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3 Rollback to Revision 1

EDIT:

The story is too good to miss, so I am adding the text to my answer. I know a lot of people don't like to click on links.

It was a snowy Christmas Eve. As the wind howled outside thecastle tower, the Wizard was playing his flute by the fireside,meditating. All the final exams were graded, the apprenticeshad all gone home, and he had spent an intense week working on Lie n-groupoids and a thought experiment which showed that quantumgravity effects made it impossible to measure distances withan uncertatinly less than Planck length. Now it was time torelax! The fire crackled, the wind whistled down the chimney,the flute's melody swirled like the smoke... it was all very peaceful until he heard someone pounding on his door.

"Whoever it is, go away!" shouted the Wiz.

A muffled voice came from behind the door. "It's me! Eric!"

"All the more reason!" growled the Wiz under his breath,putting down his flute and walking to the door after givingthe fire a poke.

"I come seeking knowledge!"

"Oh yeah?" said the Wiz, heaving the oak door open on a crackand sticking his nose out. "What exactly do you want to know?"

"All I'm asking for is reasons!" cried Eric.

"Reasons for what?!"

"You keep saying I am just being ignorant, but I'm holding out until I see some honest to goodness reasons why pseudoforms are necessary."

The Wizard began to fume. "Not that again!" he said. He triedto slam the door shut, but Eric had already slid his foot in thecrack.

"Ow! You're not getting rid of me that easily!"

"Hmm," said the Wiz and pondered a while. He couldn't helpadmiring Eric's persistence. He'd once had an apprentice namedOz, whose education was woefully inadequate in every way, butwho managed to make real progress nonetheless, thanks to his almostinsane persistence. Unfortunately, Oz had gotten swallowed by another universe as part of a disastrous experiment, and so far all the Wizard's attempts to recover him had failed. Out of pity and fond memories, the Wiz decided to give Eric one more chance.
It was Christmas Eve, after all....

Peering through the crack, he started with a sneaky question: "What sort of thing can we integrate over any smooth n-dimensional manifold?"

Eric cried: "An n-form! This is why differential forms areso great! Everything we actually measure, physically, is theintegral of some quantity over some region, or curve, or...."

"Wait," interrupted the Wiz. "Before you launch into yourpersonal philosophy again, let's get the facts straight. It's NOTTRUE that you can integrate an n-form over a smooth n-dimensionalmanifold. What you can integrate is a PSEUDO n-form."

Eric's mouth snapped shut in mid-rant, and then fell open againin shock. "Hmm... you've got my attention now!" he said.

"Good," growled the Wiz. "So, listen up: You canonly integrate an n-form over a smooth n-dimensional manifoldif it is equipped with an ORIENTATION. You may be so usedto this that you've come to accept the orientation as an inevitableprerequisite for integration. But it's not true! Integrationof pseudo n-forms works perfectly fine on any smooth manifold,even an unoriented or unorientable one. It's only if you makethe mistake of trying to integrate an N-FORM" - he practicallyspat the term out in disgust - "that you'll need an orientation.And all the orientation does is let you convert your n-form toa pseudo n-form! Correcting one bad move with another...." He trailed off, grimacing at the folly of the world.

"Hmm... this sounds very interesting," said Eric.

"Good. More to the point, it's the truth," said the Wiz.

"Why didn't any of the 10+ books I have on differential geometryand forms mention this?!"

"What do you expect if you read so few books?!" fired back theWiz. "You get what you pay for. But surely even those miserabletomes said you could only integrate an n-form on an n-manifoldthat was oriented. No? And if so, surely it was your jobto wonder whether it was possible or not to do integrals on anunoriented manifold! You should have pondered a Moebius strip, and asked yourself: What's to prevent me from doing integralson this thing?' You'd chop it up into coordinate charts, dothe integral on each piece with the help of a partition of unity,and then add up the results.... And after a little thought, you would have discovered the answer: it all works perfectly fine, if you stick in an extra minus sign to describe how your n-formtransforms under a change of coordinates whose Jacobian has negative determinant! In other words, you should use the absolute valueof the Jacobian, just like the change of variables formula inmultivariable calculus tells you to! But when you do this, you'vereinvented pseudo n-forms.

And then, after asking around a bit, you'd have discovered thateveryone... everyone who counts, that is... has already realized this!"

Eric pondered a moment. "But... but if it's that simple, whydon't my textbooks talk about it?"

Grinning evilly, the Wiz replied: "They certainly drop thenecessary clues. As for why they don't emphasize this stuff, well, this is just one of those tricks we Wizards use to distinguish the people who think for themselves from those who fall for any plausible line of claptrap. In fact, every time Halloween falls on a full moon - like this year - we get together and agree on what facts like this we will keep secret, precisely to see whorediscovers it for themselves. This year we...." At this point the absent-minded Wiz caught himself. "Whoops! This is toosecret for you! Anyway, you just failed one of these tests."

Mortally offended, but (admirably) more interested in the math than his own dignity, Eric replied "You might shoot me, but you are actually starting to make me think that perhaps it ispseudoforms that are more fundamental than regular forms!"

"Great!" said the Wiz. "Actually, this will relieve mefrom the need to shoot you, or, more likely, turn you into a toad.Anyway, I hope you see now that people who think they're really integrating n-forms are just slightly less naive than the people who think they're really integrating functions. You can only integrate these things if your n-manifold is equipped with extra structure... and this extra structure is only being used to convert these things to pseudo n-forms!"

"Well," admitted Eric ruefully, "I certainly consider myself to be naive. Honestly, I never knew this. I would really like to understand this better. Got any references?"

"References?!" cried the Wiz. "What do you think I am, a walkingjournal article? I give you all the clues you need to figure everythingout for yourself, and you want references?? Here - HERE's yourmuch-beloved REFERENCES!" he said, flinging a fireball at Eric.

The fireball was large and blue-white, hissing furiously.Luckily for Eric, his reactions were quick and he ducked in time to miss being fried. It hit the wall behind him and exploded in a bang. Simultaneously, the Wiz pushed Eric'sfoot out from the door and slammed the door shut.

Then he opened a small flap and poked his face out for amoment. "Oh, and by the way: Merry Christmas!" He tossed out a small candy-cane and snapped the flap shut.

Eric was left with his questions. Sucking mournfully on the candy, he stumbled down the castle steps and out into the chilly night.

"Oh, oh!" he wailed. "Can't ANYONE give me some references??"

2 added 7710 characters in body

EDIT:

The story is too good to miss, so I am adding the text to my answer. I know a lot of people don't like to click on links.

It was a snowy Christmas Eve. As the wind howled outside thecastle tower, the Wizard was playing his flute by the fireside,meditating. All the final exams were graded, the apprenticeshad all gone home, and he had spent an intense week working on Lie n-groupoids and a thought experiment which showed that quantumgravity effects made it impossible to measure distances withan uncertatinly less than Planck length. Now it was time torelax! The fire crackled, the wind whistled down the chimney,the flute's melody swirled like the smoke... it was all very peaceful until he heard someone pounding on his door.

"Whoever it is, go away!" shouted the Wiz.

A muffled voice came from behind the door. "It's me! Eric!"

"All the more reason!" growled the Wiz under his breath,putting down his flute and walking to the door after givingthe fire a poke.

"I come seeking knowledge!"

"Oh yeah?" said the Wiz, heaving the oak door open on a crackand sticking his nose out. "What exactly do you want to know?"

"All I'm asking for is reasons!" cried Eric.

"Reasons for what?!"

"You keep saying I am just being ignorant, but I'm holding out until I see some honest to goodness reasons why pseudoforms are necessary."

The Wizard began to fume. "Not that again!" he said. He triedto slam the door shut, but Eric had already slid his foot in thecrack.

"Ow! You're not getting rid of me that easily!"

"Hmm," said the Wiz and pondered a while. He couldn't helpadmiring Eric's persistence. He'd once had an apprentice namedOz, whose education was woefully inadequate in every way, butwho managed to make real progress nonetheless, thanks to his almostinsane persistence. Unfortunately, Oz had gotten swallowed by another universe as part of a disastrous experiment, and so far all the Wizard's attempts to recover him had failed. Out of pity and fond memories, the Wiz decided to give Eric one more chance.
It was Christmas Eve, after all....

Peering through the crack, he started with a sneaky question: "What sort of thing can we integrate over any smooth n-dimensional manifold?"

Eric cried: "An n-form! This is why differential forms areso great! Everything we actually measure, physically, is theintegral of some quantity over some region, or curve, or...."

"Wait," interrupted the Wiz. "Before you launch into yourpersonal philosophy again, let's get the facts straight. It's NOTTRUE that you can integrate an n-form over a smooth n-dimensionalmanifold. What you can integrate is a PSEUDO n-form."

Eric's mouth snapped shut in mid-rant, and then fell open againin shock. "Hmm... you've got my attention now!" he said.

"Good," growled the Wiz. "So, listen up: You canonly integrate an n-form over a smooth n-dimensional manifoldif it is equipped with an ORIENTATION. You may be so usedto this that you've come to accept the orientation as an inevitableprerequisite for integration. But it's not true! Integrationof pseudo n-forms works perfectly fine on any smooth manifold,even an unoriented or unorientable one. It's only if you makethe mistake of trying to integrate an N-FORM" - he practicallyspat the term out in disgust - "that you'll need an orientation.And all the orientation does is let you convert your n-form toa pseudo n-form! Correcting one bad move with another...." He trailed off, grimacing at the folly of the world.

"Hmm... this sounds very interesting," said Eric.

"Good. More to the point, it's the truth," said the Wiz.

"Why didn't any of the 10+ books I have on differential geometryand forms mention this?!"

"What do you expect if you read so few books?!" fired back theWiz. "You get what you pay for. But surely even those miserabletomes said you could only integrate an n-form on an n-manifoldthat was oriented. No? And if so, surely it was your jobto wonder whether it was possible or not to do integrals on anunoriented manifold! You should have pondered a Moebius strip, and asked yourself: What's to prevent me from doing integralson this thing?' You'd chop it up into coordinate charts, dothe integral on each piece with the help of a partition of unity,and then add up the results.... And after a little thought, you would have discovered the answer: it all works perfectly fine, if you stick in an extra minus sign to describe how your n-formtransforms under a change of coordinates whose Jacobian has negative determinant! In other words, you should use the absolute valueof the Jacobian, just like the change of variables formula inmultivariable calculus tells you to! But when you do this, you'vereinvented pseudo n-forms.

And then, after asking around a bit, you'd have discovered thateveryone... everyone who counts, that is... has already realized this!"

Eric pondered a moment. "But... but if it's that simple, whydon't my textbooks talk about it?"

Grinning evilly, the Wiz replied: "They certainly drop thenecessary clues. As for why they don't emphasize this stuff, well, this is just one of those tricks we Wizards use to distinguish the people who think for themselves from those who fall for any plausible line of claptrap. In fact, every time Halloween falls on a full moon - like this year - we get together and agree on what facts like this we will keep secret, precisely to see whorediscovers it for themselves. This year we...." At this point the absent-minded Wiz caught himself. "Whoops! This is toosecret for you! Anyway, you just failed one of these tests."

Mortally offended, but (admirably) more interested in the math than his own dignity, Eric replied "You might shoot me, but you are actually starting to make me think that perhaps it ispseudoforms that are more fundamental than regular forms!"

"Great!" said the Wiz. "Actually, this will relieve mefrom the need to shoot you, or, more likely, turn you into a toad.Anyway, I hope you see now that people who think they're really integrating n-forms are just slightly less naive than the people who think they're really integrating functions. You can only integrate these things if your n-manifold is equipped with extra structure... and this extra structure is only being used to convert these things to pseudo n-forms!"

"Well," admitted Eric ruefully, "I certainly consider myself to be naive. Honestly, I never knew this. I would really like to understand this better. Got any references?"

"References?!" cried the Wiz. "What do you think I am, a walkingjournal article? I give you all the clues you need to figure everythingout for yourself, and you want references?? Here - HERE's yourmuch-beloved REFERENCES!" he said, flinging a fireball at Eric.

The fireball was large and blue-white, hissing furiously.Luckily for Eric, his reactions were quick and he ducked in time to miss being fried. It hit the wall behind him and exploded in a bang. Simultaneously, the Wiz pushed Eric'sfoot out from the door and slammed the door shut.

Then he opened a small flap and poked his face out for amoment. "Oh, and by the way: Merry Christmas!" He tossed out a small candy-cane and snapped the flap shut.

Eric was left with his questions. Sucking mournfully on the candy, he stumbled down the castle steps and out into the chilly night.

"Oh, oh!" he wailed. "Can't ANYONE give me some references??"

1

The problem is that there is no way to figure out signs - It would be like trying to integrate a function from $\mathbb{R}$ to $\mathbb{R}$ without knowing whether you were moving forward or backward.

What you CAN actually integrate are pseudo-differential forms. The whole point of choosing an orientation is to turn a differential form into a psuedo-differential form. For those, I recommend the wonderful short story by John Baez found here: