Flux and the divergence theorem | MIT 18.02SC Multivariable Calculus, Fall 2010
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- เผยแพร่เมื่อ 2 ม.ค. 2011
- Flux and the divergence theorem
Instructor: Joel Lewis
View the complete course: ocw.mit.edu/18-...
License: Creative Commons BY-NC-SA
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More courses at ocw.mit.edu
I teach mathematics and I've learnt so much from the his lessons. This professor is great. Thank you so much.
I dont know what i enjoy more, him working the problem or listening to him while he is solving. Reminds me of a contestant from BGT Graham Blackledge.
love the way he walks out when we are told to pause the vid xD 1:38
Nice video, I can see why they hired him at MIT: clear, concise explanation, nice and slow, so people have enough time to digest the info. I really like my instructor, but when he explained the divergence theorem, I was lost. Joel Lewis makes it super easy. Thanks!!
I don't even understand what I don't understand
@Just Me 5:11 how did he end up with just 2z?
@@piglink10 Let me try to explain this. He calculated the divergence of F, which implies the sum of the derivatives of each component like: F(x,y,z) = (a, b, c) -> div F = da/dx + db/dy + dc/dz. So, in his example, he choose wisely the F field so when you calculate its divergence, the first two terms cancel out, leaving the third them which is 2z.
@@arufuredo what does "choosing the F field mean" and why does this cancel out the x and y components?
@@nonya1119 By the F field this guy chose, you can see that the first and second term are kinda similar. The derivative of the the first term respect to x and the derivative of the second term respect to y are equal. Hence, when you calculate the divergence of F, they cancel out, leaving just the derivative of the third term respect to z.
What I mean with "choose the F field" was that to OP selected this specifical vectorial field so that the example was easy to solve! :D
@@piglink10 because 4x^3y-4x^3y = 0. The divergence is a dot product, so the vector components reduce to a summation.
a slicker way to do this is to just integrate over z right after using div theorem.. the remaining double integral will just be the area of the circle
11 years later and this video is still really helpful , thanks so much for the useful content you provided to us💙
Very well done Joel! This video was extremely helpful and straight forward. Thanks!
There are 2 notations for vector fields. Yes there's the i,j,and k. But you can use function notation F=(Fx,Fy,Fz), which Fx,Fy, and Fz would be the same as being coefficients to a unit vector (i, j, or k). (:
fantastic video joel you're the man! I've learned a lot taking this course.
The triple integral just simplifies to 2 times the average value of z in the region (ie h/2) multiplied with the volume of the cylinder .
The triple integral of 2z, can be easily computed noticing that 1/Volume * Triple_Integral(z dV) = h/2. This is true because the z component of the center of mass is h/2
Just by inspection, that exercise begs to be solved using the divergence theorem along with cylindrical coordinates.
This video is brilliant. If my grandma saw this video, she would know the divergence theorem for sure.
He reminds me of Gale from Breaking Bad
Haha, I thought the exact same thing while watching this.
Ruven Pinkhasov Please don't disrespect this guy.
DONT DISRESPECT GALE, RIP
I learned it as d-sigma instead of dS on the surface integral. I'm guessing there is no difference? Just |ru x rv| du dv right? where r(u,v) is the surface parameterization and ru, rv are the partial derivatives
Why isn't the anti derivative of the inner integral = 1/2 x^2 ?
I am assuming that if the x and y terms had survived the calculation of div F, you would simply have converted them to their cylindrical equivalents?
I wish my university had such clear and simple explanations. Thank you sir and MIT for helping me to understand where my University fails to. I was starting to think maybe my math skills were just capping out with this last bit of calc3 stuff but nope my teachers are just terrible lols :(
+Ben P. Same. Still struggling to figure out which bounds to use on triple integrals.
Good demo of the divergence theorem simplifying the problem, but this problem is trivial without the div thm if you look at the symmetries (and anti-symmetries) of the X and Y components of the field w.r.t. those of the cylinder. As someone already mentioned, possibly pedagogical purposes for the set-up, but it might be useful for students to recognize when symmetry can simplify a problem even more than applying the div thm (or when it can simplify the math after applying the div thm).
You use the determinant of the Jacobian matrix to transform the dV into its cylindrical equivalence. Remember: dV=dxdydz
I'm pretty sure most people can solve this particular problem mentally, it was chosen for its simplicity so that when people who need concept help come to it, they can focus on the calculus rather than the arithmetic.
lol, you dont have to evaluate the triple integral. You will notice that it is 2 * centroid Z * volume of cylinder. = 2 * h/2 * pi R^2 h = pi R^2 h^2
Who knew triple integrals could be such fun.
I will die peacefully one day knowing that I never will be able to figure whatever that is he is talking about.
What should i take unit normal vector
I would like to see videos on how to find the normals to surfaces and how to use the projection technique for projecting in different planes with varying objects like the sphere etc. Can you project the whole sphere for example into a plane. How to find the bounds that may be a bit complicated. thanks
Y u take div f= 2z???
yeah it paralyzed one side of understanding.
I guess if 'div F' consist of x, y components other than z component, AND if we are going to use cylindrical CDT system, then we have to convert the x, y components into cylindrical cdt form first before doing the integral to get the flux :|. am I correct?
th-cam.com/video/swrh0xjRXmc/w-d-xo.html
you may find this helpful
you make math easy, somehow
Because this is easy shit lol, test questions are going to be so much harder and don't simply nearly as easy as this lol
1995a1995z this is just trivial computational engineering type crap
The divergence theorem requires a differentiable vector field but electric field from Coulomb's law diverges at the origin.
Consequently, Gauss's flux theorem is not applicable to the divergence of the electric field.
sites.google.com/view/physics-news/home/updates
What if we have to do an integral of this type when the cylinder is not at the origin. What would be the limits for the radius R.
I am comparing this to my Calc 3 school teacher, who explains the same concept and I am thinking "that's why MIT is the best".
Me encantó la forma en que explicas todod
I guess if 'div F' consist of x, y components other than z component, AND if we are going to use cylindrical CDT system, then we have to convert the x, y components into cylindrical cdt form first before doing the integral to get the flux :|. am I correct?
th-cam.com/video/swrh0xjRXmc/w-d-xo.html
Thanks a lot for, it really helped me. I also have a question to clarify something, lets say we decide to solve this same problem without using the divergence theorem..What would the parameterization look like...I need this answer to figure out something.....Thanks in Advanced
(x,y,z) = xi + yj + zk
The brackets should be >< but TH-cam doesn't allow them.
When deciding what to use for dV in the triple integral with cylindrical coordinates, will you always tack on a multiplier of r to the dxdrdθ?
Celia Gonzalez yes because r is the jacobian of cylindrical coordinates
isn't that a vector field. where are the i,j and k then?
After taking div, why did he choose only 2z and not the terms of x and y from div F? is it because its normal to the surface created by x and y?
Cause they cancel out when taking the divergence.
thanks, you just help me pass my analysis test
Is it weird that I'm only 15 and I watch stuff like this that I've never seen before in my free time?
Not really.
It's a good thing, but remember you don't only watch but try to do it by yourself in paper. It's a whole new different thing
It's not weird, it's awesome. You're awesome. :)
6:45 why is there "this extra factor of R" in the dV?
I believe it can be proved by jacobian matrix. When you do the calculations you get the extra R. I dont know if this info works 7 years later that you asked :D
Good explanation
I'm actually a student at WSU in Washington but holy cow this guy speaks my language, makes me sad I can't afford MIT D=
That was very well explained. Thankfully, I have a very similar problem. The flux is then equal to the volume, correct?
Adam Chavez I believe it’s the rate of force passing through the surface along the defined vector field.
Thank you Professor!
his smile is strangely mesmerising jesus christ
What if we had divergent contain x=y+z , what we should do?
did u figure it out
Is there a proof that shows that the divergence is the sum of the partial derivatives? Why is this the definition and why does it work?
There are many different proofs for it. Just requires a bit of algebra.
Probably late answer, but the divergence of a vector field F is by DEFINITION the sum of the partial derivatives of F. It's the dot product of the vector ∇ and F, where
∇ =
so
div(F) = ∇·F = · = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z
Actually using the nabla operator to define divergence with dot product COMES from the fact that it can be proved that divergence is the sum of the partial derivatives. So that is not a proof! One way the demonstrate it is that being the function differentiable in a point, you can write down its linear approximation using Taylor's and then demonstrate from there that the divergence of the function depends only on the term with the gradient. With some Algebra that brings you to the fact that divergence is actually the trace of the Jacobian matrix!
Super late response :P
this guy just got his phd degree in mathematics from mit, so congratz!
Thank you very much for your explanation!
Thanks
MIT MEANS NO ARGUMENT
Like this it looks so simple..
2019 and still helpful lol..
Marvelous
fart at 3:27
:)
Vyacheslav Kervezee w
u have a powerful ear like dog
Thank you so much!
Clear explanation !
Well explained
THANK YOU
When calculate using surface integral i found flux zero
Thank you..
best teacher
This answer is incorrect by a factor of 1/2 because he didnt include it when he integrated in respect to z. He should have put (rz^2)/2 then all the rest is correct after that, simple error.
The 2's cancel out.. Don't confuse people..
MOAR!
Pretty good, small note the results of the first integral is wrong, he forgot to divide by 1/2
he did that sir
but because there was a 2 with z,
when he did that it will be cancelled
so its obvious not to write it
I take it Back I was mistaken, thanks for the correction.
THANK
YOU
You are awesome, man :)
Is he a matematician or a physicist? :D
really great
Perhaps engineering electromagnetics is not a comprehensible subject for a 13 year old such as myself
Helped a lot
Bring him here to be a professor at SMU. Such a nice explanation!!!
Big test today
Sidewalk chalk?
thanks for posting....!!!!
plsssss make a video on maxwell's 1st equation :)
really nice....thanks a lot for the video (y)
Amazing vid
I love ur way
To which # lecture does it correspond to?
+Mobashshir Feroz This corresponds to lecture 28 of 18.02. See the course on MIT OpenCourseWare for the complete context: ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/4.-triple-integrals-and-surface-integrals-in-3-space/part-b-flux-and-the-divergence-theorem/session-84-divergence-theorem/
Thanks, I'm already through till Lec 15.
Md Mobashshir Gauss divergence theorem from vector calculus
Genius.
Thanks, very helpful !
I Really Like The Video Flux and the divergence theorem From Your
Thank you :)
I always hope Joel would moonwalk back in before the solution.
this guy is brilliant 8)
Could have saved 3k and a lot of confusion by watching these lectures instead of going to uni.
I apologize I didnt see the 2z so I correct myself on my previous post. Sorry, I was incorrect
thank you so much I'm 10
Just wondering, do you understand what is he doing in this video?
alan harper!
Here cause of onlines courses and my teachers cant explain shit
sooo awesome :3
Cool
Not bad not bad.
Nerd! Thanks so much!
this guy is awesome, but anyone else was able to solve this mentally?
Its the same thing dude.
He actually act's like Frank Jr from Friends Series The one who was the brother of Phoebe 😂😂
Fk Yeah!!!! A video with no racist comments.
Shut up boislavic
+Dustin Allen Doohhhh!!!!!!!!
I didnt pause the video, haha owned