Yep. One benefit of the HMC notes over Paul's is that the HMC notes have more visualizations. I thought the linear algebra portion of the HMC notes (change of basis, eigenstuff) was especially useful for learners as a counterpoint to most textbook presentations.
Very good. Every so often I get despondent about the state of the world; then I'm reminded that somewhere some bright Indian girl is figuring out how to build a fusion reactor from pigshit and string with knowledge she gained from sites like this. One can hope.
I didn't take Eigen Values & Vectors until I took Linear Algebra. In his notes he went over Eigen Values in Diff Eq. Not Vectors, but rather functions. Also I never learned Fourier Series in Diff Eq. We did go over Laplace transformations. Where ever he took this material, they were insanely thorough.
I must also confess that his ability in helping me understand and fall in love with Calculus two years ago was the main impetus for me to select Mathematics as my major. I'm now focusing my attention on Number Theory with high hopes of one day becoming a Theoretical Mathematician.
I should also add that Princeton Companion book to Mathematics is a valuable resource for learning what is out there in Pure Mathematics.
In math, this means that we've established that he exists, but we haven't found him yet. Last I checked, they established a lower bound on his street address of 6325.
My first degree was in "Theoretical Math". I took classes in subjects like Number Theory, Topology, Analysis, Non-commutative Ring Theory. The subjects were about Math not how to do Math to solve other problems. We studied proofs not applying math to solve word problems.
As an example, I wasn't very good at solving differential equations, a very important part of math used to solve many real-life problems. Instead I studied things like the Lebesgue Integration which extends the notion of integration to a larger set of functions for which the more familiar Riemann integral wouldn't be defined. That was taught in my second semester real analysis course that had these prerequisites: Real Analysis I, Complex Analysis I, Differential Equations, Calc I and Calc II. A lot of work to get to an interesting subject, but a subject of interest to Mathematicians not engineers using math.
An undergraduate degree in Applied math, in contrast to theoretical math, would probably have involved learning more about, say, differential equations.
Usually this is called pure mathematics, as opposed to applied mathematics.
On the other hand, applied mathematics might also involve some pretty technical "theoretical maths", which though being applied, can be studied for its own interest without the application.
So in some sense, "applied maths" sometimes means maths that is applied, rather than maths that can be applied. For that reason it perhaps makes sense to instead use the term "theoretical maths", though I am not sure if that is standard terminology or not.
Paul's Math Notes got me through all of my undergrad math courses. My professor was fantastic, but having these to reference when the professor wasn't around was a much-needed asset to learning and finishing homework. I recommend this site to everyone I know taking Calculus or higher courses.
I had Prof. Dawkins for Calculus a good decade and a half ago. He was a fantastic teacher, and the reason I finally 'got' a lot of the concepts I had struggled with up to then.
I had this guy as my math teacher. He was an absolute machine. Always came in at exactly the same time, put his bag down, took his watch off, and started transmitting knowledge. Loved his classes.
All homework problems were really realistic with messy answers and the test questions were easy. Only a few problems for both.
Once, when talking about higher dimensional math, a student asked if string theory said there are 13 dimensions. He quickly replied "yeah" and I'm sure he knew it was more complicated than that but it was an irrelevant question. Without thinking I blurted out it was consistent with up to 13 dimensions and the whole class turned around to look at this kid who corrected The Machine. He quickly admitted that was true and moved on. Highlight of my education.
Great notes, I remember using these years ago for calculus and differential equations. Very straight-forward and example-based. Remember for most of us math is about solving problems!
Those cheat sheets baffle me. I cannot imagine that someone who needs help remembering, to pick a few examples, that
a^m a^n = a^(m+n)
or that
y = mx + b
is a line will be able to use those facts in a real-world problem, with the cheat sheet in hand.
For those who used these and found them useful: did you really use them as cheat sheets, that is, to look up things while working on a problem, or did you use them more as a checklist before entering an exam, to check that you remembered most of them?
I suggest big parts of Paul's Online Notes to students who need some brush-up or remediation. When I was teaching college precalc, for instance, people usually had a reasonable grasp of y=mx+b but difficulties with exponentiation were almost universal. Some of these "cheat sheets" are useful for such students just to tape above their workspace so that while working through the rote mechanical practice problems they must do they can use the "cheat sheet" as a checklist while they work.
The review of complex numbers, on the other hand (not a cheat sheet but a condensed review) I assign to some masters' students who have not used complex numbers for 2 years and need to recall what they once learned. There it's just a concise but reasonably comprehensive refresher list.
That makes waaay more sense to me; the first uses them more as training wheels than as cheat sheet; the second is more akin to the "check that I know it" that I envisioned. Thanks.
After passing Calculus I with a C in college about 8-9 years ago, I found this site and started working with his examples. I ended up passing Calculus II (regarded at the "hardest" math class at my college) with a B+.
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[ 3.0 ms ] story [ 98.5 ms ] threadhttp://faculty.atu.edu/mfinan/nnotes.html
Another resource is https://www.youtube.com/user/patrickJMT/
Khan Academy is good if you don't know anything but I like PatrickJMT's videos better if you're reviewing.
(includes some linear algebra, despite the URL)
https://www.youtube.com/user/professorleonard57
I must also confess that his ability in helping me understand and fall in love with Calculus two years ago was the main impetus for me to select Mathematics as my major. I'm now focusing my attention on Number Theory with high hopes of one day becoming a Theoretical Mathematician.
I should also add that Princeton Companion book to Mathematics is a valuable resource for learning what is out there in Pure Mathematics.
what does that even mean ?
As an example, I wasn't very good at solving differential equations, a very important part of math used to solve many real-life problems. Instead I studied things like the Lebesgue Integration which extends the notion of integration to a larger set of functions for which the more familiar Riemann integral wouldn't be defined. That was taught in my second semester real analysis course that had these prerequisites: Real Analysis I, Complex Analysis I, Differential Equations, Calc I and Calc II. A lot of work to get to an interesting subject, but a subject of interest to Mathematicians not engineers using math. An undergraduate degree in Applied math, in contrast to theoretical math, would probably have involved learning more about, say, differential equations.
On the other hand, applied mathematics might also involve some pretty technical "theoretical maths", which though being applied, can be studied for its own interest without the application.
So in some sense, "applied maths" sometimes means maths that is applied, rather than maths that can be applied. For that reason it perhaps makes sense to instead use the term "theoretical maths", though I am not sure if that is standard terminology or not.
All homework problems were really realistic with messy answers and the test questions were easy. Only a few problems for both.
Once, when talking about higher dimensional math, a student asked if string theory said there are 13 dimensions. He quickly replied "yeah" and I'm sure he knew it was more complicated than that but it was an irrelevant question. Without thinking I blurted out it was consistent with up to 13 dimensions and the whole class turned around to look at this kid who corrected The Machine. He quickly admitted that was true and moved on. Highlight of my education.
For those who used these and found them useful: did you really use them as cheat sheets, that is, to look up things while working on a problem, or did you use them more as a checklist before entering an exam, to check that you remembered most of them?
The review of complex numbers, on the other hand (not a cheat sheet but a condensed review) I assign to some masters' students who have not used complex numbers for 2 years and need to recall what they once learned. There it's just a concise but reasonably comprehensive refresher list.
Highly recommended!