Not the person you are responding, but I also used Spivak for single variable. For multivariable calculus I used Marsden and Tromba.
The other Spivak is a great book, but maybe too much for a first approach to the subject.
As others have argued, one can learn mathematics without its history, but it is very helpful to motivate many of its abstractions. A good example of this are some of the textbooks of David Bressoud. In a "Radical approach to Analysis", concepts are introduced in historical order. The book starts with the heat equation and its solution by Fourier series which was rejected at the time, in constrast to most books that start with the real numbers and limits.
"All of statistics" by Larry Wasserman. I took a bunch of courses that taught stats as a bunch of independent tools for different problems. AOS helped me to build some a strong foundation.
"Fundamental university physics Volume 1: Mechanics" by Alonso and Finn. This book seems to be not very well known in the USA, but it is very popular in Spanish and Portuguese speaking countries. It is your classical introductory physics/mechanics course with a very high emphasis on calculus.
"Computational partial differential equations" by Hans Peter Langtagen. A book on numerical methods for solutions of PDEs. It has the right amount of rigour (so you are able to tackle the literature), but it also includes code and plenty of practical advice.
"Nonlinear dynamics and chaos" by Strogatz. I think this book is really well known and I can't add much.
A very formulaic article, start with an anecdote that could fit many other articles (it seems Feynmann always shows up when we are talking about learning), refer to some Twitter conversation, add a formula or graph which provides no insight, sprinkle memes everywhere and that's all.
In the end, it seems the entire article can be summarised to, "it is all good to google for some recipes to solve your problems, but you also should care about some deeper understanding". Is there anything else to it?
As soon as I finished my PhD in computational science, I left to the Bay Area. I did consider some positions in academia, but as others have described, the low impact of the problems I was working on and the path to permanent position didn't really motivate me.
I have been one year in industry and though I am doing something different, filtering and sensor fusion, I would describe my overall experience as very positive. I still get to read a lot of papers, work with very intelligent and motivated people and every now and then I get some praise from my coworkers on my maths knowledge, which is nice.
Deadlines are tighter and the metrics in which my work is judged on are very different.
Only to the undesirables students. Quoting the paper,
"The Mathematics Department of Moscow State University, the most prestigious mathematics school in Russia, was at that time actively trying to keep Jewish students (and other 'undesirables') from enrolling in the department.
One of the methods they used for doing this was to give the unwanted students a different set of problems on their oral exam."
Another point of view on why it is not striking that we can model heat conduction well. The heat equation is well-posed, which among other things, means that the solution depends continuously of the input data. And we observe this in many physical systems, that small changes in the cause produces small changes in the effect, therefore, we should expect from good mathematical models of reality to display this property.
It is about triangles. The idea that sine is the solution of this ODE for f(0)=0 and f'(0)=1 is quite modern.
I would say a course on trigonometry usually covers (my experience):
trigonometric functions
exact value of them for the angles 30, 45, 60, 90 ... degrees
Trigonometric formulas for the sum and difference of angles. A formuka for the double and the half angle.
Law of sine and law of cosine
Lots of relations derived from the Pythagoras theorem (sin^2+cos^=1)
how to solve trigonometric equations
With all this, you are equipped to completely determine a triangle, knowing some of its and the length of some its sides. As as application, I was taught, how to measure heights and distances provided you can measure angles.
Thus, without trigonometry, it would be fairly hard to take a course on analytic geometry.
Now, how would the course be enhanced by introducing sine as the solution of an ODE?
As young students, we are taught that these functions are ratios of sides on right angles. It is fairly intuitive for them that provided they can draw a right triangle, they just need to measure the sides. And it is easy to accept that someone computed sines and cosines for a lot of angles and put it in a table.
My point being, that we didn't need (in the past) calculus to compute trigonometric functions, and I don't see how it is a burden for students to be introduced to those functions without the Calculus definition.
And the reality is, that the definition of sine as a ratio of the catheti and hypotenuse is a rigorous definition of the function. Strictly, this sine is different from the sine of calculus. The first, the sine from Euclidean geometry, assigns a real to pair of rays, while the calculus sine, is function from the real numbers to the reals. And it does take some work to link them formally.
I think I am missing something because, I am unable to see why it is huge burden to introduce sine and cosine without their rigorous definition. At which age, are students taught trigonometry? And what does a course on trigonometry covers? What would you think they would be able to do without it?
When we were introduced the sine and the cosine function, we were already familiar with Thales theorem, so therefore we could show that this ratio was a constant.
I am quite sure historically as well sine and cosine predate the more formal construction of those functions, be it as a series, solution of an ODE or inverse of arc sin (and this defined as an integral)...
I never considered this possibility. Could you recommend some link where I could read about the field and see if I have chance?
It is not really what motivates me, but it is worth informing myself before discarding it.
This path is really appealing to me. Now, knowing that there is a demand for the people between the two worlds, I will spend some time looking for offers, and prepare a CV accordingly.
Moving to US seems a bit harder, but I wouldn't mind if I achieve this in the long term.
> Now, 30+ years later, it's a bit rusty, but I can still read it pretty well, and can sort-of understand written Portuguese as well.
I don't want to be picky. But I would not attribute your ability to read Portuguese to your musical background, but to your Spanish. My experience is that most Spanish speakers can read Portuguese to a certain degree and vice-versa. Never having studied French, I was able to read some child comics.
I have a similar story to tell. As an undergrad in a course of robotics, we had to program a wheeled robot and all the robots would be place in an environment with obstacles. The robot should avoid crashing with the obstacles and the other robots. Extra points if the robots moved smoothly.
Me and my friend implemented a very simple algorithm. All the sensors measured distances to objects, and to every reading we would assign a vector whose direction oppose the one of the sensor and length, inversely "proportional" to the distance. Add all the vectors and move in this direction with a speed proportional to the length.
This turned out to work very well to avoid static obstacles and other robots. Most students implemented finite state machines. They crashed quite a lot and their movement was very clumsy, which I suppose was due to the fact that the transition of states was not very smooth.
To be fair, our success was a combination of luck and laziness too. If we had more time, we would have implemented a FSM too.