives a pretty decent introduction to tensors and the notion of general coordinates (i.e not necessarily straight lines). For topology and some deeper geometry, the youtubechannel njwildberger seems useful.

Lastly in the most interesting parts of mechanics, where particles in some changing force field are concerned(electricity, magnetism, fluids, gases), people will introduce hamiltonians. Those are a set of equations derived from a certain mathematical operation, that let you work with basically any coordinate system you wish, while preserving the area of the figure formed by those arbitrary coordinates. As the hamiltonian is the legendre transform of the lagrange equation you need to learn about this laplace equation and then get some deeper understanding. See the links.

Book:

Modern Physics by Randy Harris

Links:

http://ocw.mit.edu/resources/res-18-006-calculus-revisited-single-variable-calculus-fall-2010/

(Herb Gross Single Variable Calculus)

http://ocw.mit.edu/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/

(Herb Gross Multi Variable Calculus)

http://ocw.mit.edu/resources/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/

(Herb Gross Complex Variables)

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/

(MIT 18.01)

http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/

(MIT 18.02)

http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/

(MIT 18.03)

http://theoreticalminimum.com/

(Leonard Susskind: Theoretical Minimum)

Mathview

(Youtube Channel: Mathview, Differential Geometry)

Njwildberger

(Youtube Channel: Njwildberger , Topology, Hyperbolic Geometry, Differential Geometry)

(Lagrange Equation)

(Hamiltonian Mechanics)

Update: I recently found out about this blog by Peteris Krumins, outright science enthousiast. There you can find lots of further education in physics, also available as video lectures. Check out his site for reviews, I can’t give them, as I’m not yet that far myself!

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Thank you for the very kind words about my “Calculus Revisited” video courses. The title “Calculus Revisited” reflects the aim of my courses; namely they were meant to be refresher courses for engineers and scientists who were entering the MIT Center for Advanced Engineering Study (CAES) to pursue post graduate studies. For that reason some topics were not treated in the same degree of depth as they would have been if the course had been meant for beginning students. In my mind, the remarkable thing is that students who are taking these courses for the first time are among the most enthusiastic commentators about my work and how much these courses have helped them. I am now 85 years of age and I get great satisfaction from seeing how work I did over 40 years ago is now helping a new generation of learners, many of whom were not yet born when the series was developed. And it gives me pleasure knowing that these courses will still be available to viewers even when I am no longer here.

Dear Herb Gross,

thank you for taking the time to comment my reaction to your class also on my blog! Your work from 40 years ago was exceptionally well done and it’s as much a lucky chance for us to have it accessible online, as it is a pleasure for you seeing it appreciated.

Meanwhile I have already found out that modern physics requires quite some more math skills to really get running, sure, but as far as basic calculus and linear algebra is concerned, I can’t immagine a more efficient introduction than yours. It’s fast, accurate and garnished with humour – so thank you again to having provided us with this treasure!

For me coming from a non-technical background and seeking understanding first, courses that are meant as refreshers by their authors in general are better suited for learning than those, who prepair you for exams and/or a career. If you don’t want to become fluent in a certain method, just knowledgeable, those are waisting much of your time. And if you decide to become more prolific, you already know where to look, but in the meantime you learned so many more things that can help you grasp most problems by analogy…

Thank you again and for the 50th anniversary of your corpus, we expect something special from you! 🙂

Thanks for yet another kind message.

I thought that you and your readers might like to know more about the “real” Herb Gross. More specifically, 40 of my 50 years in the classroom were spent at the community college teaching basic mathematics to mostly math-phobic adult learners. My goal wasn’t so much to make people “good at math” but rather to help them overcome their fear of math. What I have discovered is that math-phobic people tend to shy away from jobs or professions in which they are afraid that they might be required to use some mathematics. As a result they tend to forego opportunities that could help them attain greater upward mobility and a better quality of life.

So while I feel blessed to have been chosen by MIT’s Center for Advanced Engineering Study (CAES) to produce “Calculus Revisited”, developing the calculus video series was an aberration (but a very pleasant one) from my chosen life’s work; namely to help eradicate the problem of “innumeracy” that seems to be an international problem. For that reason, recognizing that at age 85 I will soon not be able to conduct workshops and/or teach classes live, I am currently developing my own website (www.mathasasecondlanguage.com) where I am uploading all of the work I have produced in arithmetic and basic algebra for anyone to use free of charge.

However since I do not have the same name recognition that MIT has, I worry that the people who might benefit the most from the material on my website might not even know that the website exists. For that reason I would be very grateful if you and your viewers were able to find ways to help me “spread the word”.

Once again, thanks for your very kind words. I wish you success in all of your endeavors

With warmest regards,

Herb (hgross3@comcast.net)