It is flux ACROSS the surface.. (hence it's integrated over F.n) means that flux is maximum when the normal vector of the surface is parallel to the vector field.
@4:39 there is an error for those who are hard of hearing. Pho should be "rho" the Greek letter. The captions are absolutely greatly appreciated by myself and many others. Thank you to those who put in that work on this channel.
if you are attracted to heavy mathematical physics all these theorems and techniques are absolutely necessary. Fluid dynamics done in tensors makes this look easy.
@shuffledream nope, maybe a bit too late but he did mean tangent. take a look at the first example he did ~18mins. The vector field is parallel to the normal but the answer isn't zero. and any 2 perpendicular vector dot each other would give zero, so that should explain 14:27 as well
This video gave me TREMENDOUS AMOUNT OF KNOWLEDGE!!! I liked it very much,but one thing:Why don’t they use dusters instead of keep on changing the Blackboards?(Just asking in a leisurly way).
while explaining flux in his 2nd question. the force was only directed towards the z- axis ; the question is if we look physically doesn't it happens that the flux is zero , as the amount of flux entering is also leaving . please explain
+Amar Parajuli No, because the sphere is centered at the origin, so the F vector in this case points in the negative z direction for the bottom half of the sphere.
Not sure if this makes sense to you, but the way I think about it is like this: You have z = f(x, y), and this gives rise to the level curve F(x, y, z) = f(x, y) - z = 0. We know that the gradient of a scalar function is normal to its level curves, so the gradient of F(x, y, z), Grad(F) = [f_1(x, y), f_2(x, y), -1] will be normal to the surface defined by the function z = f(x, y). The function F(x, y, z) will be a scalar valued function of three variables, so it will be a 4D surface, which is very hard to visualize. You can however compare it to, say, the half sphere z = sqrt(x^2 + y^2), and how it relates to its level curves x^2 + y^2 = C^2. The gradient of z will be normal/perpendicular to the level curves for appropriate x and y. One, possbily easier, way to looks at it is to look at the vectors [1, 0, f_1(x, y)], which when projected onto the xy-plane, is parallel to the x-axis with slope f_1(x, y) and [0, 1, f_2(x,y)], a line parallel to the y-axis with slope f_2(x, y). If you take the cross product, you'll end up with a new vector that's perpendicular to both of the aforementioned vectors, normal to the surface z = f(x, y).
@@Ptoad thanks really great use of level curves and grad vector. One thing I am still confused about is what you found with this is just the normal vector (if you scale it by its magnitude) not ndS but the answer is coming to be exactly what professor said to be the value of ndS. How is the grad vector taking in consideration of dS. can you please explain this?
15 years later, but the cameraman did not do a good job on this particular lecture. Most of the time the camera was too far zoomed in to see all of the notes on each blackboard slide. Previous lectures did a much better job of zooming out to see the full picture.
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.)
Merci Denis ! Im a very old Danish lady and i adore your expressions ❤️ Regarding math ofc and youll be proud i understand❤️
this is really amazing for students today. i wish i could rewind my current lectures
+Ryan Bennett You can if you record them :P
@@Ramix09 nah shamelessly suggesting a criminal offense is wild
that guy, Denis Auroux, is a genius in mathematics!
@E lol
i am currently engineering first year . after i get a job i will definetely donate to mit ocw
have you donated yet
@@EmpyreanLightASMR bro is in the 4th year, he can't
@@SPRINGGREEN813 omg you're right 🤥
@@EmpyreanLightASMR Bro what are you doing? Btech or BS?
It is flux ACROSS the surface.. (hence it's integrated over F.n) means that flux is maximum when the normal vector of the surface is parallel to the vector field.
@4:39 there is an error for those who are hard of hearing. Pho should be "rho" the Greek letter. The captions are absolutely greatly appreciated by myself and many others. Thank you to those who put in that work on this channel.
Yes, *f r a n c e*
Lec 27 is so interesting
Great lecturer!
I Really Like The Video Vector fields in 3D; surface integrals and flux From Your
if you are attracted to heavy mathematical physics all these theorems and techniques are absolutely necessary. Fluid dynamics done in tensors makes this look easy.
this guy is awesome
Better than my prof.
@shuffledream nope, maybe a bit too late but he did mean tangent. take a look at the first example he did ~18mins. The vector field is parallel to the normal but the answer isn't zero.
and any 2 perpendicular vector dot each other would give zero, so that should explain 14:27 as well
Denis: "In case you don't see it..."
Me: By Jove, you've read my empty blank mind.
Thanks MIT
When you're 100% screwed for finals....
please also write what is in the recitation of each video when you update the next time .
This is helpful ❤️🤍
Thanks ❤️🤍
32:26 not line but surface integral, 2variable, flux
@sahookah because proving is math and knowing the prove helps you understand the theorem
This video gave me TREMENDOUS AMOUNT OF KNOWLEDGE!!! I liked it very much,but one thing:Why don’t they use dusters instead of keep on changing the Blackboards?(Just asking in a leisurly way).
Probably so that he can pull up previous boards as a reference. (Ik it's been 9 years heh)
Wow♡♡
Thank you
mhömhömhö i am a french güy. Actually this is just the video i needed right now.
that cameraman job tho
Camera man should zoom out and hold it steady on what the prof is writing on the blackboard.
@shuffledream I'm pretty sure he meant parallel, but no one bothered to correct him.
black boards looking for a good home.
So that any slow writers have time to finish copying it down for their notes.
how come in the first example of the sphere F.nhat = a and not 1/a and in the second example H.nhat = z^2/a instead of az^2
while explaining flux in his 2nd question. the force was only directed towards the z- axis ;
the question is if we look physically doesn't it happens that the flux is zero , as the amount of flux entering is also leaving . please explain
+Amar Parajuli No, because the sphere is centered at the origin, so the F vector in this case points in the negative z direction for the bottom half of the sphere.
Well we are not measuring divergence here.
nah it's the fact that the camera man missed the miss
Where can I see the solutions to the problem sets? Thanks!
can someone shed a light on how is nds = dxdy @47:00
Not sure if this makes sense to you, but the way I think about it is like this:
You have z = f(x, y), and this gives rise to the level curve F(x, y, z) = f(x, y) - z = 0.
We know that the gradient of a scalar function is normal to its level curves, so the gradient of F(x, y, z),
Grad(F) = [f_1(x, y), f_2(x, y), -1] will be normal to the surface defined by the function z = f(x, y).
The function F(x, y, z) will be a scalar valued function of three variables, so it will be a 4D surface, which is very hard to visualize. You can however compare it to, say, the half sphere z = sqrt(x^2 + y^2), and how it relates to its level curves x^2 + y^2 = C^2. The gradient of z will be normal/perpendicular to the level curves for appropriate x and y.
One, possbily easier, way to looks at it is to look at the vectors [1, 0, f_1(x, y)], which when projected onto the xy-plane, is parallel to the x-axis with slope f_1(x, y) and [0, 1, f_2(x,y)], a line parallel to the y-axis with slope f_2(x, y). If you take the cross product, you'll end up with a new vector that's perpendicular to both of the aforementioned vectors, normal to the surface z = f(x, y).
kjQtte thanks . But I found he explained it at the beginning of next video
I didn't realize. Good luck with your studies!
@@Ptoad thanks really great use of level curves and grad vector. One thing I am still confused about is what you found with this is just the normal vector (if you scale it by its magnitude) not ndS but the answer is coming to be exactly what professor said to be the value of ndS. How is the grad vector taking in consideration of dS. can you please explain this?
15 years later, but the cameraman did not do a good job on this particular lecture. Most of the time the camera was too far zoomed in to see all of the notes on each blackboard slide. Previous lectures did a much better job of zooming out to see the full picture.
What didn't you see here? Everything is clear
@sahookah Because proving is math.
flux it continuity
2:59
It’s ok to put khoo lol
👍👍👍👍👍
The comments are edu canned laughter,not as smart as they need!!
MIT OCW is free, so there is that.
Otherwise, it is overrated.
Totally disagree with you.
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