What one fool can do, another can.
October 28, 2019 10:12 AM Subscribe
The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are... Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard.
Originally published in 1910, Silvanus P. Thompson's Calculus Made Easy has never been out of print.
Originally published in 1910, Silvanus P. Thompson's Calculus Made Easy has never been out of print.
I know more than one person who has taught or retaught themselves enough Calc from Silvanus to be ready to learn more and then go on to graduate work in a physical science. And because it’s always in print, you can probably find a really cheap copy.
posted by clew at 10:46 AM on October 28, 2019 [4 favorites]
posted by clew at 10:46 AM on October 28, 2019 [4 favorites]
Since I used wikipedia and other routes to get myself through higher levels of Calc and Differential Equations, because my professors were either useless or spoke English in a Indian-via-French accent (or worse), I can very much appreciate this.
Thanks.
posted by RolandOfEld at 10:47 AM on October 28, 2019 [3 favorites]
Thanks.
posted by RolandOfEld at 10:47 AM on October 28, 2019 [3 favorites]
I just learned more from page 2 than I learned in highschool pre-calc, calc, and university level calc (2 courses) and thermodynamics. Thank you so much!
posted by some chick at 10:57 AM on October 28, 2019 [9 favorites]
posted by some chick at 10:57 AM on October 28, 2019 [9 favorites]
I just learned more from page 2 than I learned in highschool pre-calc, calc, and university level calc (2 courses) and thermodynamics.
My pre-algebra teacher never explained what X was.
But he does. Also I didn't know any of this
ll through the calculus we are dealing with quantities that are growing, and with rates of growth. We classify all quantities into two classes: constants and variables. Those which we regard as of fixed value, and call constants, we generally denote algebraically by letters from the beginning of the alphabet, such as a, b, or c; while those which we consider as capable of growing, or (as mathematicians say) of “varying,” we denote by letters from the end of the alphabet, such as x, y, z, u, v, w, or sometimes t.
posted by The_Vegetables at 11:08 AM on October 28, 2019 [19 favorites]
My pre-algebra teacher never explained what X was.
But he does. Also I didn't know any of this
ll through the calculus we are dealing with quantities that are growing, and with rates of growth. We classify all quantities into two classes: constants and variables. Those which we regard as of fixed value, and call constants, we generally denote algebraically by letters from the beginning of the alphabet, such as a, b, or c; while those which we consider as capable of growing, or (as mathematicians say) of “varying,” we denote by letters from the end of the alphabet, such as x, y, z, u, v, w, or sometimes t.
posted by The_Vegetables at 11:08 AM on October 28, 2019 [19 favorites]
I just started reading Naomi Novik’s Temeraire books (the Napoleonic Wars, but with dragons) and the tone of this feels like an in-world extension. Love it!
posted by sixswitch at 11:09 AM on October 28, 2019 [5 favorites]
posted by sixswitch at 11:09 AM on October 28, 2019 [5 favorites]
"I. TO DELIVER YOU FROM THE PRELIMINARY TERRORS"
"∫ which is merely a long S"
Yes, this man is definitely speaking my language.
posted by MCMikeNamara at 11:44 AM on October 28, 2019 [32 favorites]
"∫ which is merely a long S"
Yes, this man is definitely speaking my language.
posted by MCMikeNamara at 11:44 AM on October 28, 2019 [32 favorites]
Neat. I'm going to hang onto this as a supplemental reading assignment for future classes. It does assume some knowledge of things like "the sum of," and the use of squares rather than lines in II seems like an odd choice. . . but, it looks quite valuable and thoughtfully written. Thanks!
posted by eotvos at 11:56 AM on October 28, 2019 [2 favorites]
posted by eotvos at 11:56 AM on October 28, 2019 [2 favorites]
The way my brain works is by being able to say things and being able to visualize things. I am often brought to a screeching halt by symbols or letters from long dead languages that I cannot pronounce without reference, these days easier to find but always tedious and derailing.
posted by Pembquist at 11:57 AM on October 28, 2019 [3 favorites]
posted by Pembquist at 11:57 AM on October 28, 2019 [3 favorites]
The "d" id "dx" stands for "delta" does it not? As in, change? So dx is the change in x?
posted by grumpybear69 at 12:48 PM on October 28, 2019 [2 favorites]
posted by grumpybear69 at 12:48 PM on October 28, 2019 [2 favorites]
This is an interesting enough text and if a person finds it valuable, fantastic! But it's a bit out of date, and just plain wrong pretty much from the get-go. This probably isn't the right place to enumerate that, exactly, but modern calculus pedagogy is much better about the distinction between dx and Δx, for example. The former definitely doesn't have a finite value that you can point to, and the latter does. Apart, the terms dy and dx are pretty tough to pin down; the latter is defined in terms of the former as dy = f'(x) dx. They are called differentials and are actually pretty weird and unintuitive! A relevant previous post on the blue illustrates this nicely, in that there is a vibrant discussion about how to even use differentials to denote the second derivative of a function. Here is that previous post.
All of Thompson's early chapters would be helped immensely by replacing his dxs and dys by Δxs and Δys. It's a great intuitive tool and we use it all the time when we teach calculus: What happens when this value (Δx) gets really really really small ? But that's not the definition of the derivative, it's just sort of the best job we can do of understanding it when we're starting out. When Thompson says "Here's all the derivative is", he's lying, a little bit.
It's worth asking ourselves, at least, whether lying to our students a little bit like this is in fact the right approach from time to time. The reader can guess where I come down on that question in this instance, though.
I haven't read the whole book, just the first few chapters over the last hour or so, hence the harping on this one example. I'm probably being too hard on it, to be honest, because my default mode is analytical. But teaching mathematics is what I do, so I've thought about these questions a lot! And so have a lot of other people in the intervening 100 years since Calculus Made Easy was published. I wonder if Thompson would call them all fools?
To those of you lamenting how little you learned in class, and by contrast extolling this textbook as lucid and so much better than the ones you used? I seriously doubt it. More likely, your approach in reading this textbook is different, your state of mind is different, and you have more years of wisdom and context to aid you as you read. The current standard is Stewart, and it has been for many years. He has all kinds of these well-written, intuitive discussions. Seriously, they're all over the book-- I just opened my copy to a random page, and there he's talking about Zeno's Paradoxes and a lovely intuitive description of series summation.
I think the mystique of Thompson comes from his iconoclasm. "Those guys are just doing things to confuse you, I'll show you what really matters". It's appealing. It's the opposite of the common wisdom around calculus. And I'll be happy to help fix any misconceptions that come from a close study of the book, because any math is still better than no math.
posted by dbx at 12:48 PM on October 28, 2019 [48 favorites]
All of Thompson's early chapters would be helped immensely by replacing his dxs and dys by Δxs and Δys. It's a great intuitive tool and we use it all the time when we teach calculus: What happens when this value (Δx) gets really really really small ? But that's not the definition of the derivative, it's just sort of the best job we can do of understanding it when we're starting out. When Thompson says "Here's all the derivative is", he's lying, a little bit.
It's worth asking ourselves, at least, whether lying to our students a little bit like this is in fact the right approach from time to time. The reader can guess where I come down on that question in this instance, though.
I haven't read the whole book, just the first few chapters over the last hour or so, hence the harping on this one example. I'm probably being too hard on it, to be honest, because my default mode is analytical. But teaching mathematics is what I do, so I've thought about these questions a lot! And so have a lot of other people in the intervening 100 years since Calculus Made Easy was published. I wonder if Thompson would call them all fools?
To those of you lamenting how little you learned in class, and by contrast extolling this textbook as lucid and so much better than the ones you used? I seriously doubt it. More likely, your approach in reading this textbook is different, your state of mind is different, and you have more years of wisdom and context to aid you as you read. The current standard is Stewart, and it has been for many years. He has all kinds of these well-written, intuitive discussions. Seriously, they're all over the book-- I just opened my copy to a random page, and there he's talking about Zeno's Paradoxes and a lovely intuitive description of series summation.
I think the mystique of Thompson comes from his iconoclasm. "Those guys are just doing things to confuse you, I'll show you what really matters". It's appealing. It's the opposite of the common wisdom around calculus. And I'll be happy to help fix any misconceptions that come from a close study of the book, because any math is still better than no math.
posted by dbx at 12:48 PM on October 28, 2019 [48 favorites]
The previous comment, written by a little bit of bx, is harsher than my opinion. I think this is a good read --- but I haven't looked at a modern intro calculus textbook for a long time now, and I didn't struggle with it the way some of my friends did.
posted by fantabulous timewaster at 12:56 PM on October 28, 2019 [13 favorites]
posted by fantabulous timewaster at 12:56 PM on October 28, 2019 [13 favorites]
From Chapter 18
posted by jedicus at 1:07 PM on October 28, 2019 [6 favorites]
No one, even to-day, is able to find the general integral of the expression,Out of curiosity I threw that into Wolfram Alpha and got a doozy of an answer involving the square root of π and the error function.
dy/dx = a−x2,
because a−x2 has never yet been found to result from differentiating anything else.
posted by jedicus at 1:07 PM on October 28, 2019 [6 favorites]
And the error function is itself written in terms of the integral of something to the power of -x², so Thompson's answer is true. The best we can do with that general integral is define some new function and say "this is the answer by definition".
posted by wanderingmind at 1:10 PM on October 28, 2019 [4 favorites]
posted by wanderingmind at 1:10 PM on October 28, 2019 [4 favorites]
I'm a little surprised they didn't correct the trillionth fraction labelled as a billionth. A millionth of a millionth is a trillionth.
posted by sydnius at 1:16 PM on October 28, 2019
posted by sydnius at 1:16 PM on October 28, 2019
The only delta I understand is delta V, and when I run out of it, something horrible is about to happen to a Kerbal.
posted by quillbreaker at 1:17 PM on October 28, 2019 [10 favorites]
posted by quillbreaker at 1:17 PM on October 28, 2019 [10 favorites]
I remember hearing about this book when I was an undergrad -- whoa, it's been around since 1910, must be relevatory! -- and I was completely underwhelmed when I got my hands on it. If I recall, it was trying to be intuitive but it still was, fundamentally, a textbook. From my experience, that format, in and of itself, can throw up a road block for non-math people trying to gain a foothold with the subject. It felt like I was reading "Calculus framed differently, but still for math folks."
Contrast with my experience when I discovered Professor E-McSquared's Calculus Primer. My brain was thinking WAAAY differently about calculus! And not the deltas and epsilons, sure that was the same. Just... I was engaging with it all in a wholly different, creative manner.
The medium is the message.
posted by Theophrastus Johnson at 1:19 PM on October 28, 2019 [7 favorites]
Contrast with my experience when I discovered Professor E-McSquared's Calculus Primer. My brain was thinking WAAAY differently about calculus! And not the deltas and epsilons, sure that was the same. Just... I was engaging with it all in a wholly different, creative manner.
The medium is the message.
posted by Theophrastus Johnson at 1:19 PM on October 28, 2019 [7 favorites]
Interesting comments.
As someone who has often shared a hallway with math faculty but spent my training mostly trying to avoid taking math classes as much as possible after the first few, the distinctions raised seem really subtle. Maybe this is a "calculus for people who don't want to become mathematicians" text. (Which, I'd claim is a non-trivial fraction of the people who need to use calculus on a daily basis.) Still, I am very intrigued by the criticism and happy to read it.
posted by eotvos at 1:19 PM on October 28, 2019 [4 favorites]
As someone who has often shared a hallway with math faculty but spent my training mostly trying to avoid taking math classes as much as possible after the first few, the distinctions raised seem really subtle. Maybe this is a "calculus for people who don't want to become mathematicians" text. (Which, I'd claim is a non-trivial fraction of the people who need to use calculus on a daily basis.) Still, I am very intrigued by the criticism and happy to read it.
posted by eotvos at 1:19 PM on October 28, 2019 [4 favorites]
Yeah, any solution to integrating the exponential of -x2 in terms of the error function is fundamentally circular, because erf is simply defined in terms of that integral.
As I understand it, it's been proven that the error function cannot be written purely in terms of the more conventional transcendental functions, but I'm not very strong in analysis and I'm not sure I've ever been exposed to the proof.
posted by jackbishop at 1:23 PM on October 28, 2019 [1 favorite]
As I understand it, it's been proven that the error function cannot be written purely in terms of the more conventional transcendental functions, but I'm not very strong in analysis and I'm not sure I've ever been exposed to the proof.
posted by jackbishop at 1:23 PM on October 28, 2019 [1 favorite]
This is great! And I realized that I actually have a 1944 copy on my bookshelf (which clearly I had not opened). I'm teaching Calc I this semester, so it's especially entertaining. The decimal points in the middle of the numbers (at least in my edition), as in "However big you make n, the value of this expression grow nearer and nearer to the figure 2•71828... a number never to be forgotten." (Interesting, he calls that constant "epsilon", but I've never seen it called anything but e.)
posted by leahwrenn at 1:32 PM on October 28, 2019 [1 favorite]
posted by leahwrenn at 1:32 PM on October 28, 2019 [1 favorite]
for what it's worth: I hadn't heard of this book before yesterday, when I found a 1950s edition (inscribed to Joseph, from Dad) at an antique store. I was instantly charmed by PRELIMINARY TERRORS et cetera. Please do not design any life-critical systems using information from this Edwardian-era textbook.
For a more modern intro I really like the Essence of Calculus video series by 3blue1brown.
posted by theodolite at 1:38 PM on October 28, 2019 [8 favorites]
For a more modern intro I really like the Essence of Calculus video series by 3blue1brown.
posted by theodolite at 1:38 PM on October 28, 2019 [8 favorites]
Metafilter: they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are
posted by bleep at 1:49 PM on October 28, 2019
posted by bleep at 1:49 PM on October 28, 2019
I also strongly endorse The Joy of X, which isn't solely focused on calculus but which is at least more modern and similarly wry and plainspoken about the ethereal absurdities of complicated math.
posted by Scattercat at 1:50 PM on October 28, 2019 [4 favorites]
posted by Scattercat at 1:50 PM on October 28, 2019 [4 favorites]
he's lying, a little bit.
math and science instruction in a nutshell. Every year is a "you know that thing we told you was true? we were lying, It's a little more complex than that"
posted by Dr. Twist at 2:10 PM on October 28, 2019 [22 favorites]
math and science instruction in a nutshell. Every year is a "you know that thing we told you was true? we were lying, It's a little more complex than that"
posted by Dr. Twist at 2:10 PM on October 28, 2019 [22 favorites]
I'm a little surprised they didn't correct the trillionth fraction labelled as a billionth. A millionth of a millionth is a trillionth.
It's not an error; this book is so old it uses the long scale.
posted by aws17576 at 2:18 PM on October 28, 2019 [11 favorites]
It's not an error; this book is so old it uses the long scale.
posted by aws17576 at 2:18 PM on October 28, 2019 [11 favorites]
I'm a little surprised they didn't correct the trillionth fraction labelled as a billionth. A millionth of a millionth is a trillionth.
That depends:
That depends:
1,000,000,000,000, i.e. one million million, or 1012 (ten to the twelfth power). This is the historic definition of a billion in British English.posted by clawsoon at 2:20 PM on October 28, 2019 [5 favorites]
Other countries such as the United States use the word billion (or words cognate to it) to denote the billions as 1,000,000,000. For details, see Long and short scales – Current usage.
The long scale is cleaner in the Latin+math sense: 106n, where n is the number indicated by the prefix (bi=>2, tri=>3, quad=>4).
The short scale has an off-by-one problem. It's much more common where I live, though, probably due to American influence.
posted by clawsoon at 2:25 PM on October 28, 2019 [8 favorites]
The short scale has an off-by-one problem. It's much more common where I live, though, probably due to American influence.
posted by clawsoon at 2:25 PM on October 28, 2019 [8 favorites]
Learn You A Calculus For Great Good.
posted by acb at 3:28 PM on October 28, 2019 [2 favorites]
posted by acb at 3:28 PM on October 28, 2019 [2 favorites]
One of the standard formulations of the Normal distribution is essentially 'a-x2':
φ(x) = e(-πx2)
When we integrate the Normal distribution, we are basically adding up the probabilities that the value of a strictly random variable will be between two values, or will be less than or greater than a certain value.
I'd like to say the fact that we have no simple formula for that integral, which means we can't calculate those probabilities in a straightforward way, has something to do with the intractability of randomness.
But I can't quite see how to do it.
posted by jamjam at 4:02 PM on October 28, 2019
φ(x) = e(-πx2)
When we integrate the Normal distribution, we are basically adding up the probabilities that the value of a strictly random variable will be between two values, or will be less than or greater than a certain value.
I'd like to say the fact that we have no simple formula for that integral, which means we can't calculate those probabilities in a straightforward way, has something to do with the intractability of randomness.
But I can't quite see how to do it.
posted by jamjam at 4:02 PM on October 28, 2019
the distinction between dx and Δx
So if I'm confused about this distinction, will it make me bad at using calculus to solve specific problems?
Or will it just get me in trouble if I ever take advanced math / if an evil math teacher ever makes me try to write proofs for any of this stuff / if I want to understand the history of calculus / stuff like that?
posted by nebulawindphone at 4:11 PM on October 28, 2019 [3 favorites]
So if I'm confused about this distinction, will it make me bad at using calculus to solve specific problems?
Or will it just get me in trouble if I ever take advanced math / if an evil math teacher ever makes me try to write proofs for any of this stuff / if I want to understand the history of calculus / stuff like that?
posted by nebulawindphone at 4:11 PM on October 28, 2019 [3 favorites]
Isn't the distinction between dx and Δx just the distinction between Newton's and Leibniz' (independently discovered) notations?
posted by acb at 4:41 PM on October 28, 2019
posted by acb at 4:41 PM on October 28, 2019
So if I'm confused about this distinction, will it make me bad at using calculus to solve specific problems?
Nah. That’s why teachers and students get away with the hand waving.
Isn't the distinction between dx and Δx just
No, it’s more what dbx said above. Δx is a finite difference of x values, dx is more of a symbolic monolith, but the answer depends a bit upon why you’re asking and what’s the underlying framework and context.
posted by SaltySalticid at 4:46 PM on October 28, 2019
Nah. That’s why teachers and students get away with the hand waving.
Isn't the distinction between dx and Δx just
No, it’s more what dbx said above. Δx is a finite difference of x values, dx is more of a symbolic monolith, but the answer depends a bit upon why you’re asking and what’s the underlying framework and context.
posted by SaltySalticid at 4:46 PM on October 28, 2019
"I'd like to say the fact that we have no simple formula for that integral, which means we can't calculate those probabilities in a straightforward way, has something to do with the intractability of randomness."
Nah, it's just an integral that doesn't have a closed form. There are LOTS of these, and it has nothing to do with randomness.
The derivative is the 'nice' operation, with lots of nice properties: like if two functions f and g are differentiable, then so are their composition f(g(x)) and product f(x)*g(x).
The integral is more like an 'inverse' operation, which sometimes has a nice answer, and other times requires lots of work or just falling back to numerical approximation...
Aaaand just to make things even worse, integrals are often EASIER in the 'real world,' where noise exists. Integrals count how things accumulate, often over time. Measuring accumulations of stuff is easy, and (unbiased) noise in the process generally goes away as you measure more of it. On the other hand, measured derivatives tend to be noisy as hell, because there's no accumulation to wash out the noise.
posted by kaibutsu at 4:52 PM on October 28, 2019 [8 favorites]
Nah, it's just an integral that doesn't have a closed form. There are LOTS of these, and it has nothing to do with randomness.
The derivative is the 'nice' operation, with lots of nice properties: like if two functions f and g are differentiable, then so are their composition f(g(x)) and product f(x)*g(x).
The integral is more like an 'inverse' operation, which sometimes has a nice answer, and other times requires lots of work or just falling back to numerical approximation...
Aaaand just to make things even worse, integrals are often EASIER in the 'real world,' where noise exists. Integrals count how things accumulate, often over time. Measuring accumulations of stuff is easy, and (unbiased) noise in the process generally goes away as you measure more of it. On the other hand, measured derivatives tend to be noisy as hell, because there's no accumulation to wash out the noise.
posted by kaibutsu at 4:52 PM on October 28, 2019 [8 favorites]
For some reason I think this is the right place to drop these.
Physics Major vs Math Class
Mathematicians vs. Physics Classes be like...
posted by zengargoyle at 5:17 PM on October 28, 2019 [8 favorites]
Physics Major vs Math Class
Mathematicians vs. Physics Classes be like...
posted by zengargoyle at 5:17 PM on October 28, 2019 [8 favorites]
I had pretty good teachers for calc (for which, from this thread, it seems I should be grateful. But biostatistics, is there a sequel on biostatistics?????? I just recently got a full grasp on the chi square test and can’t believe how straightforward it really is....I’m convinced it’s just the preliminary terrors keeping me from more deft use of biostatistics in everyday work....
posted by Tandem Affinity at 5:39 PM on October 28, 2019 [3 favorites]
posted by Tandem Affinity at 5:39 PM on October 28, 2019 [3 favorites]
Coming from a computer science background, I feel like math would make so much more sense if we used full words for function and variable names, rather than single characters. Sure, it might be more compact to use obscure notation, but it’s also much more arbitrary.
Maybe we should move to teaching math as part of an early introduction to computing, where everything is represented as functions, and people can play around with graphs, simulations, etc (as their own individual learning style dictates) until each concept makes intuitive sense. Why should math live on pencil and paper (and expensive graphing calculators) anymore?
posted by mantecol at 6:34 PM on October 28, 2019 [4 favorites]
Maybe we should move to teaching math as part of an early introduction to computing, where everything is represented as functions, and people can play around with graphs, simulations, etc (as their own individual learning style dictates) until each concept makes intuitive sense. Why should math live on pencil and paper (and expensive graphing calculators) anymore?
posted by mantecol at 6:34 PM on October 28, 2019 [4 favorites]
I took a several year break during my undergraduate education (to go and join the work force for a time) and because I didn't really plan for things to go that way I did not complete the calculus sequence required by my engineering program before the start of my hiatus.
I have a decent knack for discrete math but never quite internalized calc well enough to retain it through multiple years of minimal or no use, but with the help of Thompson's book to bring me back up to where I had left off (and then some..) I managed to finish the calculus series required by my program and all was well. Thanks, Silvanus, for saving me from having to repeat classes..
I probably should acquire another copy against need -- by this point it has been so many years since I used calc that my deriver's license [sorry!] should definitely be revoked.
posted by Nerd of the North at 8:25 PM on October 28, 2019
I have a decent knack for discrete math but never quite internalized calc well enough to retain it through multiple years of minimal or no use, but with the help of Thompson's book to bring me back up to where I had left off (and then some..) I managed to finish the calculus series required by my program and all was well. Thanks, Silvanus, for saving me from having to repeat classes..
I probably should acquire another copy against need -- by this point it has been so many years since I used calc that my deriver's license [sorry!] should definitely be revoked.
posted by Nerd of the North at 8:25 PM on October 28, 2019
the distinction between dx and Δx
So if I'm confused about this distinction, will it make me bad at using calculus to solve specific problems?
Δx is a finite expression, something you could give to a computer to get a numerical approximation of something.
dx is when you ask "what if I shrank Δx down to an infinitesimal and then added up those infinitesimals? Could I get a closed form analytical expression for what I'm trying to calculate?"
Basically, Δx means you're not using calculus at all. (Which depending on the context, is okay, or is your only actual choice.)
posted by ocschwar at 8:58 PM on October 28, 2019
So if I'm confused about this distinction, will it make me bad at using calculus to solve specific problems?
Δx is a finite expression, something you could give to a computer to get a numerical approximation of something.
dx is when you ask "what if I shrank Δx down to an infinitesimal and then added up those infinitesimals? Could I get a closed form analytical expression for what I'm trying to calculate?"
Basically, Δx means you're not using calculus at all. (Which depending on the context, is okay, or is your only actual choice.)
posted by ocschwar at 8:58 PM on October 28, 2019
I was reading this while my awesome mathematician wife was at a colloquium this evening, and told her about it when she got home, expecting her to be well familiar with it, but she wasn't! And her passion is to get people over their hangups about math (especially higher math) being something only for those with the "talent for it." (She was herself planning to become a drama teacher before falling in love with math thanks to a friend taking her out to Harvey Mudd when she was in high school, and the wool fell from her eyes seeing that it was highly collaborative rather than just the lone genius with his blackboard working alone.) She basically jumped into Math at 16 or 17 with no prior feeling that she was someone who could "do" it, and loves anything that seeks to demystify it and get people over their fear and/or animosity towards it.
This might be too early-twentieth-century British to make much of a dent on her primarily first-generation-immigrant university students, but I'm so happy for it to be another arrow in her quiver (and she does have to teach Calc from time to time, though that's not her primary field of research.)
posted by Navelgazer at 9:12 PM on October 28, 2019 [4 favorites]
This might be too early-twentieth-century British to make much of a dent on her primarily first-generation-immigrant university students, but I'm so happy for it to be another arrow in her quiver (and she does have to teach Calc from time to time, though that's not her primary field of research.)
posted by Navelgazer at 9:12 PM on October 28, 2019 [4 favorites]
I love the lighthearted, humorous tone. Pretty rare, I think, for a textbook of that period.
Just downloaded the PDF version to read offline on my tablet.
posted by ZenMasterThis at 5:22 AM on October 29, 2019 [1 favorite]
Just downloaded the PDF version to read offline on my tablet.
posted by ZenMasterThis at 5:22 AM on October 29, 2019 [1 favorite]
I'm going to chime in with dbx on this one — if you think this is good, it's either because you're reading it with a very different, more receptive mindset than you had when you were in school, or maybe your idea of what math education is is informed more by popular culture than the current actual state of math education.
To me, this feels very similar to people posting history-memes on imgur no history book will teach you about how the Pilgrims weren't actually the good guys! and what you didn't learn in school: the USA displaced the Bikini Islanders for nuclear testing@!@!@!. I don't know what school you went to and didn't pay attention in, but things like that are central to modern (say, post 1970s) US public high school history curricula, just as many of these pedagogical methods are bog-standard in math education.
Also I think Common Core is great and aligns exactly with how I (a 41 year old who predates it) do mental arithmetic, so.
posted by dmd at 5:32 AM on October 29, 2019 [2 favorites]
To me, this feels very similar to people posting history-memes on imgur no history book will teach you about how the Pilgrims weren't actually the good guys! and what you didn't learn in school: the USA displaced the Bikini Islanders for nuclear testing@!@!@!. I don't know what school you went to and didn't pay attention in, but things like that are central to modern (say, post 1970s) US public high school history curricula, just as many of these pedagogical methods are bog-standard in math education.
Also I think Common Core is great and aligns exactly with how I (a 41 year old who predates it) do mental arithmetic, so.
posted by dmd at 5:32 AM on October 29, 2019 [2 favorites]
On the subject of fun math books, what do you guys think of DIV, Grad, Curl and All That by Schey?
posted by bdc34 at 6:16 AM on October 29, 2019
posted by bdc34 at 6:16 AM on October 29, 2019
Does anyone here have an opinion on Apostol? I'm told that it also is slightly unusual in its approach, even among books that teach proofs rather than only computation, but I've only ever taken calculus the once so I wouldn't know.
posted by meaty shoe puppet at 6:43 AM on October 29, 2019
posted by meaty shoe puppet at 6:43 AM on October 29, 2019
I'm going to chime in with dbx on this one — if you think this is good, it's either because you're reading it with a very different, more receptive mindset than you had when you were in school
How about option 3, where the teachers who taught math (most of mine were men) were terrible [my college calc teacher was the best and always ended his equations writing really small in the bottom corner of the blackboard], and if you say it's gotten better, well, my kids math homework (they don't really have books yet so this is from some online printed packet I guess) in 3rd grade regularly has for multiple choice like:
a. 3
b. 4
c. none of the above
d. all of the above
And her homework regularly has 'stretch' questions that aren't covered anywhere in the week's material, just like mine always did.
posted by The_Vegetables at 6:52 AM on October 29, 2019
Does anyone here have an opinion on Apostol?
I took my first "real" math class in college out of Apostol's two books, and loved them---I think the unorthodox but useful thing he does is nicely integrate linear algebra into the exposition of multivariable calculus.
posted by rishabguha at 8:01 AM on October 29, 2019 [1 favorite]
I took my first "real" math class in college out of Apostol's two books, and loved them---I think the unorthodox but useful thing he does is nicely integrate linear algebra into the exposition of multivariable calculus.
posted by rishabguha at 8:01 AM on October 29, 2019 [1 favorite]
Respectfully The_Vegetables, I think what you're describing is a difference in mindset. The quality of your teacher has no bearing on the quality of your textbook.
I'm with you on the yuckiness of multiple choice questions, though. At any stage of mathematics. Purely a tool to make assessment easier, which is basically guaranteed to make it less effective.
posted by dbx at 8:07 AM on October 29, 2019
I'm with you on the yuckiness of multiple choice questions, though. At any stage of mathematics. Purely a tool to make assessment easier, which is basically guaranteed to make it less effective.
posted by dbx at 8:07 AM on October 29, 2019
The current standard is Stewart,
Which book by Stewart? Googling shows several. (Or rather: If Stewart is the good place to start, for someone who's mathematically inclined but knows absolutely nothing of calculus, which book is that?)
I am a person who adored math in school, didn't have access to really good math classes in rural high schools, and then got to college and was broke and dropped out of pre-calc because I couldn't afford the textbook. I like math. I like to think I'd enjoy calculus, but not having a starting point has dissuaded me from trying. (Also, I have no particular use for calculus, other than the ability to follow the occasional math joke on the internet. It's a field of study that's always been, "someday when I can afford to go back to school, I should look into it. Looks fun.")
A book that starts with "dx means 'a little bit of x'" makes sense to me, and I'm not going to believe that's the exact, literal, always definition - but it's enough to start reading the equations, enough to maybe start absorbing some of the contents. More, it's an approach I could learn on my own, without a teacher or guide, which I don't have. Most math-based Youtube channels start with the assumption that you can follow the jargon.
posted by ErisLordFreedom at 10:21 AM on October 29, 2019
Which book by Stewart? Googling shows several. (Or rather: If Stewart is the good place to start, for someone who's mathematically inclined but knows absolutely nothing of calculus, which book is that?)
I am a person who adored math in school, didn't have access to really good math classes in rural high schools, and then got to college and was broke and dropped out of pre-calc because I couldn't afford the textbook. I like math. I like to think I'd enjoy calculus, but not having a starting point has dissuaded me from trying. (Also, I have no particular use for calculus, other than the ability to follow the occasional math joke on the internet. It's a field of study that's always been, "someday when I can afford to go back to school, I should look into it. Looks fun.")
A book that starts with "dx means 'a little bit of x'" makes sense to me, and I'm not going to believe that's the exact, literal, always definition - but it's enough to start reading the equations, enough to maybe start absorbing some of the contents. More, it's an approach I could learn on my own, without a teacher or guide, which I don't have. Most math-based Youtube channels start with the assumption that you can follow the jargon.
posted by ErisLordFreedom at 10:21 AM on October 29, 2019
MetaFilter: This probably isn't the right place to enumerate that, exactly, but
posted by tonycpsu at 11:54 AM on October 29, 2019 [2 favorites]
posted by tonycpsu at 11:54 AM on October 29, 2019 [2 favorites]
The "d" id "dx" stands for "delta" does it not? As in, change? So dx is the change in x?
d is supposed to suggest delta, but as others have said above, dx is not the same idea as Δx.
Δx just means some arbitrarily small change in x, but dx is something like the grin on the Cheshire Cat: it's a placeholder for where the Δx would go if the result you were after was whatever an expression containing Δx would approach if you worked it out over and over again, making Δx smaller each time.
So after chewing on that for a bit you'd probably wonder why we need to bother with all that. Couldn't we just replace Δx with zero everywhere, and call it Job Done and go home?
Which is a good question and a good thought and would obviously save work if it worked, but it runs into trouble when Δx turns up as something you're trying to divide a thing by, which it almost always does when you're trying to compute rates of change. You can't divide things by zero.
So if, for example, we're trying to get an expression for the place where
((x + Δx)2 - x2) / Δx
would be headed if we worked it out over and over and over making Δx smaller every time, we can reason it out like this:
Expanding the (x + Δx)2 part gets us (x2 + 2xΔx + (Δx)2). But the top line of our fraction gets x2 subtracted from the bit we just expanded, so what we're left with overall comes out to
(2xΔx + (Δx)2) / Δx
We can split that into
2xΔx / Δx + (Δx)2 / Δx
and because we know Δx, though deliberately very small, is not zero, it's legitimate to divide through by it and cancel it out, which yields
2x + Δx
and it's pretty obvious now that as Δx gets smaller and smaller, this is going to get closer and closer to 2x.
And we summarize that result by writing
((x + dx)2 - x2) / dx = 2x
which is a proper equality, not some kind of half-assed approximation, which would not be correct for any value of Δx put in dx's place. Because if Δx is nonzero we always get
((x + Δx)2 - x2) / Δx = 2x + Δx
which has that pesky Δx hanging on like a dingleberry at the end; but if Δx is zero we're not allowed to divide by it and our derivation doesn't work. Writing an equation with dx in it instead of Δx is essentially just shorthand for "we worked this out legitimately, honest, no cheating or nothing, and this result is the limit as Δx goes to zero".
posted by flabdablet at 11:56 AM on October 29, 2019 [6 favorites]
d is supposed to suggest delta, but as others have said above, dx is not the same idea as Δx.
Δx just means some arbitrarily small change in x, but dx is something like the grin on the Cheshire Cat: it's a placeholder for where the Δx would go if the result you were after was whatever an expression containing Δx would approach if you worked it out over and over again, making Δx smaller each time.
So after chewing on that for a bit you'd probably wonder why we need to bother with all that. Couldn't we just replace Δx with zero everywhere, and call it Job Done and go home?
Which is a good question and a good thought and would obviously save work if it worked, but it runs into trouble when Δx turns up as something you're trying to divide a thing by, which it almost always does when you're trying to compute rates of change. You can't divide things by zero.
So if, for example, we're trying to get an expression for the place where
((x + Δx)2 - x2) / Δx
would be headed if we worked it out over and over and over making Δx smaller every time, we can reason it out like this:
Expanding the (x + Δx)2 part gets us (x2 + 2xΔx + (Δx)2). But the top line of our fraction gets x2 subtracted from the bit we just expanded, so what we're left with overall comes out to
(2xΔx + (Δx)2) / Δx
We can split that into
2xΔx / Δx + (Δx)2 / Δx
and because we know Δx, though deliberately very small, is not zero, it's legitimate to divide through by it and cancel it out, which yields
2x + Δx
and it's pretty obvious now that as Δx gets smaller and smaller, this is going to get closer and closer to 2x.
And we summarize that result by writing
((x + dx)2 - x2) / dx = 2x
which is a proper equality, not some kind of half-assed approximation, which would not be correct for any value of Δx put in dx's place. Because if Δx is nonzero we always get
((x + Δx)2 - x2) / Δx = 2x + Δx
which has that pesky Δx hanging on like a dingleberry at the end; but if Δx is zero we're not allowed to divide by it and our derivation doesn't work. Writing an equation with dx in it instead of Δx is essentially just shorthand for "we worked this out legitimately, honest, no cheating or nothing, and this result is the limit as Δx goes to zero".
posted by flabdablet at 11:56 AM on October 29, 2019 [6 favorites]
As for the posted book: yeah, nicely folksy and approachable as mathematics texts go, but not a patch on the best computer programming textbook ever written, which is of course A FORTRAN Coloring Book by Dr. Kaufman.
posted by flabdablet at 12:08 PM on October 29, 2019
posted by flabdablet at 12:08 PM on October 29, 2019
She was herself planning to become a drama teacher before falling in love with math thanks to a friend taking her out to Harvey Mudd when she was in high school, and the wool fell from her eyes seeing that it was highly collaborative rather than just the lone genius with his blackboard working alone.
I LOVED how collaborative Mudd was
posted by flaterik at 2:26 PM on October 29, 2019
I LOVED how collaborative Mudd was
posted by flaterik at 2:26 PM on October 29, 2019
This book seems to spend a lot of time on intuition, which in my experience did not come up a whole lot in my calculus classes. Intuition is really nice, and if you go to seminar talks by mathematicians, they will be all about intuition, because it does work as a way to get to the essence of what's going on without having to spend a ton of time. And maybe not so surprisingly, a lot of talks are colloquial in the way that this book is written also.
I'm not a mathematician, but my impression is that a lot of the issues with relying too much on intuition is that if you're not careful, you can be wrong. Of course, nothing in this book is wrong. But it is possible that if your only exposure to calculus is this book, you can come to various conclusions about calculus that are wrong. I do think there is an issue where, it is very easy to get the impression that intuition plays no role in math when that is not the case at all, and this book could do a good job of showing that.
posted by chernoffhoeffding at 3:42 PM on October 29, 2019
I'm not a mathematician, but my impression is that a lot of the issues with relying too much on intuition is that if you're not careful, you can be wrong. Of course, nothing in this book is wrong. But it is possible that if your only exposure to calculus is this book, you can come to various conclusions about calculus that are wrong. I do think there is an issue where, it is very easy to get the impression that intuition plays no role in math when that is not the case at all, and this book could do a good job of showing that.
posted by chernoffhoeffding at 3:42 PM on October 29, 2019
a lot of the issues with relying too much on intuition is that if you're not careful, you can be wrong.
Yep. Which is why the right way to employ intuition is to use it to come up with plausible-looking stuff far faster than you could do any other way, then check that shit with your powers of reason. That way, you give your intuition (which you can think of as a vast and mysterious machine-learning engine) the feedback it needs in order to train up in useful directions.
If you just give your intuition free rein, it's going to come up with all kinds of fabulous epiphanies that feel absolutely wonderful but increasing amounts of which will just be garbage; you can eventually end up at Time Cube, or perhaps arrested for getting naked at Changi Airport depending on your predilections.
posted by flabdablet at 6:32 AM on October 30, 2019 [1 favorite]
Yep. Which is why the right way to employ intuition is to use it to come up with plausible-looking stuff far faster than you could do any other way, then check that shit with your powers of reason. That way, you give your intuition (which you can think of as a vast and mysterious machine-learning engine) the feedback it needs in order to train up in useful directions.
If you just give your intuition free rein, it's going to come up with all kinds of fabulous epiphanies that feel absolutely wonderful but increasing amounts of which will just be garbage; you can eventually end up at Time Cube, or perhaps arrested for getting naked at Changi Airport depending on your predilections.
posted by flabdablet at 6:32 AM on October 30, 2019 [1 favorite]
Cool Navelgazer. I do think any fool (well anybody) can get into the math if they find that one hint of a thing that makes it interesting. It might just be having something explained a bit differently once or twice to get over the hump. I learned calculus by learning it with physics, not like taking physics and calculus classes at the same time, but physics-using-calculus which made everything a bit more concrete and shall we say fun and interesting.
Then my senior year of high school when I was supposed to be taking the normal physics class and the calculus class... well, the physics teacher was terrified of my solutions and the calculus teacher sent me on errands and let me sleep in class. At least until I was asked to tutor an underclassman in algebra because we sorta knew each other via sports-team. It took like a couple of months of a couple of hours a week to get her amused and interested and passing and me getting laid-off. From what I heard later, she went on to get a masters in math. All student surpasses the teacher like. Maybe the fun and interesting has to be tailored to the individual a bit, but that's beside the point. I don't doubt anybody could get pretty far in math if they found that point of view that engaged them and made it not a chore but something enjoyable.
posted by zengargoyle at 3:08 PM on November 2, 2019
Then my senior year of high school when I was supposed to be taking the normal physics class and the calculus class... well, the physics teacher was terrified of my solutions and the calculus teacher sent me on errands and let me sleep in class. At least until I was asked to tutor an underclassman in algebra because we sorta knew each other via sports-team. It took like a couple of months of a couple of hours a week to get her amused and interested and passing and me getting laid-off. From what I heard later, she went on to get a masters in math. All student surpasses the teacher like. Maybe the fun and interesting has to be tailored to the individual a bit, but that's beside the point. I don't doubt anybody could get pretty far in math if they found that point of view that engaged them and made it not a chore but something enjoyable.
posted by zengargoyle at 3:08 PM on November 2, 2019
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posted by KleenexMakesaVeryGoodHat at 10:17 AM on October 28, 2019 [3 favorites]