#88 overall
Single Variable Calculus
Coursera
5 recommenders · 5 threads · recommended in 4 years, 2014–2022 · score 5.1
A single-variable calculus course from the University of Pennsylvania on Coursera, taught by Robert Ghrist. It covers limits, derivatives, and integrals, with some attention to practical issues.
Commenters call it the best calculus course they've taken online and say it is good for refreshing single-variable calculus.
Recommendations by year
20062026
I'm going to plug Calculus: Single Variable from the University of Pennsylvania on Coursera (https://www.coursera.org/learn/single-variable-calculus).
This was the best Calculus course I've taken online.
I will try to list resources in a linear fashion, in a way that one naturally adds onto the previous (in terms of knowledge)
[PREREQUISITES]
First things first, I assume you went to a highschool, so you don't have a need for a full pre-calculus course. This would assume you, at least intuitively, understand what a function is; you know what a polynomial is; what rational, imaginary, real and complex numbers are; you can solve any quadratic equation; you know the equation of a line (and of a circle) and you can find the point that intersects two lines; you know the perimiter, area and volume formulas for common geometrical shapes/bodies and you know trigonometry in a context of a triangle. Khan Academy website (or simple googling) is good to fill any gaps in this.
[BASICS]
You would obviously start with calculus. Jim Fowlers Calculus 1 is an excellent first start if you don't know anything about the topic. Calculus: Single Variable https://www.coursera.org/course/calcsing is the more advanced version which I would strongly suggest, as it requires very little prerequisites and goes into some deeper practical issues.
By far the best resource for Linear Algebra is the MIT course taught by Gilbert Strang http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebr... If you prefer to learn through programming, https://www.coursera.org/course/matrix might be better for you, though this is a somewhat lightweight course.
[SECOND STEP]
After this point you'd might want to review single variable calculus though a more analytical approach on MIT OCW http://ocw.mit.edu/courses/mathematics/18-01sc-single-variab... as well as take your venture into multivariable calculus http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable...
Excellent book for single variable calculus (though in reality its a book in mathematical analysis) is Spivaks "Calculus" (depending on where you are, legally or illegally obtainable here http://libgen.org/ (as are the other books mentioned in this post)). A quick and dirty run through multivariable analysis is Spivaks "Calculus on Manifolds".
Another exellent book (that covers both single and multivar analysis) is Walter Rudins "Principles of Mathematical Analysis" (commonly referred to as "baby rudin" by mathematicians), though be warned, this is an advanced book. The author wont cradle you with superfluous explanations and you may encounter many examples of "magical math" (you are presented with a difficult problem and the solution is a clever idea that somebody magically pulled out of their ass in a strike of pure genius, making you feel like you would have never thought of it yourself and you should probably give up math forever. (Obviously don't, this is common in mathematics. Through time proofs get perfected until they reach a very elegant form, and are only presented that way, obscuring the decades/centuries of work that went into the making of that solution))
At this point you have all the necessery knowledge to start studying Differential Equations http://ocw.mit.edu/courses/mathematics/18-03sc-differential-...
Alternativelly you can go into Probability and Statistics https://www.coursera.org/course/biostats https://www.coursera.org/course/biostats2
[FURTHER MATH]
If you have gone through the above, you already have all the knowledge you need to study the areas you mentioned in your post. However, if you are interested in further mathematics you can go through the following:
The actual first principles of mathematics are prepositional and first order logic. It would, however, (imo) not be natural to start your study of maths with it. Good resource is https://www.coursera.org/course/intrologic and possibly https://class.stanford.edu/courses/Philosophy/LPL/2014/about
For Abstract algebra and Complex analysis (two separate subjects) you could go through Saylors courses http://www.saylor.org/majors/mathematics/ (sorry, I didn't study these in english).
You would also want to find some resource to study Galois theory which would be a nice bridge between algebra and number theory. For number theory I recommend the book by G. H. Hardy
At some point in life you'd also want to go through Partial Differential Equations, and perhaps Numerical Analysis. I guess check them out on Saylor http://www.saylor.org/majors/mathematics/
Topology by Munkres (its a book)
Rudin's Functional Analysis (this is the "big/adult rudin")
Hatcher's Algebraic Topology
[LIFE AFTER MATH]
It is, I guess, natural for mathematicians to branch out into:
[Computer/Data Science]
There are, literally, hundreds of courses on edX, Coursera and Udacity so take your pick. These are some of my favorites:
Artificial Intelligence https://www.edx.org/course/artificial-intelligence-uc-berkel...
Machine Learning https://www.coursera.org/course/ml
The 2+2 Princeton and Stanford Algorithms classes on Coursera
Discrete Optimization https://www.coursera.org/course/optimization
Convex Optimization https://itunes.apple.com/itunes-u/convex-optimization-ee364a... https://itunes.apple.com/us/course/convex-optimization-ii/id...
[Physics]
http://theoreticalminimum.com/courses
http://ocw.mit.edu/courses/physics/
computer science is an extremely broad area. Narrow it down, maybe?
1. Introduction to Computational Thinking at MIT was a great one.
2. Robert Ghrist has the best set of SV Calculus courses on Coursera.
3. Gilbert Strang's LA courses are the best.
4. For Algorithms course, I would recommend Sedgewick and Wayne's course on Coursera.
5. For PL course, go with the WashU course on Coursera.
6. For Machine Learning, you should start with Andrew Ng's Machine Learning.
7. Cryptography I, II by Dan Boneh is a great one.
8. Introduction to Programming with Python at MIT on edX is a good course. And so is CS50 by David J. Malan.
9. The nand2tetris course on Coursera.
These are some that I can vouch for. You did not mention your level or what topic you want to learn.
The best way is to do a google search for moocs, and try several of them out, and pick one with the teacher whose teaching style you like the most.
In MOOCs that are not mandated, the personality of the teacher matters a lot. In my opinion- it is the most important factor.
You should also look up courses where all materials are publicly available- tests, assignments, homework problems, etc. If some parts are not, you can always email a prof, and they will almost never turn you down.
Follow discussions on Reddit. Engage in discussions, too.
You should look up https://teachyourselfcs.com and check out the recorded courses there. Do the ones that you would like.
The Linear Algebra OCW is great if you do all the exercise and read the book as you study. For Calculus it's not so great, though the ODE course is fine.
This kind of courses from UPenn are really good if you need to refresh single variable, though I did an older version were all the courses were given together than I can't find in the current Coursera: https://www.coursera.org/learn/single-variable-calculus
The OSU Coursera course you recommended isn't bad, but the UPenn course I feel is higher quality.
[1]https://www.youtube.com/playlist?list=PLKc2XOQp0dMwj9zAXD5Ll... [2]https://www.coursera.org/learn/single-variable-calculus [3]https://www.math.upenn.edu/~ghrist/