Great lecture again, you are treasure for mankind! I find the most interesting with the mass continuity equation is the physical interpretation that we can derive for div (F) by rewriting the continuity equation in terms of material derivatives
I'm so lucky to have discoverd your channel while self-learning multi-variable calc! Abosolutely recommend to anyone (even non-math majors who hasn't touched calculus in 4 years).
@stevebrunton Is the gauss’s divergence theorem also relevant to transitions of chemical states? (Water turning to ice, dry ice to co2… etc? As it does flux mass to and from the volume through the surface area.)
Some facts about Gauss: Gauss could divide by 0 Gauss squared the circle He knew the last digit of pi He could construct lines with a compass and circles with a straight edge
I actually looked it up. It looks like he is using a Lightboard that is basically a piece of glass and lights. He writes behind that glass and flips the vid horizontally after filmming.
At 20:25 Steve oversimplified by moving the time derivative inside the volume integral just like that. It can only be done if the volume being integrated wont change over time, invalid assumption in fluid dynamics. Taking this into account leads us to another beautiful theorem called Reynolds Transport Theorem (RTT), which interestingly naturally leads to the right-hand-side on Steve’s board (if F is a velocity field).
There is something I seriously don't see starting at 10:00 to about 13:30 when talking about the little boxes filling the volume. The claim is that with an interior filled with boxes with positive divergences, only the boxes on the surface contribute to the surface integral, with the internal ones cancelling along their apposing surfaces with other adjacent boxes. This can't be true; here are two lines of reasoning which, for the sake of the first argument assume a constant, positive, divergence throughout the volume. One: Compare two cases that have the same surface, but different volumes. The integral of the divergence over the volume is proportional to the volume, but the surface integral of the flux would not increase to match if there was a cancellation of the field F at the apposing adjacent surfaces of the interior boxes. Two: Since the integrals are linear, the surface integral flux for a volume with a single box with positive divergence must be half of that with two interior boxes with the same positive divergence. But if they are adjacent, and something cancelled, it wouldn't be. I'm thinking of this a la Gauss's law for the electrical field at the surface as proportional to the enclosed charge. Am I missing something??
Hi, I think there is some problem with the "explanation' of canceling off between fluxes in the video. The cancelling off takes place because the flux through an interfacial control surface will have different signs when taken in two adjacent control volumes.Anyone thought the same?
Ahammed - I also have a problem with the cancellation, and just posted a comment about it today (Sep 24, 2022), and after doing it I thought I should see if anyone else saw this too. I'm not sure if we have the same issue, but it sounds similar...
In this equation I see that units do not match: ∫∫_S ρF • n dS = ∫∫∫_V div(ρF ) dv on the LHS: [kg/m^3][m/s][m^2] = [kg/s] on the RHS: [kg/m^3][m/s][m^3] = [kg m / s] Please let me know what I am missing.
I'm confused regarding a basic idea. If there are no sources in the volume, I don't understand why the flux as defined isn't always equal to zero. Where the field enters the volume the flux contribution F dot n would be negative and where the field exits the volume F dot n would be positive. So, the net flux would be zero.
Watching this video, I remembered being totally fascinated (for the first time in my life) by theoretical electrical theory. Thanks for the passion you bring presenting this math.
Great video. Excellent for showing the intuition of the volume built as a union of smaller volumes, for divergence theorem. Just few comments: - Divergence Thm has some assumptions. Broadly speaking, everything inside of the statement of the theorem must be meaningful, as an example if you write a divergence of F, that function F must be regular enough (differentiable) for the divergence to exist; REMEMBER that PDEs of Physics translate into "regular enough local regions" all the general Principles of Physics holding for all the physical systems, differentiable or not; - Mass equation example: > when you put time derivative inside the volume integral, you're doing right only if that volume does not change in time (i.e. you're implicitly considering a fixed control volume, otherwise that manipulation is WRONG). Anyway the conclusion you reach is right, but your derivation only holds for a steady volume for integration. Integral laws and then differential laws can be easily translated from the very statement of the Physical Principles if you firs consider Lagrangian volumes (i.e. those volumes moving with the continuum), and then transformed to fixed control volumes (some math required) > maybe I missed that, but the physical meaning of F is not explicit here. In order to have the right physical dimension, it must have the dimension of a velocity. Indeed, it is the velocity field of the continuum under investigation in most of cases (few times it's a bit more tricky, i.e. in diffusion problems that vector field contains both a local average - averaged on the species velocity, maybe - velocity contribution and a drift velocity, likely due to gradient in the specie concentration, see Fick's law for diffusion). Keep going. I'm very curious how this series evolves.
Outstanding lecture, professor. Defining first in words, providing an intuition and then releasing the math! Shock and Awe. Anyone can deliver the symbols. Gifted educators deliver intuition and genuine understanding.
@@Eigensteve such a blessing to have your intuitive explanations. Even with the lecture notes from Oxford Uni, the systematic proofs and equations were insufficient for a student to fully appreciate the utility of the material. Every university (even the top ones) shall learn from your pedagogy sir.
Hi Steve, I have to say the tiny boxes analogy is a bit confusing. Because when you integrate over the volumn, you are integrate the divergence, so at each point the integrand is positive since each point is a source, which does not reflect any 'cancellation'. (If it does, then at the points in central region the integrand should become 0 since they are 'cancelled'.) Whilist the 'cancellation' happens between the vector field F itself. So it might not be the right intuition for the theorem.
@@Eigensteve I loved 💖💖💖the example of all volume integrals cancelling except the outer skin. Wish I had this visualization in class. I always used Gauss's thm as simply a mathematical tool. 1. What I am wondering is did they call it "divergence" before Gauss's thm? Or when Gauss proved it did they coin the term "divergence". 2. Gauss doesn't get as much recognition in statistics even though it's called a Gaussian distribution. For example, if I Google 'who is the father of statistics' it says Fisher, not Gauss. Why is this? Thank you Steve!
11:15 Why do divergences in neighboring cells cancel out? I can see why fluxes would cancel, but isn't that different from divergence? Wouldn't neighboring divergences work in the same direction?
23:57 dro/dt - Density can't become smaller or be infinitesimally small - it's a constant property of a mater even idealized for the sake to keep the conversation theoretical and abstract.
Hi Steve [URGENT!] Shouldn’t F be the velocity field V in this case, I think that is intuitive from dimensional matching and have also seen it written in the standard text book instead of F. Please correct me if I am wrong. Otherwise awesome lecture. Thanks and Regards Vinayak
Hi Steve [URGENT!] Shouldn’t F be the velocity field V in this case, I think that is intuitive from dimensional matching and have also seen it written in the standard text book instead of F. Please correct me if I am wrong. Otherwise awesome lecture. Thanks and Regards Vinayak
Hi Steve [URGENT!] Shouldn’t F be the velocity field vector V in this case, I think that is intuitive from dimensional matching and have also seen it written in the standard text book instead of F. Please correct me if I am wrong. Otherwise awesome lecture. Thanks and Regards Vinayak
Wait, does…. does this guy know how to *fluently write in mirrored English ?* Or does he just write & talk to his board like if it wasn’t there & an audience was in its place, with the board being able to let invisible/special light to pass through for a camera behind it to see the guy, & his inscriptions, and then mirror it around so as to make the text visible ? Does he just add the little sounds of a marker squealing on a whiteboard just for the effect ? And he‘s just writing in the air (and is really good at it) ? Does the marker indicate its position when it crosses a certain invisible plan ? Can he see the reflexion of a light source traversing that plane to indicate he’s putting his pen in the right plane of ambient space ? Or does he just know by instinct & habit, if it’s just not that hard in the first place ? Or is it just a very clean “glassboard” ? [* Edit made 2 minutes later * :] Oh, ok, it’s both a “glass-board” (i.e. “mirroboard”) & post-production horizontal flip
You are confusing me because your are constantly using Volume and Surface area interchangeably ! Your quickly drawn sketch on the board indicates to me that it is a typical Surface, typically developed in calc 3, over a domain Region in a typical x - y plane. Is this the Volume you are often referring to ?? Then you bring in the vector field , F hat,. Is this generic vector field passing over the Surface or actually through the porous surface ?? Often teachers don't state how a field passes through a Surface. If a If a vector field passes through a Surface, how is this done ? Is it radiation ? I would appreciate a response.
It is thanksgiving eve and I am learning some quality vector calc from these lectures. They are so greatly made!! Every detail is explained and is wrapped so elegantly together. A joy to watch.
If this were done with a physical example it would be easier to visualize. Nothing would have to be changed. The explanations are very clear but if they referred to real physical variables. That would be the dream of scientists in, say biochemistry, who love this and are watching for entertainment and need to have the physics or physical chemistry reviewed at the same time as the mathematics. Is that too much to ask? (Joke).
26:39 Rho can't be continuously varying even if that term adheres to us looking how mass enters end exits particular volume. Maybe I collide density with hardness together, yet still...
Some thing I didn't understand is why we are assuming that each tiny boxes would work identically either as a source or sink. Isn't the divergence induced by the vector filed on the surface going to change from section to section
What an outstanding explanation! I'm so surprised that my Calc textbook left out the Mass Continuity Equation when going over the Divergence Theorem. It's really motivating to hear how powerful this equation is in applied math and physics. I love hearing the real-world applications.
Suppose that you have 2 volumes in a vector field, and the volumes share a common interface. Then the flux through this interface will have a positive contribution to the flux through the closed surface around the first volume, and a negative contribution for the second volume. Then the flux through the closed surface around the two volumes combined will be the sum of the two fluxes. The contribution of the "interface flux" to the "combined flux" is cancelled out. This idea can be extended to any number of volumes of any size and shape.
@@HD141937, you nailed it! I really love Prof. Bruton's lectures on this channel, but the picture he drew is not a good representation of this elegant idea. Instead of having two arrows hit each box wall, it might be better to imagine a single arrow going through the wall.
So, it means that, thanks to this equivalence between what happend in a volume and its surface, we can intuitively feel what is the conept of continuity. It's not the coninuity of the mathematcians, but rather of the physicians. What is the continuity ? To check at every scale and at every shape, this equivalence. If this rule is true then it means that the medium is continous. It's very surprising that for knowing something locally, we need to look at globally.
Dr. Brunton, you used "Gauss's Divergence Theorem"(GDT) to derive the conservation of mass(CVM). Could you show how GDT relates to the "Reynolds Transport Theorem"(RTT) & also derive the CVM using RTT? Thank you! Dr. Brunton, for taking the time to teach all of us.
I love your content. Your followers are brainy people. They love ur style. They are bored by Netflix, autodidacts. I love your lectures about Compressed Sensing.
Amazing video, sir. Any1..Correct me if I'm wrong : He was able to interchange the order of derivative and integral because of the following>>> derivative of an integral= integral of a derivative. Just thought it might help some1 like me cause i was wondering how twas possible for a couple of minutes.
Sir love you I am immensely thankful to you you are great My teacher didn't give me any concept any amount of concept of that topic❤ Love from Pakistan 🇵🇰🇵🇰 may Allah bless you ❤
Hi Dr. Brunton. As obscure as this seems is it scientifically useful to somehow perturb Guass's Divergence Theorem with an arbitrary differentiable function to see what would happen if non-conservation were to ever take place, and the consequence on the derived PDE?
I'm not disrespecting my professor, but I wish I had you teaching vector calculus concepts to me. I enjoyed your machine learning series. I'm looking forward to your next videos
At 21:35 or so, does anyone know how we formally justify changing the total derivative in respect to time with a partial derivative with respect to time when we move the derivative operator inside the triple integral? Also, in this particular exemple, I get the feeling that this could only be true if the volume is not a quantity that depends on time, but I know that conservation on mass is always true and does not rely on such assumptions... How can I convince myself that this is true no matter what happens to the volume? And this is closely related to my last question : what does happen if we consider that the volume does depend on time? Can we still switch the integral with the total derivative? Thank you Steve Brunton for these videos! It's been a long time since I saw these topics (if ever for some of them!) and I really appreciate your enthusiasm and the quality of your work :)
Dear Sir, I cannot understand the part at 12:35 sec. If there is a perimeter which has continuous outward emerging arrows, then what about the arrows that are in +z direction (emerging in 3D), as flux F is coming/flowing out (not expanding).
Great , very neat, clear and didactical explanation Professor Steve of Gauss´s Divergence Theorem. I really enjoy it !! I am following You on the networks and also in the University of Washigton UW Internet Sites. I am thinking about going back to Graduate School [second round from 58 to 100 !!] and besides the quality of the university I believe the Advisor is crucial. Not only that He has abroad knowledge and background on the subject matters but also his ability to motivate. Your lessons are highly motivational.
I am so much amazed how excited you are teaching this theorem. Wished to have teachers like you at uni too.
Great lecture again, you are treasure for mankind!
I find the most interesting with the mass continuity equation is the physical interpretation that we can derive for div (F) by rewriting the continuity equation in terms of material derivatives
This is the best explanation of the Gauss's Divergence theorem I have heard till now. ☺ Thanks, Steve.
Great explanation! Gauss truly was a super genius for figuring this out.
I'm so lucky to have discoverd your channel while self-learning multi-variable calc! Abosolutely recommend to anyone (even non-math majors who hasn't touched calculus in 4 years).
I can't thank you enough, this playlist alone made taking the first step to start serious mathematics and physics for me so damn easy!!
I'm glad to hear it! Thanks for watching :)
Thank you sir...I needed these explanations!! Respect💯
Such a good video! Love your teaching style! Keep up the good work, I’m such a fan of it!
The lecture is very well organized and superbly delivered!
Makes a complicated subject clear and attractive.
I am a fluid dynamics researcher at IIT Bombay. I want to do my Ph.D. at WashU. Now I am modeling fluid vortex around a Mobius Theorem.
Thank you so much! I don't even know what to say. You did an amazing job explaining this!
Thats a brilliant intuitive explanation
really broadened my mind . thanks !
That was really interesting. Thanks for such a fascinating lecture.
"If it's Gauss' Divergence Theorem, it's probably the best divergence theorem."
Can't wait to watch the next one.
A good refresher. Could you also show contuinty eqn for deformable control volume (i.e. V=V(t))?
@stevebrunton Is the gauss’s divergence theorem also relevant to transitions of chemical states? (Water turning to ice, dry ice to co2… etc? As it does flux mass to and from the volume through the surface area.)
Some facts about Gauss:
Gauss could divide by 0
Gauss squared the circle
He knew the last digit of pi
He could construct lines with a compass and circles with a straight edge
Love this comment!
Very intelligent Gauss but never shared the rationale or the thought process 😒
Can the Arabic language be included in the translation options please!
Sir you put d/dt inside integral but if volume is changing with time then also can we put d/dt inside integral
excellent!
I just realized he is writing mirrored. That's insane!
How does he write in reverse?
I actually looked it up. It looks like he is using a Lightboard that is basically a piece of glass and lights. He writes behind that glass and flips the vid horizontally after filmming.
wow!
hero
I could feel the flux emanating from this person.. Beautiful
Every math teacher feels it is his duty to say that he is not a fancy artist or so when he draws some kind of diagram
This channel has single-handedly rekindled my interest in math. Absolutely love it. Thanks so much for making these videos!
Awesome, so nice to hear!
At 20:25 Steve oversimplified by moving the time derivative inside the volume integral just like that. It can only be done if the volume being integrated wont change over time, invalid assumption in fluid dynamics. Taking this into account leads us to another beautiful theorem called Reynolds Transport Theorem (RTT), which interestingly naturally leads to the right-hand-side on Steve’s board (if F is a velocity field).
> I see a new video has been posted
>> I put "like"
>>> I watch the video
For the Respect's sake!
These lectures are fantastic, thank you for taking the time to produce and share them for free.
You are so very welcome!
There is something I seriously don't see starting at 10:00 to about 13:30 when talking about the little boxes filling the volume. The claim is that with an interior filled with boxes with positive divergences, only the boxes on the surface contribute to the surface integral, with the internal ones cancelling along their apposing surfaces with other adjacent boxes. This can't be true; here are two lines of reasoning which, for the sake of the first argument assume a constant, positive, divergence throughout the volume.
One: Compare two cases that have the same surface, but different volumes. The integral of the divergence over the volume is proportional to the volume, but the surface integral of the flux would not increase to match if there was a cancellation of the field F at the apposing adjacent surfaces of the interior boxes.
Two: Since the integrals are linear, the surface integral flux for a volume with a single box with positive divergence must be half of that with two interior boxes with the same positive divergence. But if they are adjacent, and something cancelled, it wouldn't be.
I'm thinking of this a la Gauss's law for the electrical field at the surface as proportional to the enclosed charge. Am I missing something??
Hi,
I think there is some problem with the "explanation' of canceling off between fluxes in the video. The cancelling off takes place because the flux through an interfacial control surface will have different signs when taken in two adjacent control volumes.Anyone thought the same?
Ahammed - I also have a problem with the cancellation, and just posted a comment about it today (Sep 24, 2022), and after doing it I thought I should see if anyone else saw this too. I'm not sure if we have the same issue, but it sounds similar...
In this equation I see that units do not match:
∫∫_S ρF • n dS = ∫∫∫_V div(ρF ) dv
on the LHS: [kg/m^3][m/s][m^2] = [kg/s]
on the RHS: [kg/m^3][m/s][m^3] = [kg m / s]
Please let me know what I am missing.
I'm confused regarding a basic idea. If there are no sources in the volume, I don't understand why the flux as defined isn't always equal to zero. Where the field enters the volume the flux contribution F dot n would be negative and where the field exits the volume F dot n would be positive. So, the net flux would be zero.
have you figured out the answer to your question here? because I'm wondering the exact same thing
Nice video but I do not understand the concept of the divergence in the volume, how they cancel out and how there was a surface without a divergence
its funny cus i saw this video and was like hmm this sounds interesting then i realized i already knew this from work i did like 4 years ago
Watching this video, I remembered being totally fascinated (for the first time in my life) by theoretical electrical theory. Thanks for the passion you bring presenting this math.
Great video. Excellent for showing the intuition of the volume built as a union of smaller volumes, for divergence theorem. Just few comments:
- Divergence Thm has some assumptions. Broadly speaking, everything inside of the statement of the theorem must be meaningful, as an example if you write a divergence of F, that function F must be regular enough (differentiable) for the divergence to exist; REMEMBER that PDEs of Physics translate into "regular enough local regions" all the general Principles of Physics holding for all the physical systems, differentiable or not;
- Mass equation example:
> when you put time derivative inside the volume integral, you're doing right only if that volume does not change in time (i.e. you're implicitly considering a fixed control volume, otherwise that manipulation is WRONG). Anyway the conclusion you reach is right, but your derivation only holds for a steady volume for integration. Integral laws and then differential laws can be easily translated from the very statement of the Physical Principles if you firs consider Lagrangian volumes (i.e. those volumes moving with the continuum), and then transformed to fixed control volumes (some math required)
> maybe I missed that, but the physical meaning of F is not explicit here. In order to have the right physical dimension, it must have the dimension of a velocity. Indeed, it is the velocity field of the continuum under investigation in most of cases (few times it's a bit more tricky, i.e. in diffusion problems that vector field contains both a local average - averaged on the species velocity, maybe - velocity contribution and a drift velocity, likely due to gradient in the specie concentration, see Fick's law for diffusion).
Keep going. I'm very curious how this series evolves.
Thanks!!
Sir explain why divergence of electric field line is positive,,and negative,plse
Outstanding lecture, professor. Defining first in words, providing an intuition and then releasing the math!
Shock and Awe.
Anyone can deliver the symbols. Gifted educators deliver intuition and genuine understanding.
I'm so grateful to hear you like it!!
@@Eigensteve such a blessing to have your intuitive explanations. Even with the lecture notes from Oxford Uni, the systematic proofs and equations were insufficient for a student to fully appreciate the utility of the material. Every university (even the top ones) shall learn from your pedagogy sir.
woow!cool explanation ! integral sum of dV finally make sense! Thanks!
Fantastic lectures. Please increase microphone volume level next time.
Hi Steve, I have to say the tiny boxes analogy is a bit confusing. Because when you integrate over the volumn, you are integrate the divergence, so at each point the integrand is positive since each point is a source, which does not reflect any 'cancellation'. (If it does, then at the points in central region the integrand should become 0 since they are 'cancelled'.) Whilist the 'cancellation' happens between the vector field F itself. So it might not be the right intuition for the theorem.
I found this confusing as well. You can't cancel a bunch of sources.
Before this I had no idea fluid mechanics can be so intuitive and interesting. Great work sir, Thank you so much for your effort.
I'm so grateful for living at this time, so I can learn this theorem in 25 minutes.
Great lecture, professor! As always, very enlightening!
Thank you so much!
@@Eigensteve I loved 💖💖💖the example of all volume integrals cancelling except the outer skin. Wish I had this visualization in class. I always used Gauss's thm as simply a mathematical tool.
1. What I am wondering is did they call it "divergence" before Gauss's thm? Or when Gauss proved it did they coin the term "divergence".
2. Gauss doesn't get as much recognition in statistics even though it's called a Gaussian distribution. For example, if I Google 'who is the father of statistics' it says Fisher, not Gauss. Why is this?
Thank you Steve!
@@electrolove9538 Thank you -- that is a great question. I don't know the history of this, but I'll look into it!
11:15 Why do divergences in neighboring cells cancel out? I can see why fluxes would cancel, but isn't that different from divergence? Wouldn't neighboring divergences work in the same direction?
23:57 dro/dt - Density can't become smaller or be infinitesimally small - it's a constant property of a mater even idealized for the sake to keep the conversation theoretical and abstract.
Hi Steve [URGENT!] Shouldn’t F be the velocity field V in this case, I think that is intuitive from dimensional matching and have also seen it written in the standard text book instead of F. Please correct me if I am wrong. Otherwise awesome lecture. Thanks and Regards Vinayak
Hi Steve [URGENT!] Shouldn’t F be the velocity field V in this case, I think that is intuitive from dimensional matching and have also seen it written in the standard text book instead of F. Please correct me if I am wrong. Otherwise awesome lecture. Thanks and Regards Vinayak
Hi Steve [URGENT!] Shouldn’t F be the velocity field vector V in this case, I think that is intuitive from dimensional matching and have also seen it written in the standard text book instead of F. Please correct me if I am wrong. Otherwise awesome lecture. Thanks and Regards Vinayak
Wait, does…. does this guy know how to *fluently write in mirrored English ?*
Or does he just write & talk to his board like if it wasn’t there & an audience was in its place, with the board being able to let invisible/special light to pass through for a camera behind it to see the guy, & his inscriptions, and then mirror it around so as to make the text visible ?
Does he just add the little sounds of a marker squealing on a whiteboard just for the effect ? And he‘s just writing in the air (and is really good at it) ? Does the marker indicate its position when it crosses a certain invisible plan ? Can he see the reflexion of a light source traversing that plane to indicate he’s putting his pen in the right plane of ambient space ? Or does he just know by instinct & habit, if it’s just not that hard in the first place ?
Or is it just a very clean “glassboard” ?
[* Edit made 2 minutes later * :] Oh, ok, it’s both a “glass-board” (i.e. “mirroboard”) & post-production horizontal flip
Excellent lecture, thank you for posting!
You are welcome!
You are confusing me because your are constantly using Volume and Surface area interchangeably ! Your quickly drawn sketch on the board indicates to me that it is a typical Surface, typically developed in calc 3, over a domain Region in a typical x - y plane.
Is this the Volume you are often referring to ??
Then you bring in the vector field , F hat,. Is this generic vector field passing over the Surface or actually through the porous surface ??
Often teachers don't state how a field passes through a Surface. If a
If a vector field passes through a Surface, how is this done ? Is it radiation ?
I would appreciate a response.
It is thanksgiving eve and I am learning some quality vector calc from these lectures. They are so greatly made!! Every detail is explained and is wrapped so elegantly together. A joy to watch.
If this were done with a physical example it would be easier to visualize. Nothing would have to be changed. The explanations are very clear but if they referred to real physical variables. That would be the dream of scientists in, say biochemistry, who love this and are watching for entertainment and need to have the physics or physical chemistry reviewed at the same time as the mathematics. Is that too much to ask? (Joke).
26:39 Rho can't be continuously varying even if that term adheres to us looking how mass enters end exits particular volume. Maybe I collide density with hardness together, yet still...
Some thing I didn't understand is why we are assuming that each tiny boxes would work identically either as a source or sink. Isn't the divergence induced by the vector filed on the surface going to change from section to section
At mis 81years l'm fascinated by grate young teachers
What an outstanding explanation! I'm so surprised that my Calc textbook left out the Mass Continuity Equation when going over the Divergence Theorem. It's really motivating to hear how powerful this equation is in applied math and physics. I love hearing the real-world applications.
In terns of computation, which integral is more effective. The surface or the volume integral?
Well, I wasn't expecting to have my entire outlook on the world around me changed today but it happened.
Rendering so beautiful to replace some of the MIT's teachers on its edX online teaching platform
Thanks for this excellent lecture , I pray
For you to be happy and long live.
i can't do the marker squeaking... but props for learning to write backwards
Awesome explanation!🙏
Thank you -- glad you liked it!!
what id really mass energy and momentum created then how will gauss divergence theorem work?
What a great lecture!! I am truly looking forward to more videos from Professor Brunton.
2:38 The generalized Stoke's theorem would like a talk.
wonderful explanation thank you.
amaaaaazing
How is he writing backwards ❔
the name of the theorem is Gaus-Green Theorem
Very excited to watch every update on this series！
Awesome, I'm excited too!
Your lectures are so inspiring! 😊
this video is helping me a lot, thanks
Brilliant explanation , thank you
The explanation of the small volumes canceling each other out is not very clear to me- will look it up in books
Suppose that you have 2 volumes in a vector field, and the volumes share a common interface. Then the flux through this interface will have a positive contribution to the flux through the closed surface around the first volume, and a negative contribution for the second volume.
Then the flux through the closed surface around the two volumes combined will be the sum of the two fluxes. The contribution of the "interface flux" to the "combined flux" is cancelled out. This idea can be extended to any number of volumes of any size and shape.
@@HD141937 this explanation makes sense. Thankyou !
@@HD141937, you nailed it! I really love Prof. Bruton's lectures on this channel, but the picture he drew is not a good representation of this elegant idea. Instead of having two arrows hit each box wall, it might be better to imagine a single arrow going through the wall.
So, it means that, thanks to this equivalence between what happend in a volume and its surface, we can intuitively feel what is the conept of continuity. It's not the coninuity of the mathematcians, but rather of the physicians.
What is the continuity ? To check at every scale and at every shape, this equivalence. If this rule is true then it means that the medium is continous.
It's very surprising that for knowing something locally, we need to look at globally.
“physicians”? “physicists” sound more likely!
@@sib5th Sorry, I'm french ... ;)
How does this board work
gotta love this man
tim cook?
Dr. Brunton, you used "Gauss's Divergence Theorem"(GDT) to derive the conservation of mass(CVM). Could you show how GDT relates to the "Reynolds Transport Theorem"(RTT) & also derive the CVM using RTT?
Thank you! Dr. Brunton, for taking the time to teach all of us.
thank you
very good
Thank you
I love your content. Your followers are brainy people. They love ur style. They are bored by Netflix, autodidacts. I love your lectures about Compressed Sensing.
Amazing video, sir.
Any1..Correct me if I'm wrong : He was able to interchange the order of derivative and integral because of the following>>> derivative of an integral= integral of a derivative.
Just thought it might help some1 like me cause i was wondering how twas possible for a couple of minutes.
Sir love you I am immensely thankful to you you are great
My teacher didn't give me any concept any amount of concept of that topic❤
Love from Pakistan 🇵🇰🇵🇰 may Allah bless you ❤
Hi Dr. Brunton. As obscure as this seems is it scientifically useful to somehow perturb Guass's Divergence Theorem with an arbitrary differentiable function to see what would happen if non-conservation were to ever take place, and the consequence on the derived PDE?
I'm not disrespecting my professor, but I wish I had you teaching vector calculus concepts to me. I enjoyed your machine learning series. I'm looking forward to your next videos
Such great explanations and a highly quality channel. Great for building a strong intuition of concepts rarely explained in a straightforward manner.
At 21:35 or so, does anyone know how we formally justify changing the total derivative in respect to time with a partial derivative with respect to time when we move the derivative operator inside the triple integral?
Also, in this particular exemple, I get the feeling that this could only be true if the volume is not a quantity that depends on time, but I know that conservation on mass is always true and does not rely on such assumptions... How can I convince myself that this is true no matter what happens to the volume? And this is closely related to my last question : what does happen if we consider that the volume does depend on time? Can we still switch the integral with the total derivative?
Thank you Steve Brunton for these videos! It's been a long time since I saw these topics (if ever for some of them!) and I really appreciate your enthusiasm and the quality of your work :)
Dear Sir,
I cannot understand the part at 12:35 sec. If there is a perimeter which has continuous outward emerging arrows, then what about the arrows that are in +z direction (emerging in 3D), as flux F is coming/flowing out (not expanding).
Great , very neat, clear and didactical explanation Professor Steve of Gauss´s Divergence Theorem. I really enjoy it !! I am following You on the networks and also in the University of Washigton UW Internet Sites. I am thinking about going back to Graduate School [second round from 58 to 100 !!] and besides the quality of the university I believe the Advisor is crucial. Not only that He has abroad knowledge and background on the subject matters but also his ability to motivate. Your lessons are highly motivational.
Being slow to get it
Will watch again
A 3D simulation would be perfect for full visibility
Cant thank you enough for the giant efforts
Great visualized explanation of Gauss's Divergence theorem