The conservation of charge, div(J)=-dp/dt, is obtained in the same way as the heat equation. Such a beatiful lecture... with such a delight in those chalk cleanings.
34:30 You might want to search RUclips for time invariant diffusion. The diffusion equation can be simplified by noticing that it looks the same if time goes forward as when time goes backward and a situation like that you can reverse the dye dillution. There are fantastic videos about how it's done. Do not say that partial differential equations are boring.
"23:37 Wow... Respect to MIT. In my school everyone would have burst out in laughter and it would have taken like 15 minutes to shut everyone up again." - $180k education comes to about $1.72 per minute to sit in, and let's be honest, joking around for 15 minutes isn't worth $26.
I cant understand why avg (divf) in D= avg(-u sub t) in D ,then Divf= -u sub t? Because if we actually plug a coordintate in the equation, it is no guaranteed the equation will be true.
in the diffusion equation, if u=u(x,y,z,t), why when taking the gradient we only take the partial derivatives of x,y,z? (for F in the left hand equation min. 49:10). I suppose it's because they're spatial dimensions, but then you need more constraints to make the result in the right hand of eq. not depend on time.. Thanks.
Yes, the gradient only operates on the spatial coordinates. The equation you're referencing doesn't need to be constrained because a function differentiated with respect to one variable doesn't necessarily get rid of another one. Similarly, if you differentiate with respect to a single variable, that variable will not necessarily vanish. Consider differentiating F with respect to x when F(x,y,z,t) = tx+y+z^2. The answer you would get is t. A partial derivative can depend on the same variables as the function you're differentiating. Sorry for such a poor explanation. :/
Thanks for the reply. I mean, in the equation both sides are equal. The divergence of F doesn't depend on t, so partial derivative of u with respect of t must not do it neither, i.e.: u must be linear with respect to t for each spatial component, something like u(x,y,z,t)=t*g(x)+t*h(y)+t*k(z).
Mauricio Barda The divergence of F is the sum of all it's spatial partial derivatives so it may depend on t. u can be any function composed of the spatial coordinates and t. e.g. u(x,y,z,t)=f(x,t)+g(y,t)+h(z,t). And u doesn't necessarily need to be the sum of functions like in the example. u can be any combination of the variables, separable or not.
Example where F's divergence is dependent on t: F(x,y,z,t)=tx+t^3y^2+z+t^5 If you find the divergence of the function I just provided, you'll notice its divergence does depend on t.
Yes, I think you are right. My mistake was thinking that the Del operator is an operator over all the variables, not only the spatial ones. Thanks again.
Could anyone please help me with this? At about 15'30'', the professor says "vertically simple". Does he mean that the sides of the solid is vertical (all equal to the height)? So the graph of the top is the same as the graph of the bottom? Thank you very much!
+John Lau I think a "vertically simple" region (in 3D) refers to a region that has a well-defined top surface (defined as a function of x, y), a well-defined (in x, y) bottom surface, and has a volume that includes all the space between those top and bottom surfaces. This entails two necessary conditions I identified: 1) The top surface and bottom surface should have identical projections ("shadows") onto any horizontal plane. 2) The sides must be completely vertical (if the sides even exist at all - a sphere has a top surface and a bottom surface with no vertical sides, but I suppose it is still considered "vertically simple" by this definition).
My only problem with the MIT lectures is they spend to much time on theory, rather than working problems. Although I'm sure they do problems during the recitation.
The point of a top level professor lecturing is to share understanding and conceptual connections. And the actual mechanics are the same thing going on about a year now (this is the 2nd semester of calc). That said, Khan and other sources are mechanics of solving heavy if you prefer.
I understand that this course is supposed to be for engineers/undergrads, but this level of math is unacceptable. The whole proof is inconsistent, lacking plausibility and even the conclusions under the given simplifications are false. I like this series, but this lecture is a joke.
+Python Ruß I think it's reasonable. Going into further detail would only serve to confuse students who haven't been exposed to mathematical rigor. It's much more fruitful to get the intuition across, and develop an understanding of how to tackle these types of problems. Rigor should be left to papers and written proofs. His argument on the Divergence Theorem is quite understandable and it follows intuitively that linearity should be satisfied when summing the x, y, and z components of F. Proving that this is valid requires more work, and likely mathematics beyond the scope of this class. Some mathematicians may enjoy "proving the obvious", but I bet you that most engineers and physicists do not.
+Sicarius Noctis You are absolutely right and I totally agree with you! Giving the full proof of the divergence theorem wouldn't benefit the students in the slightest, they would probably just be confused. It is more beneficial to give simpler statements and come to the conclusion by plausible rather than deductive reasoning. That was not my problem when I wrote that comment. Something about his reasoning was not plausible and it really upset me. I have to go through the lecture in detail again to find what it was.
10:56 "With new notations comes new responsibility." ~Auroux
The conservation of charge, div(J)=-dp/dt, is obtained in the same way as the heat equation. Such a beatiful lecture... with such a delight in those chalk cleanings.
Paul is a lucky guy.
he finally broke character and laughed on the blackboard cleaning trick.
37:17 or the groundwater flow equation =) So cool that the same equation can be applied to so many different things
Theres only 3 kinds of mathematicians: those who know how to count and those who don't.
you must be the 5th type
@@theblinkingbrownie4654 δx but δ is big
so excited to see how easily it's proved. Great.
Lecture 1: Dot Product
Lecture 2: Determinants
Lecture 3: Matrices
Lecture 4: Square Systems
Lecture 5: Parametric Equations
Lecture 6: Kepler's Second Law
Lecture 7: Exam Review (goes over practice exam 1a at 24 min 40 seconds)
Lecture 8: Partial Derivatives
Lecture 9: Max-Min and Least Squares
Lecture 10: Second Derivative Test
Lecture 11: Chain Rule
Lecture 12: Gradient
Lecture 13: Lagrange Multipliers
Lecture 14: Non-Independent Variables
Lecture 15: Partial Differential Equations
Lecture 16: Double Integrals
Lecture 17: Polar Coordinates
Lecture 18: Change of Variables
Lecture 19: Vector Fields
Lecture 20: Path Independence
Lecture 21: Gradient Fields
Lecture 22: Green's Theorem
Lecture 23: Flux
Lecture 24: Simply Connected Regions
Lecture 25: Triple Integrals
Lecture 26: Spherical Coordinates
Lecture 27: Vector Fields in 3D
Lecture 28: Divergence Theorem
Lecture 29: Divergence Theorem (cont.)
Lecture 30: Line Integrals
Lecture 31: Stokes' Theorem
Lecture 32: Stokes' Theorem (cont.)
Lecture 33: Maxwell's Equations
Lecture 34: Final Review
Lecture 35: Final Review (cont.)
U can take the screenshot of lectures in the website
There are 10 types of people. Those who understand binary and those who don't ;)
And those who weren't expecting a ternary joke.
23:37 Wow... Respect to MIT. In my school everyone would have burst out in laughter and it would have taken like 15 minutes to shut everyone up again.
Brilliant lecture!
34:30 You might want to search RUclips for time invariant diffusion. The diffusion equation can be simplified by noticing that it looks the same if time goes forward as when time goes backward and a situation like that you can reverse the dye dillution. There are fantastic videos about how it's done. Do not say that partial differential equations are boring.
You know a math professor is good when they make fun of a different field.
@@bacobjeatty2777 especially if that field is PDE's
I Really Like The Video From Your Divergence theorem (cont.): applications and proof.
I really like this prof. Wish he was the one teaching 18.03 :(
Likex100! These lectures are simply amazing.
"23:37 Wow... Respect to MIT. In my school everyone would have burst out in laughter and it would have taken like 15 minutes to shut everyone up again." - $180k education comes to about $1.72 per minute to sit in, and let's be honest, joking around for 15 minutes isn't worth $26.
Obviously it's not as amusing as the professors speed erasing
Gradient, from functions to vectors.
Divergence, from vectors to functions.
Curl, from vectors to vectors.
Oh wow 🤩 , almost get there oonly stroke’s thm left over hahahah finish all of 18.01 and 18.02 , learn more than I expected
Amazing. Cannot be explained better.
This is helpful ❤️🤍
I cant understand why avg (divf) in D= avg(-u sub t) in D ,then Divf= -u sub t? Because if we actually plug a coordintate in the equation, it is no guaranteed the equation will be true.
Thanks ❤️🤍
Letting *F* be the velocity field? Oh come on, can't we just use *v* ?
very good lecture
What sort of camera do they use........YOu get sucha gud picture when zoomed in
in the diffusion equation, if u=u(x,y,z,t), why when taking the gradient we only take the partial derivatives of x,y,z? (for F in the left hand equation min. 49:10). I suppose it's because they're spatial dimensions, but then you need more constraints to make the result in the right hand of eq. not depend on time.. Thanks.
Yes, the gradient only operates on the spatial coordinates.
The equation you're referencing doesn't need to be constrained because a function differentiated with respect to one variable doesn't necessarily get rid of another one. Similarly, if you differentiate with respect to a single variable, that variable will not necessarily vanish. Consider differentiating F with respect to x when F(x,y,z,t) = tx+y+z^2. The answer you would get is t. A partial derivative can depend on the same variables as the function you're differentiating.
Sorry for such a poor explanation. :/
Thanks for the reply. I mean, in the equation both sides are equal. The divergence of F doesn't depend on t, so partial derivative of u with respect of t must not do it neither, i.e.: u must be linear with respect to t for each spatial component, something like u(x,y,z,t)=t*g(x)+t*h(y)+t*k(z).
Mauricio Barda The divergence of F is the sum of all it's spatial partial derivatives so it may depend on t. u can be any function composed of the spatial coordinates and t. e.g. u(x,y,z,t)=f(x,t)+g(y,t)+h(z,t).
And u doesn't necessarily need to be the sum of functions like in the example. u can be any combination of the variables, separable or not.
Example where F's divergence is dependent on t:
F(x,y,z,t)=tx+t^3y^2+z+t^5
If you find the divergence of the function I just provided, you'll notice its divergence does depend on t.
Yes, I think you are right. My mistake was thinking that the Del operator is an operator over all the variables, not only the spatial ones. Thanks again.
Has to be vertically simple in all three directions.
I cant understand why is the du/dt not in the equation (which is at 36:54) even with the explanation that provided on the comment lol
35:25 you kick ass!!
@interxavierxxx How did they save your life?
28:05
No, I don't think so.
There are 10 types of people in the world: those who understand hexadecimal and F... the rest.
Could anyone please help me with this? At about 15'30'', the professor says "vertically simple". Does he mean that the sides of the solid is vertical (all equal to the height)? So the graph of the top is the same as the graph of the bottom? Thank you very much!
+John Lau I think a "vertically simple" region (in 3D) refers to a region that has a well-defined top surface (defined as a function of x, y), a well-defined (in x, y) bottom surface, and has a volume that includes all the space between those top and bottom surfaces. This entails two necessary conditions I identified:
1) The top surface and bottom surface should have identical projections ("shadows") onto any horizontal plane.
2) The sides must be completely vertical (if the sides even exist at all - a sphere has a top surface and a bottom surface with no vertical sides, but I suppose it is still considered "vertically simple" by this definition).
+Quanxiang Loo Thank you so much for addressing my question.
What's the third kind?
Third kind is one who don't care counting only.
You didn't get the joke. There are two kinds only but those who don't know how to count mistakes 2 as 3 (because they don't know how to count XD)
@@sohamgadhave99 lol, I guess it flew over my head back then xD
the one who doesn't understand jokes
@@kthegreat69420 Yeah, I appreciate the joke now, haha.
I love you.Edna..Typical Bosniak name..23:33
Not a fan of him proving the diffusion equation. I will use Chain Rule.
If anybody fails with this guy as a teacher.. you should automatically be kicked out of school
You don't know what his problem sets look like ;)
@@Nikifuj908strongly agree
there are three kinds of mathematicians 1) who know how to count 2) who don't.
My only problem with the MIT lectures is they spend to much time on theory, rather than working problems. Although I'm sure they do problems during the recitation.
Working problems is just applied theory, if you know the theory really good you basically knows how to solve all work problems
The point of a top level professor lecturing is to share understanding and conceptual connections. And the actual mechanics are the same thing going on about a year now (this is the 2nd semester of calc).
That said, Khan and other sources are mechanics of solving heavy if you prefer.
I understand that this course is supposed to be for engineers/undergrads, but this level of math is unacceptable. The whole proof is inconsistent, lacking plausibility and even the conclusions under the given simplifications are false. I like this series, but this lecture is a joke.
+Python Ruß sir may be you missed the previous lectures...
+Maruf Hossen I did watch the whole series...did I miss something? If so, please tell me.
+Python Ruß you said ,the given simplifications are false.which simplification you think false.could you tell me ?
+Python Ruß I think it's reasonable. Going into further detail would only serve to confuse students who haven't been exposed to mathematical rigor. It's much more fruitful to get the intuition across, and develop an understanding of how to tackle these types of problems. Rigor should be left to papers and written proofs.
His argument on the Divergence Theorem is quite understandable and it follows intuitively that linearity should be satisfied when summing the x, y, and z components of F. Proving that this is valid requires more work, and likely mathematics beyond the scope of this class. Some mathematicians may enjoy "proving the obvious", but I bet you that most engineers and physicists do not.
+Sicarius Noctis You are absolutely right and I totally agree with you! Giving the full proof of the divergence theorem wouldn't benefit the students in the slightest, they would probably just be confused. It is more beneficial to give simpler statements and come to the conclusion by plausible rather than deductive reasoning.
That was not my problem when I wrote that comment. Something about his reasoning was not plausible and it really upset me. I have to go through the lecture in detail again to find what it was.