I tend to have a weird population of physics students; some are taking Calc 3 (multivariable and vector) while others are taking Calc 1 (derivatives). Possibly all will be deficient in algebra compared to when I took physics [especially thanks to The Unprecedented Situation We All Find Ourselves In (COVID)].
So, what I am working for is some sort of ... well, not an easy way out, but something where students can know just enough to start; knowing they will get the proper math methods later. Last year I think I did ok with algebra -- get to students to solve systems of equations that they don't retain or don't cover. This year I want to add calculus, as would be needed for physics.
Anyway, this is what I want to develop on. Define the differential: for y = f(x), dy = f'(x) dx, determined by replacing y --> y + dy, x --> x + dx, and then taking the first-order approximation in dx, dropping all higher powers. I think that that might be more accessible; explain that what we want in physics is to see rates of change; if we make a little change (x+dx), how does the function change (y+dy) and our derivative dy/dx really is a division of two small changes. [note: vid linked below, one must be careful with differentials manipulation, things are only approximately equal unless everything is at one one point, dx is infinitesimal.]
To be clear, this is just changing perspective on the math method, but the goal is to deliver it in a useful method for what i'm trying to do. The bits I want to remember are from these references:
Also in my lazy research, I found these lectures from MIT OCW on calc. Hanev't looked at them yet, these are just for later reference for me. I think this approx and infin. is what I'm looking for above, but i can only read transcript with tonight's internet connection. Thanks, capitalism.
Rolle's theorem states that (to paraphrase) if you have a continuous function f where f(a) = f(b), then there must be some point c between a and b where f'(c) = 0.
In physics, there is often a need to find the 'turnaround point', where the motion of an object reverses direction. Think of an object tossed into the air: "What goes up, must come down". There is a function f which describes the position, and a derivative f' describing the velocity (in that direction). Since the object covers a path that will have the same position [f(a) = f(b)], there must be a point where the object turns around, and at that point, the velocity is zero (f'(c) = 0).
There is a lot of information online, and some of it is in a useful state. One of those unusual fonts of info comes from the old fashioned way of research -- asking a librarian. the Library of congress has some people who will try to answer questions, and they have a frequently asked question list.
What I want to remember out of this video is the method of detection of vibration -- Matthias uses a speaker with a weighted cone as sort of a microphone. Measuring the signal with an oscilloscope / spectrum analyzer makes finding vibration frequency fairly straightforward. Can this be used along the lines of makeshift seismometers (generally the voice coil being moved around a magnet through some resonant system, like a pendulum.)?
Ellipses are very ill defined in mathematics, but this is the first time i've seen ellipses abused to explain using math something about the 'nature' of the universe. The places where I saw this use of ellipses most commonly is
In the image above, note that a number of the form x = 2^n -1 in binary should be all 1's, if you have an infinitely long number of this sort, it will be an infinite line of 1's (...111). 2x must be an even number, which would end in 0 in binary (...110); the ending position is for 2^0 = 1 which determines parity (1=odd, 0=even). However, since it is infinitely long, all you have really done is subtract 1 from x (...111 - ...001 = ...110); 2x = x - 1 --> x = -1. In computer numbers, this would indicate that 2^N is the most-negative number you can have, your number type overflows at 2^n -1. This is the behavior of two's complement computers; see [Wikipedia -- Two's Complement] This sort of implies that ellipses to the left of the decimal point (and, semi-jokingly, algebra and the universe) work like math on a two's complement machine
While preparing this, my cat Rusty would like to share his thoughts on the matter below:
5555555555555555555555555555555555555555 j ju888888 vfffffffffff?fffffffffffffffffffffffffffffffff nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnncccccccccccc m njmhy
Also, I don't think people know when to use raincoats or umbrellas. I see people with umbrellas out and open with barely a drop falling, and people with nothing in a downpour repeatedly. I don't know where to find a good raincoat; I generally get whatever's around. Also I don't mind a little light rain. Or waiting out a downpour. I kinda think people forgot how to deal with weather. Not everyone; people know what works for them, I guess, but I can't understand the variety of responses. I don't know where rain goods are sold; maybe I haven't been to the big city to see something that isn't a crap box store.
You see, the spare I have is a 'donut' like probably everyone else, unless your car is a jeep sort of thing or from 1985 or earlier. You should not run the donut on a drive or steering wheel. (the donut shouldn't exist, but c'est la merde). I always get my flat tires in the worst ways; today it is the driver's front tire on a FWD car in a parking lot in 90 degree, no-shade, beautiful weather. I wasn't far from home and it was easier to do it wrong and safely creep the car home where I could do it right in cooler evening weather.
Anyone can have the knowledge to change a tire, but it's not an easy thing. Actually removing the tire from the hub is easy, the strenuous part is getting the lugnuts loosened and raising the car on the jack. That is the work.
I am not a mechanic; I am not your mechanic. I'm just going to list things that I found useful in a precarious situation, or intend to have next time.
I grew up in what one might refer to as 'shitkick holler', and still live there. This is the home of the 'freedom-loving rugged individualist', so let a resident tell you now that trying to be a freedom-loving rugged individualist is a great way to die alone in a ditch. No one is an island to themselves; we all need community. In the example above, a driver should have three people who can help them; without them, you would be alone changing a tire on the side of the road, or walking home.
Think of all the people who might think of themselves as 'rugged individualists' -- they generally rely the most on the community around them. People to deliver food and fuel and medicine to where they can get it. People to maintain infrastructure. People to entertain them, and to care that they live.
Now think of the people who claim they are 'freedom lovers'. What do you think they feel about your freedom? In my experience, to mangle a song, they think freedom is making sure you have nothing left to lose.
I am typing this on the tilde.club server, tmux'ed and over an ssh connection, which goes through my phone hotspot to the greater internet. This afternoon, there was significant latency, such that I could type several words, then wait ~30 seconds for them to appear. I learned that if you have vim mappings that look like <leader><keys>, the leader can timeout before the keys arrive to be read, and it can do unintended changes.
I have a keybinding, \sv that tells vim to reload .vimrc. I was editing .vimrc, and testing the changes in another document, which kept seeing a letter change to a v -- vim was seeing <leader><timeout> substitute letter (s) with v. In those cases, remapping keys to avoid the <leader> key is likely better. Or, just run the command yourself. ( :source $MYVIMRC ) Also, bitten again by trying to put literal angle brackets in html. Damn.
This page has a textwidth of 64em, blatantly copied from the tilde.club css file. I haven't changed it because the text is intended to be as readable as a book, which generally has about 65 characters in a column (this is the 'default' setting in TeX typesetting, and it matches my experience with normal books. I don't know about textbooks.)
This isn't an attempt to make a mobile-friendly website, and I don't know how it would look on a phone. Good luck, phone-internet users. I'm also not optimizing for desktop; I don;t want to install new browsers and check how it looks; it seems ok on firefox and dillo, though they disagree on rendering the page.
An aside, Debian 11, why isn't dc a default install program?
tmux worked out of the box. tmux can work, but you have to have your shell understand that it can do 256 colors. For me, that was adding
export TERM=xterm-256colorto my .bashrc. Anyway, thank you, tmux. Now I have to get used to C-b instead of C-a, but i'm also getting used to things I remember from 20 years ago being subtly different than my memory. Thanks, time!
In vim, you can run a command and then read the result into your file using :r!
nnoremap <F2> i<CR><ESC>k:r!date +\%Y-\%b-\%d<CR><ESC>kJJ<ESC> Do not add things to your .vimrc that you do not understand. This maps the F2 key to a series of commands; the extra movement commands put the text exactly where I want it, not on a new line; you split the line, add the date, then rejoin the three lines you have. I am sure there is a better way, but I'm still learning. For example, you have to enter the tag brackets with &-escaped text in HTML (see source). Thanks vim macros and stack overflow!
As an aside, my current internet connection is bad, and I can type much faster than the screen can display. Vim is maybe designed for this (I have never used a 300 baud connection on a dumb terminal); if you are editing and know you made a mistake in vim too far back, stop and go to normal mode. Then search (? or F) for the mistake in the line, and fix it, then re-enter insert mode with 'append at end of line (A)'. (If you error spans from some point to the end of the line, use substitute to eol (S). I never did learn touch typing, or the habit I saw them learn of hitting backspace, then retyping. Type only once.
A further aside, if you bold and italic text in vim, it might change the highlighting on it, be not afraid if you haven't closed your tags yet. (Do close them, I guess)
One more; don't put the first paragraph in <p> tags. It leaves unsightly space. I don't know why. Bit again by the literal < >; don't write at logical bedtime.
I can't really call it the end now, can I? I'm starting here and appending to the top of the doc.