
Big Picture of Calculus (2010) [video] - espeed
https://www.youtube.com/watch?v=UcWsDwg1XwM&index=2&list=PLBE9407EA64E2C318
======
datainplace
Now I know someone is browsing my Youtube history and posting to HN. Youtube
really is the math teacher I never had.

Probability Models and Axioms
[https://www.youtube.com/watch?v=j9WZyLZCBzs](https://www.youtube.com/watch?v=j9WZyLZCBzs)

The Exponential Function
[https://www.youtube.com/watch?v=oo1ZZlvT2LQ&t=100s](https://www.youtube.com/watch?v=oo1ZZlvT2LQ&t=100s)

Vector Space
[https://www.youtube.com/watch?v=ozwodzD5bJM&t=36s](https://www.youtube.com/watch?v=ozwodzD5bJM&t=36s)

Tree cutting fails:
[https://www.youtube.com/watch?v=JHZkR6UVegY](https://www.youtube.com/watch?v=JHZkR6UVegY)

~~~
mindcrime
_Youtube really is the math teacher I never had._

Same here. I'm on a quest to run myself through the equivalent of a standard
Calc I, Calc II, Calc III, Linear Algebra sequence, and I've managed to find
great Youtube resources for all of those things (and more). It's times like
these that you marvel at how awesome the Internet can be at its best. :-)

~~~
40acres
Hey I'd love to hear about how your learning experience is going! Care to
share your experiences?

I'm about to start a self-directed learning exercise focused on CS related
math. I've never used online resources to learn / brush up on complex topics
before so I'm a bit unsure what to expect.

~~~
mindcrime
Sure, I'm at work right now, but I could write up some stuff about my
experience later.

------
seycombi
If you want to get into the details of Calculus

Professor Leonard
[https://www.youtube.com/user/professorleonard57](https://www.youtube.com/user/professorleonard57)

Herbert Gross MIT Calculus Revisited: Multivariable Calculus
[https://www.youtube.com/playlist?list=PL1C22D4DED943EF7B](https://www.youtube.com/playlist?list=PL1C22D4DED943EF7B)

Herbert Gross MIT Calculus Revisited: Calculus of Complex Variables
[https://www.youtube.com/playlist?list=PLD971E94905A70448](https://www.youtube.com/playlist?list=PLD971E94905A70448)

Herbert Gross MIT Calculus Revisited: Single Variable Calculus
[https://www.youtube.com/playlist?list=PL3B08AE665AB9002A](https://www.youtube.com/playlist?list=PL3B08AE665AB9002A)

MIT 18.02 Multivariable Calculus, Fall 2007
[https://www.youtube.com/playlist?list=PL4C4C8A7D06566F38](https://www.youtube.com/playlist?list=PL4C4C8A7D06566F38)

MIT 18.02SC: Homework Help for Multivariable Calculus
[https://www.youtube.com/playlist?list=PLF07555F3CC669D01](https://www.youtube.com/playlist?list=PLF07555F3CC669D01)

MIT 18.01 Single Variable Calculus, Fall 2006
[https://www.youtube.com/playlist?list=PL590CCC2BC5AF3BC1](https://www.youtube.com/playlist?list=PL590CCC2BC5AF3BC1)

MIT 18.01SC: Homework Help for Single Variable Calculus
[https://www.youtube.com/playlist?list=PL21BCE50ABFF029F1](https://www.youtube.com/playlist?list=PL21BCE50ABFF029F1)

~~~
mindcrime
Second the recommendation for Professor Leonard. He does a really nice job
explaining things and making it accessible.

------
mrcactu5
I was a teaching assistant at UC Santa Barbara for 3 years, I can tell you,
students might slug through the calculations , but none -- or a precious few
-- really get the big picture.

Mostly, it is that calculus pedagogy is a disaster. And it hasn't been updated
to include recent advances in technology (such as Data Science)

Strang is an amazing lecturer. I can tell you, that far into your Masters and
PhD these same basic issues resurface. There's at least one project I can
think of where the entire problem hinges on taking one derivative.

------
billyzs
Gilbert Starng also has an excellent series on Linear Algebra on MIT
opencourseware [[https://ocw.mit.edu/courses/mathematics/18-06-linear-
algebra...](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-
spring-2010/)]

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shaftway
Honestly I couldn't slog through the video. It started off with "let's use
letters" and immediately got to things like "the right letters for this case
are 'df' and 'dt'". This just feels like "here's a bunch of magic that you'll
just have to memorize". Reminds me of my favorite quote:

> And then Satan said "put the alphabet in math".

When I learned the basics of calculus it was via simple problems that we
wanted to answer. And for a while we estimated the answers by brute forcing
approximations. I still drop back to this when I can't remember how to do
something and need to re-discover parts of calculus.

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pknerd
Thanks for sharing such an amazing thing. I am done with studies but should be
helpful to understand it properly to teach my kids in future so that they just
fill up pages with dy/dx without getting an idea behind it.

~~~
corysama
For kids I'm a fan of [http://dragonbox.com/](http://dragonbox.com/) and their
[http://www.dragonboxapp.com/](http://www.dragonboxapp.com/) They do a pretty
good job of teaching symbolic algebra to kids who don't yet know arithmetic.

~~~
pknerd
Thanks

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huangc10
I find that most people understand the general concepts of basic calculus.
What's hard to wrap around is when you start getting into proofs (ie. delta-
epsilon, trig, e,...and that's just some general ones).

I suppose that's how they differentiate the good students from the just okay
students. Personally, I don't think youtube videos will help that much when it
goes down to the nitty gritty. You really have to sit down, think, and
understand.

~~~
adeledeweylopez
Keisler has a introductory text book (available free here:
[https://www.math.wisc.edu/~keisler/calc.html](https://www.math.wisc.edu/~keisler/calc.html))
that uses non-standard analysis, which he claims is easier for beginners to
understand than the notorious delta-epsilon proofs.

I can't say how effective it is at that goal, but it is all mathematically
legitimate, and has a nice intuitive appeal to it.

~~~
rahimnathwani
Looks similar to this book, which I bought after a recommendation here:
[https://www.amazon.com/Infinitesimal-Calculus-Dover-Books-
Ma...](https://www.amazon.com/Infinitesimal-Calculus-Dover-Books-
Mathematics/dp/0486428869)

------
stephengillie
Having had a non-traditional route through learning various math topics, I'd
like to advocate the teaching of calculus (the understanding of the
accumulation of infinitesimally-small changes) before the teaching of
trigonometry (the relationships between different geometric shapes, and the
resulting association with waves).

In my (humbly biased) opinion, it's easier to learn how trigonometric concepts
operate with a calculus background, than with merely an algebra-and-geometry
background. Trigonometry complicates calculus education - knowing trigonometry
may facilitate better calculus understanding to some, but not knowing
trigonometry doesn't necessarily complicate learning calculus concepts. The
trigonometry-first tradition in math education reflects a "number-phile" bias,
not the most optimal route for humans to ingest and retain concepts.

Edit: The "number-phile" bias is clear - we want our teachers to be people
with passion on the topic, as we're more likely to get a better understanding
of the topic. Understanding sine waves as the result of an integral was much
easier for me to understand than as the result of a complicated calculation.

~~~
jacobolus
It’s somewhat misleading/unfortunate to use angle measures as the basic
characterization of an angle/rotation, when just trying to solve basic
geometry / triangle measurement problems. It’s almost always easier to use a
vector representation, written in terms of explicit coordinates if necessary
(e.g. as a complex number). [http://www.shapeoperator.com/2016/12/12/sunset-
geometry/](http://www.shapeoperator.com/2016/12/12/sunset-geometry/)

Where angle measures and sines/cosines start to be useful is with uniform
circular motion, which as you say is a calculus problem.

The key to understanding the form of trigonometry courses is to understand the
context for their creation. Namely, all calculations (e.g. for astronomy,
navigation, engineering, mapmaking, ...) used to be done by hand by humans,
which was very expensive. It was important to have someone with fluent
knowledge of trigonometric identities simplify formulas to a form with as few
arithmetic operations and table lookups as possible to save money, or just to
match the available function tables. Today in a computer age, extensive
memorization of trigonometric identities is an anachronism, and spending lots
of time on practicing their manipulation is okay algebra practice but not
anything directly useful per se.

------
peetle
I love Gilbert Strang, but _by far_ my favorite book on calculus is "Calculus
Made Easy" by Silvanus Thompson. It supplies a very elegant and powerful set
of tools to derive many of the "fundamentals" of calculus.

------
brightball
What would be some practical "daily life" applications of calculus? I've been
discussing a focus on teaching with applicable examples in a lot of subjects,
but this is one where I've been struggling a bit.

~~~
dlo
I'm going to expand the definition of "daily life application" from things you
actually use daily to things you experience daily. These could be engaging
too!

For example... what are sunspots? How does the plumbing around your house
work? How does a car battery work? How does a microwave oven work?

Another thought. If we are unable to find applications for a technique, should
we place such an emphasis on teaching it? I look around and see all the very
practical (and interesting!) math and science that is not being taught, e.g.
signal processing, just because they are nontraditional. It's a shame to me.

~~~
brightball
That's kind've where I was going with it. Basic understanding of the building
blocks of society type stuff: power, water, gas, construction, radio, biology,
growing food, vehicles, finances, etc.

Thinking back, I just realized the main reason that I was bored to tears in
high school wasn't so much that the subjects weren't worth knowing...just that
at the time I was being taught formulas without any clear understanding of why
it was useful to know it. If you don't know why it's useful to know it,
there's very little reason to remember it beyond passing a test.

~~~
dlo
You might enjoy this. I actually think this is well done.

[https://www.grc.nasa.gov/www/k-12/airplane/index.html](https://www.grc.nasa.gov/www/k-12/airplane/index.html)

Obviously, it could use more work, but the kernel of something really good is
there.

------
alfonsodev
Thank you for this video, is there a platform that you could aggregate all
this resources mapping them to certain knowledge tree ? So that you could have
the big picture and then zoom in for specific topics videos. Something like
Khan Academy's knowledge map[1], that you could customise to add you own
contents, this video for example.

[1]
[https://www.khanacademy.org/exercisedashboard](https://www.khanacademy.org/exercisedashboard)

~~~
Miserlou57
I worked pretty hard on something like this many years ago. I am not a coder
so it was a formidable challenge and I never really got it polished or
complete. The goal was to create a blank canvas where anyone could organize
instructional content from the web (i.e. KhanAcademy, YouTube, etc) into their
own class. Each class topic was structured like a subreddit for each topic,
with the best results hopefully filtering to the top

~~~
loblollyboy
I just had that idea this weekend - also like a github branch-diagram of
math/physics history

