So me and Donny was driving down to Lino the other weekend (muskie season is fast approaching and REI was having a sale on wire leaders) when we experienced an automotive emergency that got me thinking about the Jacobian.
Donny fancies himself something of a gearhead. He spent a year in automotive cert over at the vo-tech but had to drop out after my sister got pregnant (got pregnant again, I should say. That girl is more fertile than the Nile river basin). So now he mostly does piecework around town. Snowmobiles, Snowcats -- that sort of thing. The big money is in Evinrudes, but the Lino CC got a lock on that and ain't no way they'd let a Circle Pines boy like Donny into their club.
Anyhoo, Donny's pride and joy is his '71 Chevelle, which isn't much to look at but under the hood is quite the formidable piece of machinery. Sort of what you imagine Springsteen drove over to Mary's house after she got raped at graduation.
Footnote: Holy cow. I just now realized Mary got raped at graduation. That's messed up.
Lots of stuff had gone under that hood, like solid lifters and a Holly double pumper and other words I used to know before graduate school pushed them out. Donny boasted he put 350 heads on a 305 engine, just like Patterson Hood. He, also, did not have good intentions. So he opens it up a bit, what with eight lanes of light traffic and a converging horizon, although I cannot confirm nor deny any laws were broken.
And that's when it hit us. Or, rather, we hit it.
It's easy to tell where county maintenance takes over on I-94 -- roadkill dot the landscape like a piloerect Margo Robbie. That's where one of these ex-critters put itself in our harm's way, a pointless act of defiance against similar metal monsters that had no doubt gished many of its kinfolk for reasons a 12 gram brain could not possibly comprehend. Sort of like how the Dwarven spider guardians in Skyrim explode after you kill them, because what's the point if they're dead, not that I would know anything about that.
In short, a raccoon, or possibly an opossum, had become lodged in our left front wheel well with loud report.
Donny drove on, pushing the tach up another hundred to compensate for the drag.
"Think we should do something about that?" I asked after a few miles.
"About what?" Donny replied.
He gets this from my sister, who in turn got it from our father. The standard clan Kitty approach to any problem is to assume if you don't acknowledge it, it will take care of itself. They usually do, albeit the outcome not the one you would have preferred.
"The marsupial now ensconced in our undercarriage," I said. I talk like that because I'm an academic, although what I said at the time involved a great deal many more swear words. And, yes, I realize raccoons are not marsupials. The distinction seemed unimportant at the time.
"Seriously?" Donny scanned the dash, as if in addition to speedometer, tachometer, oil pressure, and engine temperature gauges, the '71 Chevelle came equipped with a check raccoon warning light.
He shrugged. "Readouts are all in the green." I was expected to reply "there goes our salvage," but I broke protocol and expressed my lingering concerns about the odd vibration, and odor, now coming from the driver's side quarter panel. Would it not be prudent to pull over and have a gander?
"Nah, it'll burn off," Donny assured me. "Besides, them things got the rabies."
Which brings us to the Jacobian.
Your first encounter with the Jacobian was probably integration in polar coordinates, even though you may have not known it at the time. Whereas in Cartesian, the area element dA is simply dx dy, switching to polar (r,θ) introduces a factor of r: dA = r dr dθ. That's a Jacobian. Omitting it is a common mistake of the calculus padawan. Doing so simplifies many an integral at the cost of obtaining a wrong answer.
The why of including r in dA comes from a picture:
At left, a little chunk of area in Cartesian coordinates. At right, the same idea in polar. If you did not include r -- that is, if you wrote ΔA = Δr Δθ -- the expression is not only wrong, it's dimensionally nonsensical. The units of Δr is length but the units of Δθ is not. Only by multiplying Δθ by r do we get a length, and only then does their product give an area. The rest then follows.
Or does it?
I'm not mental -- I can see that including r is necessary to properly define ΔA. My difficulty is what comes after. To create an integrand, we take the limit as ΔA goes to zero. During that process, I claim the factor of r should become superfluous. If the lengths of the sides are going to zero, what difference does it make if we include r or not? Lim Δ→0 r Δr Δθ = limΔ→0 Δr Δθ = 0.
In other words, when sufficiently small, a wedge and a square look exactly alike. Here, see for yourself:
This is the wedge that goes into the integral. Not the big honkin' wedge of the figure above, but a little speck of a wedge. Who cares if it has the right shape? A speck doesn't have a shape. A speck is a speck is a speck.
When I ask calculus wonks this question, they invariably inform me one must include r because it gives the right answer. "Because it gives the right answer" has been the fount of quite the number of my prior rantings. Perhaps this reflects my deep seated resentment of authority or an innate sense of fair play (LabKitty: Cleisthenes or Zack de la Rocha? You must decide). Let's see if we can do better.
We shall corner the quarry like so: I understand it is proper and correct to include r at the start. That's just geometry. Yet, I don't understand why r need appear in the result. Ergo, if I am right (spoiler: I'm not*), logic demands r must disappear during the process of converting the former into the latter. That is, while taking the limit. Does it?
(* if this seems trite, take the LabKitty challenge: Drink a quart of whiskey,** then spot the flaw in my argument.)
(** please do not drink a quart of whiskey. I already get enough shrieking 3AM phone calls from my attorney.)
To explore this, we require a problem that is both suitable for polar coordinates yet comes with an obvious answer, so we can compare two versions of the calculation. I can think of no plainer problem than computing the area of a circle. A circle with unit radius for good measure, which has area π.
As per the usual geometric-limit approach, we tile the circle and compute net tile area. Standard rules apply: We add tiles until they threaten to spill over the edge of the circle, stop, and sum their area. We then repeat the process using smaller and smaller tiles until we grok some pattern.
The final twist is we will compute tile area in two different ways: 1) using ΔA = r Δr Δθ, and 2) using ΔA = Δr Δθ. If I am right, these results should converge during refinement.
I whipped up in a little Javascript app for this, because it was Saturday night and why not:
The buttons allow you to increase and decrease the tiling resolution (up to a point -- after about 4 clicks in either direction the calculation goes wonky. At low resolution, the tiling is useless. And at high resolution, computation time of my dopey code becomes prohibitive. You'll know you hit a limit when a button deactivates).
The app displays the current area approximation using both approaches under the plot (for simplicity, I compute the area in the first quadrant and multiply by four). It also displays the number of tiles, and the average error of the tiles in the wrong approach.
Go play. Then, read on.
What you should have noticed is -- just as I predicted -- individual tile wrongness indeed goes to zero as the tiling resolution increases. A speck is a speck is a speck. The bad news, however, is we get more of them. The trade-off is what screws the pooch.
The decrease in tile wrongness is exactly offset by an increase in the number of wrong tiles. In this particular integral, a factor of 2 emerges. That is, including the Jacobian (r) reduces the calculated area by 1/2. Even as the tiles get ground to dust, r is still present. Defiant, determined, dogged. It keeps the results in check by dragging down the sum the entire time. Hinders it. Holds it back. Gums up the works.
Just like a dead raccoon stuck in your wheel well.
Donny pulled over when we got to the Lake Elmo exit. He was perfectly willing to keep grinding mammal, you understand. This wasn't some Grinchian moment where he became overwhelmed for the welfare of small woodland creatures as the result of an oversized heart (Donny's gets his oversized heart from his mother's side of the family).
The change was a result of other developments. The smoke we were trailing had taken on the bluish tinge suggestive of burning oil. This presented an affront to Donny's automotive pride, for the Chevelle had received new rings as recently as last summer. I know this because I scraped my knuckles something fierce helping him reseat the pistons because Donny refused to buy a ring compressor, instead using a contraption he rigged from a couple of empty PBR cans and some ball bungees.
So there we were, parked on the shoulder of I-94 while logging trucks and wholesome family vehicles whizzed past. Donny was still flirting with the idea nothing was amiss, but rounding the front of the Chevelle put an end to innocence.
The sight was an unnerving melding of flesh and machine, like a Chilton's manual illustrated by H.R. Giger. It had surrendered its organic nature, the raccoon had, when it accepted the Chevelle's caress of steel. The chunks that would surrender easily had already done so, yielding to centripetal force and slipstream. What remained was the tenacious part, the winner of this unholy Darwinian struggle. A tangled ball of mud and fur, guts and grease, crepuscular and varicose, twisted and weaved within and intertwined around the A-arms and coil spring. A paw hung from a rack bushing, eerily humanlike, unmistakably pointing a tiny index finger in our direction. J'accuse!
"Told ya," I said. Because I had.
Donny kicked at the carcass but it didn't budge. "Go find me a stick or something." I wandered off into the bush not understanding why Donny felt he was in charge, but searching for a stick offered the possibility the problem would be resolved without requiring my hands-on participation.
By the time I returned, a Minnesota Highway Patrol vehicle had filed in behind the Chevelle. Donny was leaning on the trunk with his arms folded across his chest while a Statey in dress blues squinted at his driver's license. When I emerged from the underbrush, she gave me a once-over from under the brim of her Smokey hat like I was a car thief who just happened upon a '71 Chevelle parked on the shoulder of I-94 with a smoldering raccoon wedged in the fender and decided it was my lucky day.
"You two a ways from Circle Pines," the officer said suspiciously.
"REI having a sale on wire leaders," I explained.
"My stars. You two going out for muskie?"
"Every year," I said.
"Fish of a thousand casts," Donny chimed in. He was off by an order of magnitude. I let it slide.
She circled around to the front of the Chevelle and grimaced at our stowaway.
"County maintenance," I said, hoping to sway her to our cause by exploiting interdepartmental rivalry in the Minnesota bureaucracy.
"Yah" she said, nodding gravely. "Welp. Let's see about getting you on your way. "
The three of us joined forces to clear the wheel well like some kind of samizdat jpop band. Donny stabbed at the carcass with the quality poking stick I had provided while the Statey bounced the bumper. Meanwhile, I sat in the driver's seat and cranked the tiller left and right.
There was much cursing from Donny, but the bond he had formed with the Chevelle lo these many years gave him the strength to power through. Chunk by fetid chunk he worked his beloved from this meat prison like Michelangelo calling forth David from a marble slab. The main bulk of the carcass finally relinquished its grip after a final stabbing frenzy. It fell to the pavement with a desiccated thud.
We gathered around the corpse, attempting to mark its passing with some measure of dignity. It was, after all, one of God's creatures, no less than the three of us.
The Statey punted it into the grass. "Yah know, these things got the rabies," she said. Donny glared at me sideways, like my sister did.
We waited until MHP was out of sight and earshot, then Donny turned the motor over. He tapped his class ring against the side of the shifter and studied approaching traffic in the rearview mirror. The Chevelle idled with the grumble of an impatient predator, threatening to stall if you didn't goose the throttle from time to time. It was the price you accepted for an aggressive camshaft.
The gap Donny was waiting for materialized and he dropped the clutch. The Chevelle roared to life. It's Thrush pipes called out to things living or otherwise in a deafening warning that cleared all from our path or who might dare to be. Fus ro dah.
Three tire chirps later we were at cruising speed. Donny let off the throttle. The tach dropped to idle but the Chevelle did not slow, a gift of Valvoline lubrication and proper tire pressure.
And an absence of Jacobian.
Donny fancies himself something of a gearhead. He spent a year in automotive cert over at the vo-tech but had to drop out after my sister got pregnant (got pregnant again, I should say. That girl is more fertile than the Nile river basin). So now he mostly does piecework around town. Snowmobiles, Snowcats -- that sort of thing. The big money is in Evinrudes, but the Lino CC got a lock on that and ain't no way they'd let a Circle Pines boy like Donny into their club.
Anyhoo, Donny's pride and joy is his '71 Chevelle, which isn't much to look at but under the hood is quite the formidable piece of machinery. Sort of what you imagine Springsteen drove over to Mary's house after she got raped at graduation.
Footnote: Holy cow. I just now realized Mary got raped at graduation. That's messed up.
Lots of stuff had gone under that hood, like solid lifters and a Holly double pumper and other words I used to know before graduate school pushed them out. Donny boasted he put 350 heads on a 305 engine, just like Patterson Hood. He, also, did not have good intentions. So he opens it up a bit, what with eight lanes of light traffic and a converging horizon, although I cannot confirm nor deny any laws were broken.
And that's when it hit us. Or, rather, we hit it.
It's easy to tell where county maintenance takes over on I-94 -- roadkill dot the landscape like a piloerect Margo Robbie. That's where one of these ex-critters put itself in our harm's way, a pointless act of defiance against similar metal monsters that had no doubt gished many of its kinfolk for reasons a 12 gram brain could not possibly comprehend. Sort of like how the Dwarven spider guardians in Skyrim explode after you kill them, because what's the point if they're dead, not that I would know anything about that.
In short, a raccoon, or possibly an opossum, had become lodged in our left front wheel well with loud report.
Donny drove on, pushing the tach up another hundred to compensate for the drag.
"Think we should do something about that?" I asked after a few miles.
"About what?" Donny replied.
He gets this from my sister, who in turn got it from our father. The standard clan Kitty approach to any problem is to assume if you don't acknowledge it, it will take care of itself. They usually do, albeit the outcome not the one you would have preferred.
"The marsupial now ensconced in our undercarriage," I said. I talk like that because I'm an academic, although what I said at the time involved a great deal many more swear words. And, yes, I realize raccoons are not marsupials. The distinction seemed unimportant at the time.
"Seriously?" Donny scanned the dash, as if in addition to speedometer, tachometer, oil pressure, and engine temperature gauges, the '71 Chevelle came equipped with a check raccoon warning light.
He shrugged. "Readouts are all in the green." I was expected to reply "there goes our salvage," but I broke protocol and expressed my lingering concerns about the odd vibration, and odor, now coming from the driver's side quarter panel. Would it not be prudent to pull over and have a gander?
"Nah, it'll burn off," Donny assured me. "Besides, them things got the rabies."
Which brings us to the Jacobian.
Your first encounter with the Jacobian was probably integration in polar coordinates, even though you may have not known it at the time. Whereas in Cartesian, the area element dA is simply dx dy, switching to polar (r,θ) introduces a factor of r: dA = r dr dθ. That's a Jacobian. Omitting it is a common mistake of the calculus padawan. Doing so simplifies many an integral at the cost of obtaining a wrong answer.
The why of including r in dA comes from a picture:
At left, a little chunk of area in Cartesian coordinates. At right, the same idea in polar. If you did not include r -- that is, if you wrote ΔA = Δr Δθ -- the expression is not only wrong, it's dimensionally nonsensical. The units of Δr is length but the units of Δθ is not. Only by multiplying Δθ by r do we get a length, and only then does their product give an area. The rest then follows.
Or does it?
I'm not mental -- I can see that including r is necessary to properly define ΔA. My difficulty is what comes after. To create an integrand, we take the limit as ΔA goes to zero. During that process, I claim the factor of r should become superfluous. If the lengths of the sides are going to zero, what difference does it make if we include r or not? Lim Δ→0 r Δr Δθ = limΔ→0 Δr Δθ = 0.
In other words, when sufficiently small, a wedge and a square look exactly alike. Here, see for yourself:
. ← wedge . ← square
This is the wedge that goes into the integral. Not the big honkin' wedge of the figure above, but a little speck of a wedge. Who cares if it has the right shape? A speck doesn't have a shape. A speck is a speck is a speck.
When I ask calculus wonks this question, they invariably inform me one must include r because it gives the right answer. "Because it gives the right answer" has been the fount of quite the number of my prior rantings. Perhaps this reflects my deep seated resentment of authority or an innate sense of fair play (LabKitty: Cleisthenes or Zack de la Rocha? You must decide). Let's see if we can do better.
We shall corner the quarry like so: I understand it is proper and correct to include r at the start. That's just geometry. Yet, I don't understand why r need appear in the result. Ergo, if I am right (spoiler: I'm not*), logic demands r must disappear during the process of converting the former into the latter. That is, while taking the limit. Does it?
(* if this seems trite, take the LabKitty challenge: Drink a quart of whiskey,** then spot the flaw in my argument.)
(** please do not drink a quart of whiskey. I already get enough shrieking 3AM phone calls from my attorney.)
To explore this, we require a problem that is both suitable for polar coordinates yet comes with an obvious answer, so we can compare two versions of the calculation. I can think of no plainer problem than computing the area of a circle. A circle with unit radius for good measure, which has area π.
As per the usual geometric-limit approach, we tile the circle and compute net tile area. Standard rules apply: We add tiles until they threaten to spill over the edge of the circle, stop, and sum their area. We then repeat the process using smaller and smaller tiles until we grok some pattern.
The final twist is we will compute tile area in two different ways: 1) using ΔA = r Δr Δθ, and 2) using ΔA = Δr Δθ. If I am right, these results should converge during refinement.
I whipped up in a little Javascript app for this, because it was Saturday night and why not:
results text #1
results text #2
results text #3
The app displays the current area approximation using both approaches under the plot (for simplicity, I compute the area in the first quadrant and multiply by four). It also displays the number of tiles, and the average error of the tiles in the wrong approach.
Go play. Then, read on.
What you should have noticed is -- just as I predicted -- individual tile wrongness indeed goes to zero as the tiling resolution increases. A speck is a speck is a speck. The bad news, however, is we get more of them. The trade-off is what screws the pooch.
The decrease in tile wrongness is exactly offset by an increase in the number of wrong tiles. In this particular integral, a factor of 2 emerges. That is, including the Jacobian (r) reduces the calculated area by 1/2. Even as the tiles get ground to dust, r is still present. Defiant, determined, dogged. It keeps the results in check by dragging down the sum the entire time. Hinders it. Holds it back. Gums up the works.
Just like a dead raccoon stuck in your wheel well.
Donny pulled over when we got to the Lake Elmo exit. He was perfectly willing to keep grinding mammal, you understand. This wasn't some Grinchian moment where he became overwhelmed for the welfare of small woodland creatures as the result of an oversized heart (Donny's gets his oversized heart from his mother's side of the family).
The change was a result of other developments. The smoke we were trailing had taken on the bluish tinge suggestive of burning oil. This presented an affront to Donny's automotive pride, for the Chevelle had received new rings as recently as last summer. I know this because I scraped my knuckles something fierce helping him reseat the pistons because Donny refused to buy a ring compressor, instead using a contraption he rigged from a couple of empty PBR cans and some ball bungees.
So there we were, parked on the shoulder of I-94 while logging trucks and wholesome family vehicles whizzed past. Donny was still flirting with the idea nothing was amiss, but rounding the front of the Chevelle put an end to innocence.
The sight was an unnerving melding of flesh and machine, like a Chilton's manual illustrated by H.R. Giger. It had surrendered its organic nature, the raccoon had, when it accepted the Chevelle's caress of steel. The chunks that would surrender easily had already done so, yielding to centripetal force and slipstream. What remained was the tenacious part, the winner of this unholy Darwinian struggle. A tangled ball of mud and fur, guts and grease, crepuscular and varicose, twisted and weaved within and intertwined around the A-arms and coil spring. A paw hung from a rack bushing, eerily humanlike, unmistakably pointing a tiny index finger in our direction. J'accuse!
"Told ya," I said. Because I had.
Donny kicked at the carcass but it didn't budge. "Go find me a stick or something." I wandered off into the bush not understanding why Donny felt he was in charge, but searching for a stick offered the possibility the problem would be resolved without requiring my hands-on participation.
By the time I returned, a Minnesota Highway Patrol vehicle had filed in behind the Chevelle. Donny was leaning on the trunk with his arms folded across his chest while a Statey in dress blues squinted at his driver's license. When I emerged from the underbrush, she gave me a once-over from under the brim of her Smokey hat like I was a car thief who just happened upon a '71 Chevelle parked on the shoulder of I-94 with a smoldering raccoon wedged in the fender and decided it was my lucky day.
"You two a ways from Circle Pines," the officer said suspiciously.
"REI having a sale on wire leaders," I explained.
"My stars. You two going out for muskie?"
"Every year," I said.
"Fish of a thousand casts," Donny chimed in. He was off by an order of magnitude. I let it slide.
She circled around to the front of the Chevelle and grimaced at our stowaway.
"County maintenance," I said, hoping to sway her to our cause by exploiting interdepartmental rivalry in the Minnesota bureaucracy.
"Yah" she said, nodding gravely. "Welp. Let's see about getting you on your way. "
The three of us joined forces to clear the wheel well like some kind of samizdat jpop band. Donny stabbed at the carcass with the quality poking stick I had provided while the Statey bounced the bumper. Meanwhile, I sat in the driver's seat and cranked the tiller left and right.
There was much cursing from Donny, but the bond he had formed with the Chevelle lo these many years gave him the strength to power through. Chunk by fetid chunk he worked his beloved from this meat prison like Michelangelo calling forth David from a marble slab. The main bulk of the carcass finally relinquished its grip after a final stabbing frenzy. It fell to the pavement with a desiccated thud.
We gathered around the corpse, attempting to mark its passing with some measure of dignity. It was, after all, one of God's creatures, no less than the three of us.
The Statey punted it into the grass. "Yah know, these things got the rabies," she said. Donny glared at me sideways, like my sister did.
We waited until MHP was out of sight and earshot, then Donny turned the motor over. He tapped his class ring against the side of the shifter and studied approaching traffic in the rearview mirror. The Chevelle idled with the grumble of an impatient predator, threatening to stall if you didn't goose the throttle from time to time. It was the price you accepted for an aggressive camshaft.
The gap Donny was waiting for materialized and he dropped the clutch. The Chevelle roared to life. It's Thrush pipes called out to things living or otherwise in a deafening warning that cleared all from our path or who might dare to be. Fus ro dah.
Three tire chirps later we were at cruising speed. Donny let off the throttle. The tach dropped to idle but the Chevelle did not slow, a gift of Valvoline lubrication and proper tire pressure.
And an absence of Jacobian.
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