"Calculus Made Easy". Originally written by Silvanus Thompson, but modernized by the inimitable Martin Gardner.
It appears there are newer editions of CME. I've been working through the Gardner (1998) edition. Got it on abebooks.com. I like it, because it has plenty of exercises. And I trust Gardner. I haven't seen the 2020 or 2022 editions.
I highly recommend anything by the late Martin Gardner, including his "Annotated Alice" books.
I highly recommend anything by the late Martin Gardner, including his "Annotated Alice" books.
Ya know, believe it or not, I've never even read Alice in Wonderland, so that'd be a good way to read it :)
Has calculus changed in the last 40 years? I still use what I learned in university back then and it still seems to work.
Right, so that's why I think I am going with the 1991 edition of James Stewart Calculus for $11 in very good condition used :) I don't have my original Swokowski Calculus book anymore.Smart move. I don't think the sort of calculus we use in electronics has actually changed a lot since the mid-late 1800's when Cauchy, Maxwell and Riemann worked in methods to deal with complex numbers. I did a 2nd degree in mathematics (my first was in physics) and touched on some pretty weird stuff, but that does not seem to have leaked very far from the ivory tower.
The calculus is very useful basis for things widely used in electronics but has little direct application in its own right. So if you are aiming for differential equations, Fourier and Laplace and others carry on. Otherwise you might be better served by boning up on linear algebra, boolean logic and other fields.
The calculus is very useful basis for things widely used in electronics but has little direct application in its own right. So if you are aiming for differential equations, Fourier and Laplace and others carry on. Otherwise you might be better served by boning up on linear algebra, boolean logic and other fields.I'm good with boolean logic. I did ace calc i, ii, ii, linear algebra and differential equations in college. I do remember the laplace transform and thought it was neat. But I forget how to do it all lol.. forget all the notation as well. After calc I want to relearn diffy q and linear algebra. Should I skip multivariable calculus or is that used a lot as well in electronics? Like Calc ii is the prerequisite for Diffy Q if I recall. I remember taking Diffy Q and Calc III the same semeseter.
Has calculus changed in the last 40 years?
There are some good refresher courses on https://brilliant.org/calculus/
You gotta love an animation!
Ȁ>> syms x
>> partfrac(x^2/(x^3 - 3*x + 2))
ans = (sym)
4 5 1
--------- + --------- + ----------
9*(x + 2) 9*(x - 1) 3*(x - 1)^2
>>
Now as to checking your work, you will find that a lot of books have practice questions but not all of them have the answers in the book itself, sometimes you have to get a second answer book to go with it. What else helps though is automatic software. The stuff of today can beat down a lot of calculus even some differential equations. You can use that software to check your answers to the practice questions.
Oh and you can also ask questions on forums like this one but im not sure if this one has a section for math.
It takes Octave less than a second
Code: [Select](%i9) partfrac(x^2/(x^3 - 3*x + 2), x);
4 5 1
(%o9) --------- + --------- + ----------
9 (x + 2) 9 (x - 1) 2
3 (x - 1)
Maxima is based on a 1982 version of Macsyma, which was developed at MIT with funding from the United States Department of Energy and other government agencies. A version of Macsyma was maintained by Bill Schelter from 1982 until his death in 2001. In 1998, Schelter obtained permission from the Department of Energy to release his version under the GPL. That version, now called Maxima, is maintained by an independent group of users and developers. Maxima does not include any of the many modifications and enhancements made to the commercial version of Macsyma during 1982–1999. Though the core functionality remains similar, code depending on these enhancements may not work on Maxima, and bugs which were fixed in Macsyma may still be present in Maxima, and vice versa. Maxima participated in Google Summer of Code in 2019 under International Neuroinformatics Coordinating Facility.
--> ratprint : false$
fpprintprec : 4$
eq1 : 0 = -V1 + Z1*I1 + Z4*(I1-I2) ;
eq2 : 0 = Z4*(I2-I1) + Z2*(I2-I4) + Z5*(I2-I3) ;
eq3 : 0 = Z5*(I3-I2) + Z3*I3 + V2 ;
eq4 : 0 = Z2*(I4-I2) + V3 ;
eq5 : VA = V1 - I1*Z1 ;
eq6 : VB = VA - V3 ;
eq7 : Z1 = 2 ;
eq8 : Z2 = -5*%i ;
eq9 : Z3 = 4 ;
eq10 : Z4 = -5*%i ;
eq11 : Z5 = 4*%i ;
eq12 : V1 = 120 ;
eq13 : V2 = 120*%i ;
eq14 : V3 = 14.14*%i + 14.14 ;
res : solve([eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10,eq11,eq12,eq13,eq14])$
results : expand(float(res))$
lngth : length(results[1])$
sorted : sort(results[1])$
print("")$
for i:1 thru lngth do
print(sorted[i])$
(eq1) 0=(I1-I2)*Z4+I1*Z1-V1
(eq2) 0=(I2-I3)*Z5+(I2-I1)*Z4+(I2-I4)*Z2
(eq3) 0=(I3-I2)*Z5+I3*Z3+V2
(eq4) 0=(I4-I2)*Z2+V3
(eq5) VA=V1-I1*Z1
(eq6) VB=VA-V3
(eq7) Z1=2
(eq8) Z2=-5*%i
(eq9) Z3=4
(eq10) Z4=-5*%i
(eq11) Z5=4*%i
(eq12) V1=120
(eq13) V2=120*%i
(eq14) V3=14.14*%i+14.14
I1=16.81-22.88*%i
I2=25.96-40.15*%i
I3=18.06-22.1*%i
I4=28.79-42.98*%i
V1=120.0
V2=120.0*%i
V3=14.14*%i+14.14
VA=45.76*%i+86.38
VB=31.62*%i+72.24
Z1=2.0" "
Z2=-5.0*%i" "
Z3=4.0" "
Z4=-5.0*%i" "
Z5=4.0*%i" "
Have you played with SageMath? It's in Python. Playign around with it.. pretty cool -- Opensource.
"Calculus Made Easy". Originally written by Silvanus Thompson, but modernized by the inimitable Martin Gardner.
I'll have to install it on another machine. It pretty much filled the disk on my main machine. I'm going to uninstall WSL anyway, I need the space. Never buy a computer with less than 2 TB of disk space...
I haven't tried SageMath with a GUI or command file and both are necessary. I think it's doable with Jupyter but I haven't gotten that far. There are limitations to WSL in terms of graphics (and there's no desktop) so I'll reinstall SageMath on one of my Linux machines. It'll be worth spending some time with the package.
Nice book collection there :)It's only the applied stuff from my past. I also have a couple of volumes of Schaum's notes on applied calculus, but you can look that stuff up yourself. I didn't bother with the more abstract real and complex analysis texts. Not really pertinent to EE.
I found what looks like a decent College Algebra textbook with table of contents in the PDF.Yes. That looks like a good compilation of applied algebra concepts. It should be fine to do on-line.
https://openstax.org/details/books/college-algebra-2e
What do you think? I can just read and do problems from my iPad. Free.
What do you think? I also see another flavor of a combined book called "Algebra and Trigonometry" which I thought was Precalculus. LOL.. too many different varieties of Algebra and Trig classes if you ask me.
I took calculus and did well, but never could grasp the concepts which has annoyed me for years.
Personally, when I don't grasp the concept, I can't retain what I'm learning. The most basic is why taking the first derivative gives you position, the second gives you velocity, etc... besides reading this in a textbook or online, I don't understand why. In fact, I had to look online just to type that sentence because it never clicked with me.
Word problems was where I couldn't do any calculus. One word problem I remember was a swimmer swimming to shore at a certain speed while the water current was pushing him sideways at another speed. Where on shore would the swimmer end up after X time (I believe this is the correct question that was asked). While I understand the problem in hand, I don't understand how to apply calculus to solve it.
In my case, with electronics, such as inductor (v = L (di/dt)) I get the linear aspect of this. If the current went from 0 to 1A in 1s, and the inductor is 10mH (milli Henries), then the voltage is 10mV [(1A/1s) * 10mH]. But what does this really tell me if it's a 60Hz sine wave (AC voltage source)?
Anyway, sometime ago I came across a video from this guy, but didn't get too far into as I found following him was easy until I closed the video and tried to think about it further:
https://www.youtube.com/watch?v=UcWsDwg1XwM&list=PLFW_V3qDH5jRyfpD9uiq6aKVTWIxpbIm3&index=1 (https://www.youtube.com/watch?v=UcWsDwg1XwM&list=PLFW_V3qDH5jRyfpD9uiq6aKVTWIxpbIm3&index=1)
In my case, with electronics, such as inductor (v = L (di/dt)) I get the linear aspect of this. If the current went from 0 to 1A in 1s, and the inductor is 10mH (milli Henries), then the voltage is 10mV [(1A/1s) * 10mH]. But what does this really tell me if it's a 60Hz sine wave (AC voltage source)?
All of AI is based on linear algebra and everybody needs to get a taste of that. Even the simple Digits Recognition problem is working in 784 dimensional space. A wee bit hard to visualize... That problem and its Neural Network solution is the "Hello World of AI".
In grad school there was a required course in Linear Algebra and we spent most of the time talking about solving simultaneous equations. If we only knew what was coming at us...
Oh, and you absolutely MUST have a command of statistics (which leaves me hurting) to get anywhere with data analysis and machine learning. I hated statistics!
Introduction to Mathematical Thinking (Standford Coursera) by Devlin -- freeI took that! A real eye-opener, and rather depressing :-(
Discrete Mathematics -- Susanna S. EppLet us know how that goes. I've had a lot of trouble with the more theoretical math classes (and books.) They seem to start by assuming a more theoretical background than I've got. "You probably recognize this result as the Golden Ratio, and can see how that makes sense." (NO!) Sigh.
That is a lot of self study! But if you have the time I am sure it will be rewarding. I just have a couple of suggestions. First, you will certainly need some multivariable calculus for electronics and AI / machine learning, so I would recommend working through those chapters of whichever calculus book to select. If you want to save some effort, you may not need to go deeply into multiple integration (setting up integrals to find volumes of weird 3-D shapes is not needed in electronics), and if you don't plan on learning engineering electromagnetics then you can skip the vector calculus chapter (with the idea that you can go back and learn it if you need it). Second, once you know calculus then you should learn calculus-based probability and statistics. I think it will actually be easier to understand that way, plus it will help reinforce your calculus knowledge.All of AI is based on linear algebra and everybody needs to get a taste of that. Even the simple Digits Recognition problem is working in 784 dimensional space. A wee bit hard to visualize... That problem and its Neural Network solution is the "Hello World of AI".
In grad school there was a required course in Linear Algebra and we spent most of the time talking about solving simultaneous equations. If we only knew what was coming at us...
Oh, and you absolutely MUST have a command of statistics (which leaves me hurting) to get anywhere with data analysis and machine learning. I hated statistics!
I recently learned about OpenAI's ChatGTP and Midjourney/Stable Diffusion, and am blown away by what AI can do now. So I have an interest in it as well as the analog synthesizer electronics. My major was in Computer Science so I'm good with programming and software engineering (as well as database design and programming).
So it looks like Linear Algebra and Probability & Statistics will perhaps be useful for both interests in AI and electronics.
Here's my plan:
1) Algebra & Trigonometery (6e by Blitzer, cheap used copy) -- started on this an am 84 pages in so far, doing all the problems
2) Introduction to Mathematical Thinking (Standford Coursera) by Devlin -- free
3) Calculus I (Stewart textbook, used copies) -- will use OpenStax and Larson as well as Khan as backups
4) Calculus II - Stewart "
5) Elementary Linear Algebra -- Howard Anton
6) Discrete Mathematics -- Susanna S. Epp
7) Probability & Statistics (the college level course with precalculus as prerequisite) -- Anthony Hayter
8 ) Differential Equations -- I dunno which one yet.. will worry about this later
9) Physics I & II (mechanical and electricity & magnetism) - Paul A. Tipler
10) The Art of Electronics
11) DSP
I don't know if Calculus III (multivariable calculus) would be that helpful with AI or electronics.
I imagine all this is going to take me a couple years studying a few hours each day.
We seem to get away with posting math questions in the Beginners forum and since that is at the top of the forum list, it's as good a place as any until the mods complain.
I don't think there is enough interest in math to have a separate forum. Personally, I like the math questions, particularly if they lend themselves to machine solutions. Mesh and nodal equations are especially easy with Octave or MATLAB.
There are an enormous number of good sites on the Internet that provide tutoring. I haven't found a courteous place out in the wild to ask questions. EEVblog would rank very high in courteous responses.
One thing that baffles me is that we can take functions and their derivative or integrate them, but what if we only had a graph?
Say it's a graph of torque of a motor measured on a dyno. I'm sure the computer can give you an area under the curve, but how is it possible to perform this without the computer if the graph isn't some function?
My friend and I were discussing torque/hp. He said the larger the area under a curve, the more torque or whatever. He showed me a graph of one engine where it was basically a wavy horizontal line (say y=3) and then another that was a curve of basically y=x (where x and y began at zero). He said because it appeared the area under the "wavy line" of y=3 is higher than y=x, then it has more area.
My argument was that the y=x could have more area depending on how far you go on the X axis.
To prove my argument, I took the y=3 to be a perfectly horizontal line and y=x to be a perfect right triangle, so I calculated the area of a rectangle versus the area of a right triangle.
Turned out in his example, the rectangle had more area proving him correct, however, I twisted things around a bit and used the same example with different numbers making the triangle have more area to prove that just because there is a horizontal line doesn't mean it will always have more area under the curve than a triangle.
Anyway, my point is, I took a wavy horiztonal line and made it perfectly flat and a curved line and made it a delta x / delta y.
Mathematically it's wrong to do this because I missed (or added) area. So how can I look at a curve and know the area under it without a computer doing the work for me; and without approximating?
Mathematically it's wrong to do this because I missed (or added) area. So how can I look at a curve and know the area under it without a computer doing the work for me; and without approximating?Assuming you mean how to estimate the area, without resorting to approximations:
I generally respond, when can, to some (first year) Calculus questions. Partially because that math catagory often (seemingly) lacks common sense or intuitive approach / description. But, with good teacher it's not so daunting. I was lucky, first couple of college semesters, to have a good Prof.I agree, first year calculus is often approached from odd angles, and can be difficult to grasp because of that. A good prof can approach it from different directions, and adjusts their approach to suit the students best. One is very lucky to have that sort of a prof; I know a few (mostly in physics, though).
One thing that baffles me is that we can take functions and their derivative or integrate them, but what if we only had a graph?You probably won't learn that in a calculus class, at least not in any useful way.
It's a field of computer science, though
You probably learn in calculus class that the derivative is just the slope of the tangent line deltaY/deltaX as deltaX approaches zero, and integration is just a sum of rectangle areas deltaY*deltaX (also as deltaZ approaches zero.)
That reminds me of a calculus exercise I liked, and many are familiar with:
What height and radius minimizes the surface area of a cylindrical container with unit volume, \$V = 1\$?
f(x) = (x**2) + sqrt(1.0 + (2.0 * x))
QuoteYou probably learn in calculus class that the derivative is just the slope of the tangent line deltaY/deltaX as deltaX approaches zero, and integration is just a sum of rectangle areas deltaY*deltaX (also as deltaZ approaches zero.)
I remember first starting calculus and thinking why can't the 'd' in dx/dt just cancel leaving you with x/y.
Of course, the volume of paint required to coat the interior surface is not a physical situation, since paint has a finite thickness and will clog the pipe past a certain small diameter.
Of course, the volume of paint required to coat the interior surface is not a physical situation, since paint has a finite thickness and will clog the pipe past a certain small diameter.
Sure, mess up the beautiful math with reality. Mathematicians live in their own version of reality unencumbered by physical constraints.
You can search the specific topics at khan academy.
...Otherwise you might be better served by boning up on linear algebra, boolean logic and other fields.This is a good suggestion. Sometimes I have to break out the books for trig or some geometry calculation that I should have remembered. Boolean logic might not apply if she is interested in instruments but I think your point is the math follows the tech.
(I plan on doing mostly analog electronics related to music synthesizers.)
I've decided I need to take a step or two backwards before tackling the James Stewart Calculus book I ordered the other day.I'd do neither. Why? Because most of what you will need to know will come back real fast and can be refreshed from a handbook or the electronics text you will use. Beyond all of that a lot of the heavy math is incorporated into various bits of software that you are likely to use.
I can't decided if I should just buy a Precalculus book or buy the combination of a College Algebra book and Trigonometry book.
I am thinking the College Algebra and Trig books because perhaps they'd each be thinner over all (easier to handle) and perhaps more comprehensive.Yes this is a huge problem. But again I think you are focusing too much on what you think you have forgotten. You probably did forget a lot but it is not completely gone, I know that I've forgotten a lot but you can get back into it, by picking up a handbook or a text focused on the electronics. Walk by any engineers desk and you will find all sorts of reference books.
The way I progressed in college back around the early 90's was, I took Intermediate Algebra followed by Trig straight into Calculus. Which was a mistake really. I should of take Precalc or college algebra as I really struggled with Calculus I despite getting an A in it. (I got an A in both Intermediate Algebra and Trig as well.)
What do you think? I also see another flavor of a combined book called "Algebra and Trigonometry" which I thought was Precalculus. LOL.. too many different varieties of Algebra and Trig classes if you ask me.
I was considering the College Algebra book by Kaufman, an older cheaper edition. I don't know which Trig book to get. But I figure I better go through both of them again before tackling Calculus I & II followed by Differential Equations and Linear Algebra. (Guessing I can skip multivariable calculus perhaps.. I only ever recall find volumes of 3D objects as well as line integrals.. don't remember the differentiation part of calc iii.)Again I wouldn't invest in a teaching text book. You have already gone through that and will just need refreshing from time to time. Spending months on getting to 100% mathematically is a big distraction when you can start engineering hardware tomorrow. I'd spend the money on breadboards, power supplies and instrumentation.
Again I wouldn't invest in a teaching text book. You have already gone through that and will just need refreshing from time to time. Spending months on getting to 100% mathematically is a big distraction when you can start engineering hardware tomorrow. I'd spend the money on breadboards, power supplies and instrumentation.
30 years is a long time.You are taking to somebody that is 62 going on to real old age. I've forgotten much from those years from college and high school, worse sometimes I forget where my keys are! >:D >:D >:D