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).
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.
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.
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.
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??
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
just came here to say that while i'm currently not watching most of your videos as you upload them, i'm still very thankful because i'm 99% certain that i'll need them again at some point in the future
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.
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).
Thank you so much Dr Brunton for making such high quality content freely available. I have recently left work to return to uni and without this channel I would be seriously underprepared. Your ability to take a difficult subject and make it seem almost like common sense in incredible. I hope that my lecturers will have half of your passion and skill.
@@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!
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.
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.
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'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
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 ❤
What are you really interested in? Calculus is quite wide topics. Are you interested in some application/example with differential operators and evaluation of integrals? Three-dimensional space is enough for you? Are you interested in time derivative of integrals over time-dependent domains?...
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.
One issue with this theorem is that we have to define how much flows through the surface at the same time. That mans that we integrate over the surface for one moment in time. After completion we can see how that result evolves over time, but we can't use the left part of the surface values of one time moment and add that to the right part of the surface for a later time moment. But this means that we have to know what simultaneity means in that case. Thanks!
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.
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.
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...
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'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.
I just realized that if it can be said that the noise of a vacuum is 92 decibel, it's thanks to the Gauss's divergence theorem... As there is a kind of conservation law of the noise, and as the noise is given by a power, it's maybe, the conservation law of the energy which allows us to measure the sound of machines and equipments ...
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
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).
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.
I think Gauss's divergence theorem is helpful when thinking about empathy flow through boundaries around emotional systems/machines. Thank you for making these 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 :)
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
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...
@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.)
Is this guy writing backwards, or is there some kind of postprocessing effect that makes it have the correct orientation to the viewer? Great video btw, takes something complicated and makes it pretty intuitive.
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?
![Div, Grad, and Curl: Vector Calculus Building Blocks for PDEs [Divergence, Gradient, and Curl]](http://i.ytimg.com/vi/lKXW7DRyyro/mqdefault.jpg)
> I see a new video has been posted
>> I put "like"
>>> I watch the video
For the Respect's sake!
I am so much amazed how excited you are teaching this theorem. Wished to have teachers like you at uni too.
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).
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
These lectures are fantastic, thank you for taking the time to produce and share them for free.
You are so very welcome!
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.
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.
This is a treasure worth 1M views. I learnt this in my college days. Understood 15 years later.
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!!
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.
ನಿಮಗೆ ಅನಂತ ಧನ್ಯವಾದಗಳು... ಗುರುಗಳೇ ಇದೊಂದು ಅದ್ಭುತ ಪ್ರದರ್ಶನ
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.
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??
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
just came here to say that while i'm currently not watching most of your videos as you upload them, i'm still very thankful because i'm 99% certain that i'll need them again at some point in the future
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.
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.
Very excited to watch every update on this series！
Awesome, I'm excited too!
Such great explanations and a highly quality channel. Great for building a strong intuition of concepts rarely explained in a straightforward manner.
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).
You have no idea how much gratitude i have towards you... Thank you soo much for uploading this...
Life saving videos for students. Awesome. Thank you so much.
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 :)
woow!cool explanation ! integral sum of dV finally make sense! Thanks!
He has 4k quality lectures, my eyes are so comfortable watching it.
Great visualized explanation of Gauss's Divergence theorem
Thank you so much Dr Brunton for making such high quality content freely available. I have recently left work to return to uni and without this channel I would be seriously underprepared. Your ability to take a difficult subject and make it seem almost like common sense in incredible. I hope that my lecturers will have half of your passion and skill.
非常感谢！这个讲解让人印象深刻，过目难忘！这是我见过的向量微积分原理最好的讲解，再次感谢
Makes a complicated subject clear and attractive.
MVP....Most Valuable Professor of the year.
I was wondering how good can be someone in explaining complex subjects in an easy way.
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.
What a great lecture!! I am truly looking forward to more videos from Professor Brunton.
Thank you sir...I needed these explanations!! Respect💯
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!
I like how you write in mirror texts along with teaching.
The lecture is very well organized and superbly delivered!
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.
Excellent lecture, thank you for posting!
You are welcome!
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.
Best explanation ever. hugely thanks.
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.
Remembering in such a good way...thank you so much
Such a good video! Love your teaching style! Keep up the good work, I’m such a fan of it!
Thank you so much! I don't even know what to say. You did an amazing job explaining this!
Beautiful content, professor. Brilliant channel, thank you.
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
Thanks for this excellent lecture , I pray
For you to be happy and long live.
Awesome explanation!🙏
Thank you -- glad you liked it!!
Thats a brilliant intuitive explanation
Your lectures are so inspiring! 😊
At mis 81years l'm fascinated by grate young teachers
I am sooooooo grateful for this video!!!
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 ❤
really broadened my mind . thanks !
Brilliant explanation , thank you
That was really interesting. Thanks for such a fascinating lecture.
Fantastic lectures. Please increase microphone volume level next time.
Perfect, please more calculus lectures.
What are you really interested in? Calculus is quite wide topics. Are you interested in some application/example with differential operators and evaluation of integrals? Three-dimensional space is enough for you? Are you interested in time derivative of integrals over time-dependent domains?...
this video is helping me a lot, thanks
Can't wait to watch the next one.
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
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.
wonderful explanation thank you.
One issue with this theorem is that we have to define how much flows through the surface at the same time. That mans that we integrate over the surface for one moment in time. After completion we can see how that result evolves over time, but we can't use the left part of the surface values of one time moment and add that to the right part of the surface for a later time moment. But this means that we have to know what simultaneity means in that case. Thanks!
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.
Well, I wasn't expecting to have my entire outlook on the world around me changed today but it happened.
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
Watching this while waiting for my next flight. Thanks
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.
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...
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 😒
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
Awesome lecture sir...🙏
Sir explain why divergence of electric field line is positive,,and negative,plse
"If it's Gauss' Divergence Theorem, it's probably the best divergence theorem."
A good refresher. Could you also show contuinty eqn for deformable control volume (i.e. V=V(t))?
amaaaaazing
gotta love this man
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
I just realized that if it can be said that the noise of a vacuum is 92 decibel, it's thanks to the Gauss's divergence theorem...
As there is a kind of conservation law of the noise, and as the noise is given by a power, it's maybe, the conservation law of the energy which allows us to measure the sound of machines and equipments ...
??
i can't do the marker squeaking... but props for learning to write backwards
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
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).
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 ... ;)
2:38 The generalized Stoke's theorem would like a talk.
I think Gauss's divergence theorem is helpful when thinking about empathy flow through boundaries around emotional systems/machines. Thank you for making these videos.
the name of the theorem is Gaus-Green Theorem
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 :)
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
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...
very good
@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.)
Rendering so beautiful to replace some of the MIT's teachers on its edX online teaching platform
Thank you
anyone else super impressed he can write in reverse so we see it normal?
Is this guy writing backwards, or is there some kind of postprocessing effect that makes it have the correct orientation to the viewer? Great video btw, takes something complicated and makes it pretty intuitive.
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?