MAN, these are just spectacularly well explained videos! I've sat here now and watched 1 through 6 straight through without getting up from my seat once (and I have a weak bladder).
Oh, Ozmo, I think you would be a good person to ask, have you heard of a man named Nassim Haramein, and his Schwarzschild Proton singularity theory? I wish I was in a position to know if any of his math and theories are mathematically sound. I know some of his ideas he advocates are way way out there, but his universal scaling theory seems really cool. If it's true. Just wondering if you could answer a bit on how sound it all is?
@innovaxl I hadn't heard of him, but I looked him up. I scanned his Schwarzschild Proton paper. I would say his ideas are not sound. As an example, if any sub-atomic particles were blackholes we wouldn't be able to smash them apart in particle colliders, yet we do that regularly.
If he claims protons are black holes, then yes. Protons and other non-fundamental subatomic particles would not be black holes, and would be able to smash perfectly fine into the smaller components. But the seemingly fundamental particles such as up/charm/electron might be black holes, as they can't be smashed.
The reason to not expect those particles to be black holes is that small black holes are expected to evaporate at an alarming rate, which these particles do not do.
The only thing I know about Nassim Haramein is that his wiki article got deleted. There are no third party articles for his 'research' to be found anywhere.
On first impressions, he raises all flags of quackery.
Thankyou for the series. I can't wait for the next one. Hopefully it won't be as long a wait from the last video. It looks like the graphs in this one might have taken a bit more time to put together than in the previous videos of the series.
On more related note, despite math being my weakest subject, I can understand this. Not in the technical details, but I understand how it's formed, what it does and why you'd need it.
I just have no idea how to apply it.
Still... that's math even I'd call 'cool/awesome'.
Thank you Oz for giving a set of videos that attempt to bridge the gap between what we think we know, what we should know, and what we don't know. Tall order but so far I think your achieving it. Good Job.
Interesting, I just came upon this topic in a totally different field: robotics! Turns out this is very useful for dynamic optimization in robotic arms, if you consider the arm's joints as making a curved coordinate system. I had no idea I already knew the basics required for relativity! Cool stuff.
@Version2PointZero Yeah, I had to muddle through the math on that myself for my own crappy little robot. I wish I'd chosen six legs instead of four though. I know some of the bigger industrial arms have quite a few joints and probably a more involved type of solution finding for movement.
@DGPR105 The idea is that "normal" 3D space is "flat" (the "regular" Pythagorean theorem works in it). The surface is 2D (it takes two numbers to specify position on the surface) but is curved. That curved surface is embedded in flat 3D space.
@ozmoroid When you say that "'normal' 3D space is 'flat'," are you referring to the fact that it is not warped in the way that gravity warps spacetime?
@DGPR105 Imagine having a mountain and then cutting it into slices. If you draw the projection of the bottom surface(the perimeter of it actually) you can get these shapes.
The longer the distance of them the less steeper the mountain is. Now can you imagine this on derivatives? ;) I don't know the english word for this but in greek it is "ισοσταθμικές curves"
So hard to visualize general relativity without using math (matrix and equation) and the basic knowledge of tensors...(Which is also complex to grasp, tensors notation I mean) But you got it covered! Nice video! And yes, feel free to use the metric system!!! :)
@animimm I used English units on purpose to emphasize the fact that coordinate systems are arbitrary. If you follow the series you will see that I primarily use seconds and light-seconds for time and distance measurements.
The best path would be the contour line, but that might not be the shortest. A compromise would allow alterations of altitude within the capabilities of the motor to give the best compromise between shortest time and best fuel usage.
I got to curved co-ordinate systems and my brain started thumping the inside of my skull in frustration. I only took Uni maths in my first year, and I only got around a 50% average for the four courses I had to take - it's the entire reason I couldn't continue taking Physics into my second year (I had a much nicer 70% average for my physics marks) - I just knew stuff like this would come up and my brain would fry.
I'm glad that I can see what you mean without understanding the math - well done.
'dx' or any other form of this nonsense, does not exist!
It is supposed to be a 'quantity' so small that it is smaller than anything else, on the other hand it also is not zero... this doesn't stand its ground in any rigorous theory. Damn you, physicists! First define 'differentials' and then we'll talk!
^^ other than this rant of a frustrated Math student, nice video!
@ImaginaryMdA Take a few semesters of calculus (and then afterwards a Classical Mechanics course) and you will see how useful differentials are in modeling real world phenomena. If you want the definition of a differential, google it, there are tons of excellent resources. If you don't have much of a math background, I recommennd the Khan Academy. They have really good explanations of mathematical concepts up through Calculus.
Excuse me? not much of a Math background? Let me introduce myself a little bit, I am in my second year studying pure mathematics at university, my total grades last year included: 2 times 20/20 and some 4/5 times 19/20, 19/20 was also given to me on the course you talk about: Calculus I and 18 for Calculus II, so I think my background is quite all right, thank you.
BTW my teachers too are frustrated with the excessive use of the outdated concept of a differential!
@ImaginaryMdA here the differential is just a one form, a concept that works really well in differential geometry, the one form space(it's a vector space) is the set of linear function from a vector space to the reals. The cool things is that the vector space are one forms for one forms, that's why they are called duel spaces to one another. btw I mean no offence
@ImaginaryMdA I was in no way attacking your math competency, I just assumed since your understanding of the mathematical branch of analysis seemed limited you might not be fully aware of just how differentials (which are certainly not an outdated concept) are defined. I'm glad that you have done well for yourself in your mathematics education so far, but you have some misperceptions about the way differentials are defined. It will make more sense to you in future math studies very soon for you.
@mage1over137 True, but the way I use differentials in this video (thinking of dx as the limit of delta x) is a little bit "loose" from a mathematical point of view.
This is amazing and SIMPLE! As a math student, this is the first time i really truly understood the thing about event horizon! Thank you very much for this video!
I next year i have exam on differential geometry, and now i can realize fundamentals of it, thanks to you and this video!
What'chou talkin' about, Willis? My brain gave up about a third of the way in, but I'm still interested to find out how this all turns out. I hope there will be more pictures next time. ...I like pictures...
Oh yeah, and since the Higgs Boson has been in the news recently, I hope you'll make a video sometime about that if you know anything about it.
Thanks for this series of videos - it took Einstein ten years and a lot of help from his friends and colleagues to find the right bit of mathematics and then move special relativity to general relativity and the mathematics is hard but your explanation was very clear and visual elements helps. Modern geometry tends to be taught without images and it makes it much harder. I understood this area of mathematics a lot more when I studied Visual Complex Analysis by Tristan Needham.
@LlamaTheory You've probably done classical electromagnetism (Maxwell's Equations). If you haven't, I expect you will soon. You can formulate Maxwell's Equations in tensor form as well (and they get smaller in the process, reducing to a single equation encompassing all electromagnetic phenomena) but the actual math used in working them is no more complex than before, and about as complex as this.
GR matrices have more terms, and some awkward links between terms, but not too bad.
@stiimuli Hopefully the viewer should be able to gloss over the specific equations and focus on the concepts visually - at least that's what I'm aiming for. Equations are only necessary when you want to get numbers out of the theory, make precise predictions and that sort of thing. But maybe I need to work in more visuals and/or animations.
@ozmoroid I understood the concepts via the mathematics. But too, I tutored freshman physics for a year and found an alarming number of my student reviews contained novel misspellings of "tutor" as "torture."
@ozmoroid i stand corrected. i checked. in my statistics class for a year my teacher called it epsilon perhaps he did this so we wouldn't confuse smaller case sigma (standard deviation) with upper case sigma (sum of listed products).
Holy crap... I didn't think you would go for general relativity, the most notoriously difficult physics topic for pedagogy... but you are doing it well
your videos always gives me a headache. and im drunk so i shouldnt have an headache untill tomorrow. luckily english is not my native language so i understand maybe every third word. i can imagine a lot of brain matter on the walls of your viewers. a lot of bangs around the world. made by a head.
I'm assuming the answer to the question "What is the shortest path" could be derived via an optimization problem using the dx, dy, and ds values. I'm still in awe at how people like Gauss could tackle this mufti-variable calculus back in the 18th and 19th centuries.
Fairly easily (in the sense that the prerequisites are low, not in it not being difficult math), since it doesn't actually depend on much in the way of scientific discoveries or the technology itself. Math and logic are pretty much the only things you can actually solve entirely as a thought experiment at any level of technology.
@rkyeun While the math is probably not that difficult the volume of calculations might quickly grow since finding the shortest path around a set of hill is probably an NP-complete problem. I would not even start without a computer and if Gauss really solved real-world problems with this method I am in awe. (well, I am anyway, but even more so).
I'm pretty sure you just apply the jacobian to convert the field into a flat plane with distorted distances, then integrate the distance formula over the field (That's your NP-complete part, but we do integration all the time), and take the derivative of the result to find where the distance formula has a minimum.
But it's been way too long since I had calculus.
If the landscape isn't trivially simple, finding the solution will be much much harder.
@rkyeun Maybe you are right, but at every "hill" you can choose to go to the left or the right and for whatever choice you have to find a minimum and to know which one is the shortest you have to calculate them both. That's why I think it is an NP-complete problem. From what I gather it is much like the "travelling salesman" problem (I assume you are familiar with that) except that it require more computing power.
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MAN, these are just spectacularly well explained videos! I've sat here now and watched 1 through 6 straight through without getting up from my seat once (and I have a weak bladder).
mphello 1 week ago
...You lost me.
ShinobuHarvester 3 weeks ago
Dafuq:/
Jaymaul009 3 weeks ago
Oh, Ozmo, I think you would be a good person to ask, have you heard of a man named Nassim Haramein, and his Schwarzschild Proton singularity theory? I wish I was in a position to know if any of his math and theories are mathematically sound. I know some of his ideas he advocates are way way out there, but his universal scaling theory seems really cool. If it's true. Just wondering if you could answer a bit on how sound it all is?
innovaxl 1 month ago in playlist Physics
@innovaxl I hadn't heard of him, but I looked him up. I scanned his Schwarzschild Proton paper. I would say his ideas are not sound. As an example, if any sub-atomic particles were blackholes we wouldn't be able to smash them apart in particle colliders, yet we do that regularly.
ozmoroid 3 weeks ago
@ozmoroid
If he claims protons are black holes, then yes. Protons and other non-fundamental subatomic particles would not be black holes, and would be able to smash perfectly fine into the smaller components. But the seemingly fundamental particles such as up/charm/electron might be black holes, as they can't be smashed.
The reason to not expect those particles to be black holes is that small black holes are expected to evaporate at an alarming rate, which these particles do not do.
rkyeun 3 weeks ago
The only thing I know about Nassim Haramein is that his wiki article got deleted. There are no third party articles for his 'research' to be found anywhere.
On first impressions, he raises all flags of quackery.
StrikaAmaru 1 week ago
Thankyou for the series. I can't wait for the next one. Hopefully it won't be as long a wait from the last video. It looks like the graphs in this one might have taken a bit more time to put together than in the previous videos of the series.
innovaxl 1 month ago in playlist Physics
G.I. Jay...
Sorry, I was too compelled to make that lame one.
On more related note, despite math being my weakest subject, I can understand this. Not in the technical details, but I understand how it's formed, what it does and why you'd need it.
I just have no idea how to apply it.
Still... that's math even I'd call 'cool/awesome'.
I want more, I say! :)
AchmeZe 1 month ago
Thank you Oz for giving a set of videos that attempt to bridge the gap between what we think we know, what we should know, and what we don't know. Tall order but so far I think your achieving it. Good Job.
Briekout 1 month ago
Please continue this series soon! I love it!
KamiKagutsuchi 1 month ago
Interesting, I just came upon this topic in a totally different field: robotics! Turns out this is very useful for dynamic optimization in robotic arms, if you consider the arm's joints as making a curved coordinate system. I had no idea I already knew the basics required for relativity! Cool stuff.
Version2PointZero 1 month ago in playlist Uploaded videos
@Version2PointZero Plus the rotation/spinning stuff is big in quantum mechanics.
ozmoroid 1 month ago
@Version2PointZero Yeah, I had to muddle through the math on that myself for my own crappy little robot. I wish I'd chosen six legs instead of four though. I know some of the bigger industrial arms have quite a few joints and probably a more involved type of solution finding for movement.
htomerif 1 month ago
One never perceives herself to be moving at all.
WaiteDavidMSPhysics 1 month ago
@ImaginaryMdA
Which concept of differentials is outdated and why?
about dx:
I always thought, that dx was defined as "dx=|x1,x2|=1/n for n->UsDebt".
Why isnt it rigorous?
FernestHall 2 months ago
@FernestHall The limit you use goes to zero, which is not usefull.
mutomeister 1 month ago
feet yards fathoms?1?1 WTF man, use SI units.
KaerFyzarc 2 months ago
@KaerFyzarc English units are used for illustrative purposes. The "real" work is done with seconds and light-seconds.
ozmoroid 2 months ago
@ozmoroid
At 1:45, do you mean to say that a 3D curved surface is embedded in 2D flat space?
I don't have much experience with mathematics, but it seems to make more sense this way.
DGPR105 2 months ago
@DGPR105 The idea is that "normal" 3D space is "flat" (the "regular" Pythagorean theorem works in it). The surface is 2D (it takes two numbers to specify position on the surface) but is curved. That curved surface is embedded in flat 3D space.
ozmoroid 2 months ago
@ozmoroid When you say that "'normal' 3D space is 'flat'," are you referring to the fact that it is not warped in the way that gravity warps spacetime?
DGPR105 2 months ago
@DGPR105 Imagine having a mountain and then cutting it into slices. If you draw the projection of the bottom surface(the perimeter of it actually) you can get these shapes.
The longer the distance of them the less steeper the mountain is. Now can you imagine this on derivatives? ;) I don't know the english word for this but in greek it is "ισοσταθμικές curves"
ironis24 2 months ago
I didn't understand a word of this. I tried, and I was enjoying the series to this point, but now I'm completely lost. Utterly and completely.
gamutman 2 months ago
So hard to visualize general relativity without using math (matrix and equation) and the basic knowledge of tensors...(Which is also complex to grasp, tensors notation I mean) But you got it covered! Nice video! And yes, feel free to use the metric system!!! :)
sorad10 2 months ago
@sorad10 Thanks. The "real" units I'm using are seconds and light-seconds.
ozmoroid 2 months ago
My favorite thing on Youtube, I can't get enough of your relativity series.
Bishop134265 2 months ago
Bugger to this, I'm building bridges & tunnelling!
MrLittletomdj 2 months ago
It couldn't be any simpler! :-(
MsWanderer1 2 months ago
yards, miles, feet, fingers, dicks...
STOP USING THIS STUPID MESUREMENT SYSTEM. The whole world is using METRIC SYSTEM
animimm 2 months ago
@animimm I used English units on purpose to emphasize the fact that coordinate systems are arbitrary. If you follow the series you will see that I primarily use seconds and light-seconds for time and distance measurements.
ozmoroid 2 months ago
@ozmoroid I watched the whole series and enjoyed it. Thank you.
animimm 2 months ago
@animimm I agree
dumbnetworks 2 months ago
Math overload... I'm going to have to watch that one again, when I have a few more minutes to let it sink in.
DSAhmed 2 months ago
The best path would be the contour line, but that might not be the shortest. A compromise would allow alterations of altitude within the capabilities of the motor to give the best compromise between shortest time and best fuel usage.
RyuDarragh 2 months ago
You don't have to know the math in this video. But ignoring the math is like watching TV and turning the sound off.
alueshen 2 months ago
My brain hurts!
Madman1234855 2 months ago
I got to curved co-ordinate systems and my brain started thumping the inside of my skull in frustration. I only took Uni maths in my first year, and I only got around a 50% average for the four courses I had to take - it's the entire reason I couldn't continue taking Physics into my second year (I had a much nicer 70% average for my physics marks) - I just knew stuff like this would come up and my brain would fry.
I'm glad that I can see what you mean without understanding the math - well done.
sirnlawson 2 months ago
Over. My. Head.
bamboo4tameshigiri 2 months ago
I love this video series!
ocdfreak 2 months ago
Cool Series
TankFridaysSeacret2 2 months ago
Wow, my 2 years of uni math ran out along the way, but I could still follow due to your excellent conceptualisation.
hilbert54 2 months ago
Beautifully done.
AuntieDiluvian 2 months ago
'dx' or any other form of this nonsense, does not exist!
It is supposed to be a 'quantity' so small that it is smaller than anything else, on the other hand it also is not zero... this doesn't stand its ground in any rigorous theory. Damn you, physicists! First define 'differentials' and then we'll talk!
^^ other than this rant of a frustrated Math student, nice video!
ImaginaryMdA 2 months ago
@ImaginaryMdA Take a few semesters of calculus (and then afterwards a Classical Mechanics course) and you will see how useful differentials are in modeling real world phenomena. If you want the definition of a differential, google it, there are tons of excellent resources. If you don't have much of a math background, I recommennd the Khan Academy. They have really good explanations of mathematical concepts up through Calculus.
ocdfreak 2 months ago
@ocdfreak
Excuse me? not much of a Math background? Let me introduce myself a little bit, I am in my second year studying pure mathematics at university, my total grades last year included: 2 times 20/20 and some 4/5 times 19/20, 19/20 was also given to me on the course you talk about: Calculus I and 18 for Calculus II, so I think my background is quite all right, thank you.
BTW my teachers too are frustrated with the excessive use of the outdated concept of a differential!
ImaginaryMdA 2 months ago
@ImaginaryMdA here the differential is just a one form, a concept that works really well in differential geometry, the one form space(it's a vector space) is the set of linear function from a vector space to the reals. The cool things is that the vector space are one forms for one forms, that's why they are called duel spaces to one another. btw I mean no offence
mage1over137 1 month ago
@ImaginaryMdA I was in no way attacking your math competency, I just assumed since your understanding of the mathematical branch of analysis seemed limited you might not be fully aware of just how differentials (which are certainly not an outdated concept) are defined. I'm glad that you have done well for yourself in your mathematics education so far, but you have some misperceptions about the way differentials are defined. It will make more sense to you in future math studies very soon for you.
ocdfreak 3 weeks ago
@ImaginaryMdA I know differentials aren't mathematically rigorous, but they are so darned useful for thinking about physics.
ozmoroid 2 months ago
@ozmoroid Isn't dx a well defined basis one form especially in this context?
mage1over137 1 month ago
@mage1over137 True, but the way I use differentials in this video (thinking of dx as the limit of delta x) is a little bit "loose" from a mathematical point of view.
ozmoroid 1 month ago in playlist Physics
This is one of the best explanations I have seen for the math. Thanks, I can actually understand it!
Fishalways 2 months ago
This is ridiculously nice series! ;) Keep up the great work.
morphles 2 months ago
I'm clad you found your thing to do on youtube ozmoroid
darkdragonsoul99 2 months ago
Great job oz, as most people learn and comprehend things differently the visual component of your vids really does help.
AussieNaturalist 2 months ago
Yes. Part 7. Finally :)
MrShyguy0 2 months ago
This is amazing and SIMPLE! As a math student, this is the first time i really truly understood the thing about event horizon! Thank you very much for this video!
I next year i have exam on differential geometry, and now i can realize fundamentals of it, thanks to you and this video!
gerilac86 2 months ago
What'chou talkin' about, Willis? My brain gave up about a third of the way in, but I'm still interested to find out how this all turns out. I hope there will be more pictures next time. ...I like pictures...
Oh yeah, and since the Higgs Boson has been in the news recently, I hope you'll make a video sometime about that if you know anything about it.
Rationalific 2 months ago
Your voice is to awesome to actually learn anything from this.
JupiterJazzjaa2 2 months ago
It sounds like part 2 of this video is going to be very ambitious when it comes to the visuals. Looking forward to it. Thanks for this!
TheFaithCheck 2 months ago
Thanks for this series of videos - it took Einstein ten years and a lot of help from his friends and colleagues to find the right bit of mathematics and then move special relativity to general relativity and the mathematics is hard but your explanation was very clear and visual elements helps. Modern geometry tends to be taught without images and it makes it much harder. I understood this area of mathematics a lot more when I studied Visual Complex Analysis by Tristan Needham.
johncrwarner 2 months ago
I'm currently a junior in engineering... haven't seen anything this complex yet. Differentiating matrices.... god that's brutal.
LlamaTheory 2 months ago
@LlamaTheory You can always let a computer algebra program do it for you. ;-)
ozmoroid 2 months ago
@ozmoroid It's nice to know how to do it by hand, but yes... Wolfram Alpha is my best friend.
LlamaTheory 2 months ago
@LlamaTheory
That's Calculus III, at my college.
And sometimes we differentiated them, and sometimes we integrated them.
And we didn't get to use graphing calculators, OH BOY!
I failed that class four times and then got a C.
rkyeun 2 months ago
@LlamaTheory
Are you talking about Jacobians? I don't remember him mentioning differentiating matrices per se, but there is a metric tensor coefficient matrix.
Tr3xKuro 2 months ago
@LlamaTheory You've probably done classical electromagnetism (Maxwell's Equations). If you haven't, I expect you will soon. You can formulate Maxwell's Equations in tensor form as well (and they get smaller in the process, reducing to a single equation encompassing all electromagnetic phenomena) but the actual math used in working them is no more complex than before, and about as complex as this.
GR matrices have more terms, and some awkward links between terms, but not too bad.
evensgrey 2 months ago
I REALLY want to understand this but every time you start to talk about equations I start to smell smoke
=(
stiimuli 2 months ago
@stiimuli Hopefully the viewer should be able to gloss over the specific equations and focus on the concepts visually - at least that's what I'm aiming for. Equations are only necessary when you want to get numbers out of the theory, make precise predictions and that sort of thing. But maybe I need to work in more visuals and/or animations.
ozmoroid 2 months ago
@ozmoroid I understood the concepts via the mathematics. But too, I tutored freshman physics for a year and found an alarming number of my student reviews contained novel misspellings of "tutor" as "torture."
alivingock 2 months ago in playlist Uploaded videos
Nice! I'm going into engineering myself, so it's good to refresh my mind every now and then :)
pythor2 2 months ago
the greek letter you used at 6:55 was not sigma, it was epsilon. but yes it means the sum of.
greycloud24 2 months ago
@greycloud24 No, it is (capital) sigma. At least that's what the entire math and science community calls it.
dXoverdteqprogress 2 months ago
@greycloud24 I'm pretty sure it's capital sigma. See Greek_alphabet on Wikipedia.
ozmoroid 2 months ago
@ozmoroid It is capital sigma. But it does look like a lowercase Epsilon I guess.
Awesome video, loved every minute of it!
MindLessWiz 2 months ago
@ozmoroid i stand corrected. i checked. in my statistics class for a year my teacher called it epsilon perhaps he did this so we wouldn't confuse smaller case sigma (standard deviation) with upper case sigma (sum of listed products).
greycloud24 2 months ago
@ozmoroid Capital Epsilon looks almost exactly like E. That's definitely Capital Sigmas in the video (just like it should be).
evensgrey 2 months ago
@greycloud24
Σ is capital sigma, Ε is capital epsilon. Small sigma is σ and ς at the end of words. Small epsilon is ε. Does that help?
Quintinohthree 2 months ago
@greycloud24
100% incorrect.
Tr3xKuro 2 months ago
I thought Gauss sounded more like house then pass ?
SaintCog 2 months ago
@SaintCog I think you're correct. Something like Gaouse.
ozmoroid 2 months ago
@SaintCog "you say tomatoe i say potato" - Ali G ;)
squizill 2 months ago
@SaintCog -- yes, Gauss like »house« or »mouse«.
The »Friedrich« part on the other hand, let's try »free-dreeχ« (Greek small letter chi, like Donald Knuth’s Teχ). Unicode for the win.
virumoz 2 months ago
I just realized that I've been watching this for 10 minutes, but I completely blanked out. Nice try though, ozmoroid. I appreciate it.
TadRaunch 2 months ago
Thank you Ozmproid for teaching me humility.
TheFeltbegone 2 months ago
Holy crap... I didn't think you would go for general relativity, the most notoriously difficult physics topic for pedagogy... but you are doing it well
Tr3xKuro 2 months ago in playlist Physics
Lay people just make roads on the north side ,in the northern hemisphere anyway..And us a big bulldozer.
no1saphead 2 months ago
You do a beautiful job of explaining these types of concepts, my thanks to you.
VapidZero 2 months ago
I thought I was going to have to mop my brain up off the floor, but I surprised myself in being able to follow along to the end. Yay me.
Locke99GS 2 months ago
Damn I miss all those math and math-o-centric science classes
skoockum 2 months ago
Hahaha! Those symbols are hilarious!
glifencible 2 months ago
Can you help me plot the shortest distance to Drew Barrymore's vagina?
RectalItching 2 months ago
@RectalItching About 13 rum & diet Cokes
GoblinXXX 2 months ago
Great stuff ozmo....less than 2 minutes in and already I've had a braingasm
skoockum 2 months ago
ugh..........
MpowerdAPE 2 months ago
I'm thankful that you are still making these videos. You are a great teacher.
Aletheia216 2 months ago
your videos always gives me a headache. and im drunk so i shouldnt have an headache untill tomorrow. luckily english is not my native language so i understand maybe every third word. i can imagine a lot of brain matter on the walls of your viewers. a lot of bangs around the world. made by a head.
and i suck at math..
LowBattMouse 2 months ago
Interesting introduction to multivar calculus, great series of videos
Keinlicht 2 months ago
I'm assuming the answer to the question "What is the shortest path" could be derived via an optimization problem using the dx, dy, and ds values. I'm still in awe at how people like Gauss could tackle this mufti-variable calculus back in the 18th and 19th centuries.
lhvinny 2 months ago
@lhvinny
Fairly easily (in the sense that the prerequisites are low, not in it not being difficult math), since it doesn't actually depend on much in the way of scientific discoveries or the technology itself. Math and logic are pretty much the only things you can actually solve entirely as a thought experiment at any level of technology.
rkyeun 2 months ago
@rkyeun While the math is probably not that difficult the volume of calculations might quickly grow since finding the shortest path around a set of hill is probably an NP-complete problem. I would not even start without a computer and if Gauss really solved real-world problems with this method I am in awe. (well, I am anyway, but even more so).
FredricF 2 months ago
@FredricF
I'm pretty sure you just apply the jacobian to convert the field into a flat plane with distorted distances, then integrate the distance formula over the field (That's your NP-complete part, but we do integration all the time), and take the derivative of the result to find where the distance formula has a minimum.
But it's been way too long since I had calculus.
If the landscape isn't trivially simple, finding the solution will be much much harder.
rkyeun 2 months ago
@rkyeun Maybe you are right, but at every "hill" you can choose to go to the left or the right and for whatever choice you have to find a minimum and to know which one is the shortest you have to calculate them both. That's why I think it is an NP-complete problem. From what I gather it is much like the "travelling salesman" problem (I assume you are familiar with that) except that it require more computing power.
FredricF 2 months ago
Thanks for making these videos
biscoito1r 2 months ago