Angels: God Hears Prayers

 This is my second entry in an effort to record what I've gleaned from a little study of angelic visitations in the scriptures.

Hagar

We open again with Hagar (and we will again later; it's one of the best angel stories in scripture). Hagar had been ejected from Abraham's household and was wandering in the desert with her son Ishmael. They were out of water were so near death that Hagar had left Ishmael under a bush so that should wouldn't see him die.

In the midst of this the scriptures tell us that "God heard the voice of the lad." And he sends an angel. An angel appears to Hagar and tells her

Fear not; for God hath heard the voice of the lad where he is.

The angel shows her where to find water but before he does that he assures her that she has been heard and is known.

Many of the angels in the scriptures repeat this message.

Zacharias

When the angel Gabriel appeared to Zacharias the soon-to-be father of John the Baptist the first thing he said was

Fear not, Zacharias: for thy prayer is heard.

 

 Cornelius

When the angel visited the devout centurion Cornelius, in preparation for the preaching of the gospel to the Gentiles, the angel says

Thy prayers and thine alms are come up for a memorial before God.

 King Benjamin

In the Book of Mormon, the angel that visited King Benjamin in the night before his great address greets him

Awake, and hear the words which I shall tell thee; for behold I am come to declare unto you glad tidings of great joy. For the Lord hath heard thy prayers, and hath judged of thy righteousness.

He follows this greeting with a long and revelatory discourse on the mission and coming of the Messiah.

Alma

Alma the Younger gets a much less happy greeting. In fact, the angel does not tell him that his prayers are heard. After ordering him to stop persecuting the church the angel tells him why God has sent an angel to him:

 Behold, the Lord hath heard the prayers of his people, and also the prayers of his servant, Alma, who is thy father.

Even though Alma the Younger has not prayed, the angel still takes time to tell him that God has heard the prayers of others.

God hears prayers

Why do these angels take time to tell the recipients of their message that God has heard their prayers? Shouldn't that be obvious from the rest of their message, or from their very presence? 

 One benefit of the repetition of the message is that we, as readers, can learn from the lesson. It is easy to feel at times like prayers go nowhere or are unheard. I'm not smart enough to know why some prayers are answered right away and some never, but it is without doubt that God hears prayers. He instructs his angels to tell it to people over and over.

My takeaway: I should remind my loved ones more often that God hears prayers remind each other as well. Alma the Elder never got direct angelic confirmation that his prayers were heard, but it must have been just as sweet to hear it from his son.

Angels: Bring Food

After a recent church discussion of Elder Godoy's talk I Believe in Angels, and also after reading an excellent response, I started to study what angels do when they appear to men. I'm trying to find every description of angelic visitations in (and sometimes out of) scripture and looking for common themes.

What's the point? One of my reasons is pure spiritual enrichment. Angelic visitations tend to be some of the most powerful displays of God's tender and personal loving-kindness. And maybe (maybe!) I can become a better minister by studying how God's angelic ministers do it.

My favorite theme so far is that angels sometimes bring food and drink with them. Not every time, of course, but there are two separate Biblical instances of angels bringing food.

 Example: Hagar

In Genesis 21 Hagar, the mother of Ishmael, concubine of Abraham, and handmaiden to Sarah, has been ejected from Abraham's household. She is left in the wilderness with the child Ishmael and a single bottle of water.

And the water was spent in the bottle, and she cast the child under one of the shrubs.

And she went, and sat her down over against him a good way off, as it were a bowshot: for she said, Let me not see the death of the child. And she sat over against him, and lift up her voice, and wept.

And God heard the voice of the lad; and the angel of God called to Hagar out of heaven, and said unto her, What aileth thee, Hagar? fear not; for God hath heard the voice of the lad where he is.
...
And God opened her eyes, and she saw a well of water; and she went, and filled the bottle with water, and gave the lad drink.

The water this angel brings was already there, but I think it hardly matters exactly how the miracle was done. The Lord heard her, knew she needed water, and gave her some.

Example: Elijah

In 1 Kings 19 Elijah is also wandering alone in the wilderness. He has just enacted one the greatest miracles of his ministry and convinced the people to kill out the priests of Baal. Unfortunately, this brought reprisal from the queen Jezebel who slew many of Elijah's fellow prophets. Distraught and distraught, Elijah flees to the wilderness. Exhausted, he lays down under a tree and gives up, saying "It is enough; now, O Lord, take away my life; for I am not better than my fathers."

And as he lay and slept under a juniper tree, behold, then an angel touched him, and said unto him, Arise and eat.

And he looked, and, behold, there was a cake baken on the coals, and a cruse of water at his head. And he did eat and drink, and laid him down again.

And the angel of the Lord came again the second time, and touched him, and said, Arise and eat; because the journey is too great for thee.

And he arose, and did eat and drink, and went in the strength of that meat forty days and forty nights unto Horeb the mount of God.

This angel doesn't have any message for Elijah. He'll get to talk with God directly on Mount Horeb. This angel's job is simply to make sure that his belly is full so that he can keep going with the bigger plan.

There are two lessons here. First, God is concerned with our bodily needs. He knows we need to eat and we need to drink, and he cares about that. He also knows that our bodies and our spirits are inextricably connected; Elijah didn't need an angelic sermon to get him to Horeb, he needed some bread.

 The second lesson is directly connected to the first, and is for people trying to minister in God's way. When God says "all things unto me are spiritual, and not at any time have I given unto you a law which was temporal" it must not mean that only spiritual things are important and temporal or bodily concerns can be ignored. It means that bodily, temporal care is spiritual care. Sometimes we just need to bring some bread.

Public Discourse Is a Trap

Note: This was originally written some time in late 2019.

Public discourse is a trap.

At any given time there are a dozen or topics that every intelligent, engaged people are supposed to form opinions on and argue about. In the past few months the list has looked something like:

• Is Donald Trump racist?
• Are the detention facilities at the US border concentration camps?
• Is it appropriate for Nike to use the Betsy Ross flag on a shoe line?
• Legal abortion: good or bad?
• Is it a good idea to ban plastic straws?
• Why are there not more women employed in the software engineering industry?

and so on. The list can be as long as you like. I've chosen particularly contentious issues, ones that start really heated arguments on the internet and in the news media. There is no limit to the amount of time you can spend arguing about any one of these topics.

Much has been said about this both in the present and the past. Many people have written about how there's too much contention, too much division, how we need to do better listening and finding common ground. But there's a more fundamental issue that's been bothering me lately: all the items on that list were chosen by a stranger.

I worry that I've spent days of my life thinking and arguing about the topics in the list above, not because I sat down and made a list of what I thought was most important, but because some stranger I don't know has been working every day for the past ten years to get everyone in the country to talk about their favorite issue. 

Sometimes I do think the issues of the day are important ones. But sometimes I think they are a waste of time; often the questions are false dichotomies. Sometimes it seems like the questions are ones that are maximally contentious while minimally substantive. And sometimes it seems like the issues are chosen just to make everyone pay attention to an individual that thrives on attention, positive or negative.

And why should I spend my time on that? Why should I fall into someone else's attention trap?

The first obvious answer to the problem of people not talking about the things I think are important is to spend more time trying to persuade everyone else that my ideas are the important ones to discuss. The second obvious answer is to disengage entirely.

The problem with the second answer is that, unfortunately, public discourse can sometimes have large real-world effects and disengaging means having even less control over what happens to me and my loved ones.

The problem with the first answer is that if before there were 1,000 people working to make everyone else listen to their opinions, now there would be 1,001. And most likely 90% of them are more dedicated to their idea than me, more willing to spend time on discourse than me, and more in love with arguing than me. It feels impossible to contribute meaningfully without making it my full-time job, and a full-time job I would hate at that.

Ramblings on Eric Newby

I've been reading a series of autobiographical books by Eric Newby, who was an English travel writer in the mid 20th century. I read the first one "The Last Grain Race" because it was about sailing ships -- as a young man just before WWII he signed on to crew on one of the last ever commercial sailing ships. I wasn't expecting to read any more of his work until I saw the summary of his book "Love and War in the Appenines," which read:

"In 1943 prisoner of war Eric Newby successfully escaped the Germans and sought refuge in the mountains and forests of the Apennines [in Italy]... But amidst the danger and risks were humour, friendship and remarkably, love - as his life became interwoven with the hopes of the local girl who rescued him and eventually became his wife"

And apparently that's a love story I can't turn down. It turned out to a be a beautiful book. It was clearly written not to aggrandize his own actions but out of deep love and gratitude to the Italian farmers who took huge risks to help him. It's strange as an adventure novel because he ends up getting captured again in the end and doesn't do anything important in the war effort. But it captures instead what's really important: people being kind and sacrificing for others because it is the right thing to do.

And then the book ends without telling you anything about the rest of his courtship with his wife, so I had to hunt down a third book "A Little Place in Italy," an account of him and his wife buying a house in the same region of Italy twenty years later. Again, nothing much of consequence happens, but he gives charming and loving portraits of the farmer neighbors who welcome them into the community, mixed in with idyllic scenes of things like working on a neighbor's vineyard and taking a break for eating a mid-morning picnic delivered by the women of the house carrying baskets on their heads.

He does have that very English superpower for absurd coincidences and lucky breaks. During his time as refugee he gets lost in the mountains one day only to stumble into the house of an eccentric old man who shelters him for a day or two before leading him back to his hiding spot. The old man is a character. He is a skilled tinkerer and laborer, but has a habit of talking to himself incessantly. He also is a traditional storyteller and knows all the old stories. In the later book when they find the house they are to buy they learn that a man has been living in two of the rooms since the last owners. When they meet him Eric is shocked to see that it is the same old man. I gasped when I read it. The old man, Attillio, continues to board with them and helps them take care of the house and their vines. They are not sure if he remembers either of them, but he takes an especial liking to Eric's wife, Wanda, calling her "mi padrona."

All the books read like the book that every returned missionary wishes they could write. It exudes the desire to share with others the special love that one gets for a specific place and set of people. You know you will never really be able to get other people to see these things and these people the way you see them simply because they weren't there when you were, but it still feels important to try.

In fact, the bit about reuniting with Attilio feels very similar to an experience in India. When I was in Bangalore an older man dropped in to church meetings one day and so we started to teach him. He was a magician and entertainer, and a Christian, and had lived all over, including in the Maldives. He enjoyed meeting with us, but told us that since he was staying for free in a hostel run by a pastor he couldn't come to church with us for fear of upsetting the pastor. Then one day he disappeared. We tried calling him to no avail until a week or so later he called us from a payphone telling us he had moved to Calcutta. This was strange and surprising, but so was everything about him. We couldn't decide if this was eccentricity, or a lie to avoid having to keep meeting with us.

Six months later I was in Visakhapatnam heading for home through a busy square when I heard someone calling "Pritchett! Pritchett!" I turned and saw my magician friend walking towards me. When I got over my shock at seeing him again I introduced him to my new companion, and he led us to his tiny rooftop apartment. There he gave us snacks let us hold his dove that he used in his magic shows. I swear I am not making this up. We started teaching him again and he came to church a few times. At one point the Branch President, Prasad Kusuma, hired him to do some magic for a branch activity. Then he moved a ways out of the city into a similar hostel / pastor arrangement and we never really saw him again. 

All the books are filled with little snapshots of characters like this. Never making fun, but always highlighting the mix oddities and vices and virtues that all real people have, from Finnish sailors playing pranks on each other to Italian farm girls analyzing their dreams from a book over breakfast to a Nazi office who cares more about collecting butterflies than capturing an English escapee. They capture the important things rather than the exciting things.

And now, selected quotes from the books:

From "The Last Grain Race," young Eric working as a clerk in a London ad agency

"When I was not speculating about what I read, I would fight with Stan, ... one of the two assistants in the department... Both Stan and Les, the second assistant, called me 'Noob'. 

"'Ere', Noob, what abaht a pummel?' Stan would croak invitingly, and we would pummel one another until Miss Phrygian banged furiously on the frosted glass of her office door to stop the din."

From "The Last Grain Race", rounding Cape Horn

As [the Moshulu] ran she surged into the sea so that it came up at us on the bowsprit as if it was trying to lick us off. ...
"Kossuri, take my bloddy byxor," Taanila yelled in my ear.
I put my hand under his oilskin coat and took a good hold on his belt and trousers as he got down on to the footrope to take the gasket that Yonny Valker and I were swinging towards him over the sail and under the bowsprit.
At that moment, it was fortunate that Taanila's mother could not see him. It was fortunate, perhaps, that none of our mothers could see us. 

 From "Love and War in the Appenines," as a local clan builds him a hidden hideaway

This took much longer than they thought because while they were digging it they uncovered a perfectly enormous rock.

... Then, as if he had been waiting for his cue, an old man appeared on the cliff above us and looked down rather critically on the party assembled below. ... He went over it with his hands, very slowly, almost lovingly. It must have weighed half a ton. Then, when he had finished caressing it, he called for a sledgehammer and hit it deliberately but not particularly hard and it broke in two almost equal halves.  It was like magic and I would not have been surprised if a toad had emerged from it and turned into a beautiful princess who had been asleep for a million years. Even the others were impressed.

From "Love and War in the Appenines"

"Signor Zanoni," I said, using one of my small store of stock phrases, "Posso dormire nee vostro fienile?" "Can I sleep in your hayloft?" 

 "Did anyone see you on the road coming here?" he said.

 I told him that I had seen no one and that I was as sure as I could be that no one had seen me.

 There was a long pause before he answered, which seemed like an age. "No," he said, finally, "you can't."

... "No, you can't sleep in my hay," he said after another equally long pause. "You might set it on fire and where would I be then? But you can sleep in my house in a bed and you will, to, but before we go in I have to finish with Bella [the cow]." And he went back to milking her."

From "A Small Place in Italy"

 Almost every day Signor Guiseppe used to take off from his new house up the hill ... and come down the hill to our dell, singing a bit on the way. This was what most men did when visiting or passing through other peoples' properties by the labyrinth of tracks and paths that were normally open to anyone who wished to use them. ... Women would cough discreetly or carry on an over-loud conversation with whoever they were traveling with.

 ... One reason for all this ceremony - the coughing and the singing - was probably so as to not catch the owners of these properties with their trousers down or their skirts up, or both.

 From "A Small Place in Italy"

Signor Guiseppe ... had got it into his head that our well needed cleaning out and being a countryman saw nothing strange in arriving on our doorstep at half-past six in the morning to discuss the matter, without telling us that he was coming and without the usual premonitory arias.

Response to John Mandlbaur

This is a response to John Mandlbaur's analysis of the conservation of angular momentum found here. Since he asked me on Quora to address his work without prejudice, I will do so. I will analyze his work with the same level of rigor I apply to my own experiments. At the end I will propose my own experiment.

Description of the experiment

I will only deal with your analysis of Walter Lewin's spinning platform demonstration, which makes up the first part of his essay, as I believe it will be sufficient to show why I don't accept your conclusion.  You measures Lewin's rate of rotation before and after he brings in his arms by timing the period of Lewin's rotation in the video. You then calculates the ratio of his measured final and initial angular velocities and compare it to the prediction Lewin calculated in the video (beginning around the 20 minute mark)

You find that the initial angular velocity divided by the final angular velocity is about 2.12 ± 0.2 while Lewin predicted a ratio of 3. You then conclude, after similar analyses of similar videos, that the law of conservation of angular momentum is not true.

What You Did Right

You were right to do an experiment. It is always good to test hypotheses, and to not take anything for granted. This experiment is sound in principle: you use the law of conservation of angular momentum to predict what the ratio of the angular velocities should be, then you measure the angular velocities and calculate their ratio and compare. If you find with high confidence that the measured and predicted ratios differ, you may begin to doubt the law of conservation of angular momentum.

You were also right to attempt to estimate the error in his your measurements. No measurement is perfect, and an imperfect measurement puts limits on how confident we can be in our experimental conclusions.

However, having read your analysis I do not share your confidence in your conclusion. I think you ignored several other sources of error that should stop us from concluding with high confidence that the prediction and measurement differ.

What You Did Not Do

There are two major sources of uncertainty in this experiment that we need to account for. The first is the uncertainty in the moments of inertia, the second is friction.

Moment of inertia

You did put error bars on your time measurements, but we also need to put error bars on the moment of inertia values. If we do not have an idea of how precise the prediction is, we can't correctly compare the measured result, and we won't know how confident to be in our conclusions.

Since Lewin was trying to show how to calculate simple moments of inertia, he did not actually measure his moment of inertia, nor did he do the incredibly complicated problem of calculating the moment of inertia of a real human body. Instead, he used some simple estimates, like modeling himself as a 75 kg cylinder with a radius of 20 cm. If we want to use Lewin's demo as a real experiment we need to understand how much error there is in this model. 

One way to estimate how much error there is in the moment of inertia value is to use a slightly more realistic model and see by how much the two models differ. Instead of a 20cm radius cylinder let's model Lewin as a cuboid with a width of 40 cm and a depth of 10 cm. These are roughly my dimensions. If he is 75 kg (which is another rough estimate) then this moment of inertia works out to 1.0625 kg m^2, compared to the 1.5 kg m^2 we get from Lewin's estimate. That's a difference of about 30%. If we want to be cute, we can treat these two estimates as data points and calculate their average and standard deviation to get the first moment of inertia to be  1.28 ± 0.22 kg m^2. This is still an incredibly rough estimate, but at least it gets us some idea of the amount of error on those values, which starts at roughly 17% at the least.

And I have not included estimates of the error of using 75 kg -- which is a fairly standard human mass physicists use in practice calculations and not an actual estimate of Lewin's mass -- or the error in the estimate of the length of his arm, or the fact that the final moment of inertia calculation ignores the weights he is holding, or any of the errors in the calculation of the final moment of inertia (which is even more complicated because some of his mass in his arms is extended).

Suppose, for example, that there is a 20% error in both moment of inertia values. Then the ratio predicted angular velocities also has an error of about 28% (this is rough statistics; the variances add, so multiply the error by square root of two). But then the predicted ratio of angular velocities is 3 ± 0.84, which overlaps significantly with the measured ratio of 2.12 ± 0.2! So we cannot conclude that angular momentum is not conserved.

Of course the 20% is a made up number, but the point is that the conclusion about angular momentum depends on the error on the predicted value. Unless we know how precise our prediction is, we can't draw any meaningful conclusions. If John wants to convince me of his conclusion he must convincingly quantify the error on the prediction.

Friction

You also neglected the fact that Lewin's demonstration has friction in it. The law of conservation of angular momentum is true in the absence of friction. A fully rigorous experiment would not predict that angular momentum is completely constant, but that it decreases at a mostly steady, hopefully small rate. Again, since Lewin is doing a demo and not a full experiment he doesn't bother to include the complication of friction. When I do a similar experiment with my students we do include friction.

Unfortunately, I can think of no great way to account for friction in this experiment. We do not have access to the platform Lewin used so we cannot measure it directly. It may be possible to compare his rotation speed at two different points in time and calculate the rate at which the speed has changed, but it looks tricky because he keeps changing his shape and the camera angle.

Without a way to quantify how much friction there is all we can say is that the angular momentum in the final position is less than the angular momentum in the initial position, and hence that the ratio of angular speeds should be somewhat less than 3.  How much less? We don't know, and until we know we can't actually make much of a conclusion. However, it is telling that your measured rate is in fact smaller than 3, consistent with some angular momentum being lost to friction.

Your original analysis essentially says "since 2.12 ± 0.2 does not overlap with 3 ± 0 so angular momentum is not conserved." The problem is that both the 3 and the 0 are not the right numbers to put there. To get the right numbers we need much better measurements or calculations of the moments of inertia and the friction. As it stands the real values could be 3 ± 0.5 or 2.23 ± 0.5 or 100 ± 99. We just don't know.

Comments on other sections

The criticisms above apply to the other experiments mentioned in your essay. In all of the referenced demonstrations the moment of inertia is either estimated or calculated using simple models with no estimate of the error. Friction is also not taken into account (except for the second video with the ping-pong ball, where the experimenter finds that reducing the time of the experiment --- and hence the amount of time friction can act --- makes the measured ratio more nearly match the friction-less prediction). The paper linked to in the third section is not published in any journal. It does only basic estimates of the error on the angular momentum or moments of inertia. The fact that Figures 3 and 6 do not show error bars on the data points is telling.

If you really want to stick to analyzing these videos you should start by figuring out ways to carefully quantify the error in the moment of inertia measurements and find ways to account for the friction in each case. Without doing that your analysis will not be persuasive.

A Proposed Experiment

I perform an angular momentum experiment with my 12th grade physics students nearly every year. The purpose is slightly different than your, but with slight re-purposing it can be made to work. I don't have any leftover data that I can use right now, but I can propose an experiment.

I have a spinning arm that can carry weights that slide in and out along each side of the arm. The arm has a photogate speedometer that can be connected to a digital device that tracks angle to measure the angular speed of the arm. There is a pulley in the center of the arm. String can be attached to the sliding weights to make their change their radius of rotation, hence changing the moment of inertia of the arm.

Here is my proposed experiment.

1. Measure the amount of friction in the arm. Assume that the frictional torque is constant. Get the arm spinning with the weights in a fixed position and measure the angular speed over time with the digital readout. The slope of the angular speed tells us the ratio of the torque to the angular momentum. Repeat this for several different configurations of the arm to determine if the torque depends on the moment of inertia or not.
2. Measure the moment of inertia of the arm with weights in several different positions. I can do this by attaching a hanging mass to a pulley so that a string turns the arm as the mass falls. Once the mass has fallen a fixed distance, the arm will have gained a fixed amount of rotational kinetic energy. Measuring the rotational speed once the mass has fallen allows us to calculate the moment of inertia of the arm. Repeating this procedure several times for each arm position and accounting for friction gives us error values for all the moment of inertia values we will use.
3. Get the arm spinning with the weights in an extended position. Collect several turns of angle data to measure the initial angular velocity. Pull on the pulley string to pull the arm into a compact position. Collect several turns of angle data to measure the final angular velocity. Record the time along with each angular velocity measurement.
4. Repeat step 3 multiple times and with multiple different weight positions.
5. Using the angular velocities for 3-4 and the moments of inertia from 2, calculate the initial and final angular momenta. Calculate the mean angular momenta and standard deviations.
6. Subtract the torque from step 1 times the time between the two speed measurements from the initial angular momentum to account for friction. Take the difference between the corrected initial angular momentum and final angular momentum. Calculate error bars for this difference using the error bars for all the values that go into it. Do this for each different arrangement of extended and contracted positions.
7. If angular momentum is conserved, the differences calculated in 6 should be consistent with zero within an error bar. The more times I repeat each measurement the closer the average should be to zero.

My proposed experiment accounts for all of my issues with your experiment. It gets actual measurements of the moments of inertia and their errors instead of simplified calculations. It accounts for friction. It also gives well-quantified error bars on the velocity measurements by repeating the experiment multiple times.

My experiment still has some limitations. It does not account for air resistance, which is another source of error. As part of my experiment I would give some back-of-the-envelope estimations of the air resistance torque. If my estimations got me a significant number, I would come up with some ways to measure it experimentally. I expect that the main uncertainty will be either my measurement of the friction or the angular speeds. I don't know the precision limits of the digital reader off-hand.

My question to you is this: if I do this experiment and document my results thoroughly, and I find that the answers are consistent conservation of angular momentum, will it change your mind? If not, are there any changes I could make to this proposed experiment that would make it more convincing to you?

Intuition for the fundamental theorem of calculus

I've never really understood the fundamental theorem of calculus. I mean, I passed Calc I, but I just memorized the proof and then forgot it. Since then I've felt alternately that it's either too obvious or too "analysis-y" to prove.

That's why I was happy to discover some intuition for it while designing an activity for my 12th grade physics students. Now I have a proof, or close enough to it, that makes sense to me and fills in the gaps I've always had.

The idea

The basic idea is using successive linear approximation to predict future values of a function. In terms of physics, if you know the position of an object and its velocity you can approximate its position in $0.1$ seconds via $x(0.1) = x(0)+v(0) \times 0.1$. This is only an approximation for $x(0.1)$, but if $0.1$ is small compared to the rate at which the velocity is changing then it can be a pretty good one.

If we want to predict the position at more distant times we can simply iterate the procedure. Take our approximation for $x(0.1)$ and we can predict $x(0.2) = x(0.1)+v(0.1)\times 0.1$. We can take this as far as we want.

And if the approximation isn't good, we can take intermediate steps. Go in steps of $0.01$ seconds instead of $0.1$ so that 
$$x(0.1) = x(0)+0.01\times\left(v(0)+v(0.01)+\dots +v(0.99)\right)$$.
You can see how this procedure works to approximate a curve in the image below. You start at $(a,f(a))$, and then use the slope of $f$ at $a$ to move to an approximation for $f(a+\Delta x)$. You then move the slope there to move on to the next step, and so on.

An illustration of the step-forward technique for approximating $f(b)$ from $f(a)$ with two steps, five steps, and twenty steps. The approximation gets better as the number of steps increases.

The sum on the right hand side is a Riemann sum with $\Delta t = 0.01$, so in the limit that the step-size goes to zero this becomes $x(a) = x(b) + \int_a^b v(t) dt$. Identifying $v(t)$ with $dx/dt$ shows that this is the fundamental theorem of calculus.

The thing to prove is that this approximation procedure does actually work when we let the step-size go to zero. That part is non-obvious and is the real content of the fundamental theorem of calculus. That it should work has always felt obvious to me in physics and so I've often felt confused as to what is actually there to prove.

The lemma

The proof I came up with relies on the fact that the derivative of $f$ at a point is the slope of the best affine approximation for $f$ around that point. The process I described uses successive affine approximations of $f$ at intermediate points. This process only converges if we use the best such approximation at each point.

To be formal, by "best affine approximation" I mean the $A,B$ such that in neighborhoods around $x_0$, $f(x) = A+B(x-x_0)+R(x-x_0)$ where $R(x-x_0)\in o(x-a)$. The little $o$ notation means that $R(x-x_0)/(x-x_0) \rightarrow 0$ as $x\rightarrow x_0$. (There is a slightly more formal definition, but this is good enough.)

This Math StackExchange answer proves that the best affine approximation has $A=f(x_0)$ and $B = f'(x_0)$. Thanks for doing my work for me. It also proves that $f$ has a best affine approximation at every point in an interval as long as it is differentiable on that interval.

The proof

Consider a function $f$ that is differentiable on the interval $[a,b]$. Define $\Delta x_N = (b-a)/N$ and
$$f_N(b) = f(a)+\Delta x_N\times \sum_{i=0}^{N-1} f'(a+i\times\Delta x_N)$$.
The sum on the right is a Riemann sum, so $\lim_{N\rightarrow \infty} f_N(b) = f(a) + \int_a^b f'(x) dx$.

Since $f'$ gives the best affine approximation at every point, $|f(b) - f(a+(N-1)\Delta x_N)-f'(a+(N-1)\Delta x_N)\Delta x_N| \in o( \Delta x_N)$. Likewise for $|f(a+(N-1)\Delta x_N)-f(a+(N-2)\Delta x_N) - S(a+(N-2)\Delta x_N)\Delta x_N|$ and so on. We successively approximate $f$ at each intermediate step by the affine approximation for $f$ at the previous step. Each time we do so we pick up an error term that is in $o(\Delta x_N)$. Hence,
$$|f(b)-f_N(b)| \in o(\Delta x_N)$$.
The only catch is that since we pick up $N$ error terms we can say that there is an $m$ such that $|f(b)-f_N(b)| \le m N\Delta x_N^2$.

Since $N\Delta x_N^2 = (b-a)^2/N$ in the limit that $N\rightarrow \infty$ we have $\lim_{N\rightarrow \infty} |f(b)-f_N(b)|= 0$, and hence $f_\infty(b) = f(b) = f(a) +\int_a^b f'(x) dx$, QED

Alternative proof

I originally came up with a slightly different proof using Taylor series. This is less mathematically elegant and limited to analytic functions, but it's slightly more intuitive for me.

Consider that

$$f(b) = f(b-\Delta x_N) + \sum_{m=1}^\infty \frac{\Delta x_N^m}{m!}f^{(m)}(b-\Delta x_N) \\

= f(b-2\Delta x_N) + \sum_{m=1}^\infty \frac{\Delta x_N^m}{m!}(f^{(m)}(b-\Delta x_N)+f^{(m)}(b-2\Delta x_N))\\

= f(a) + \sum_{m=1}^\infty \frac{\Delta x_N^m}{m!}\sum_{i=0}^{N-1} f^{(m)}(a+i\Delta x_N)$$

In the limit that $N\rightarrow \infty$ the $m=1$ term in the sum becomes the integral from $a$ to $b$ of $f'$. The other terms in the sum vanish as long as all the other derivatives of $f$ are bounded on $[a,b]$. When they are bounded $|f(a+i\Delta x_N)| < M$ for all $i$, and so $|\sum_{i=0}^{N-1} f^{(m)}(a+i\Delta x_N)| < M\times N$ for all $m > 1$. But then $\Delta x_N^m \times M \times N = M(b-a)^m/N^{m-1}$, which vanishes as $N\rightarrow \infty$, and hence all the sums vanish for $m \ge 2$.

This proof is slightly different because when we use the full Taylor series the calculation of $f(b)$ is exact at every $N$, but for every finite $N$ there non-zero higher-derivative corrections. In the limit that $N\rightarrow \infty$ the calculation stays exact for $f(b)$ but the higher-derivative corrections vanish and only the integral term is necessary.

Boston Haikai 189 -- Illegal lane changes

With punk rock playing
I change lanes illegally
Cut through Somerville

Sept 23 -- Somerville, Hwy 28