The Divergence Theorem // Geometric Intuition & Statement // Vector Calculus

[@timondalton8731]

I find it rare to both understand an equation intuitively and how to calculate it after watching a video. You are raising the bar of education everywhere.

[@prostatecancergaming9531]

Intuition and rigor are the two most important things in getting better at math. Not intelligence and creativity…

[@DarinBrownSJDCMath]

Thanks for making all these public this weekend!

[@DrTrefor]

You're welcome, I figured a bunch of people would be studying for exams right now and might need them:)

[@anilram1000]

Dr , Could you make a lecture series on conformal mapping?

[@tomatrix7525]

It’s amazing how these are just translations of 2d concepts into 3d. Great presentation!

[@DrTrefor]

Indeed. Sometimes like in stokes theorem new complexity manifests in higher dimensions, others it is really just an exact clone of the idea

[@kanjunior1019]

That what I figured this out today..after years of learning..

[@sarvasvkakkar2545]

A Really Big Thanks to you Sir for giving an amazing source to have a crystal clear understanding of concepts in multivariable calculus!

[@carmelpule8493]

Congratulations for your videos, I am now a very old man and when I was younger about 65 years ago, I tried to clarify in my mind the different activities that the few derivatives and associated integrals, contribute to the following set of particular "activities/ functions" they create I could see all this as an Engineering function in my own mind, but never drew my concepts on papers. They seem to have the same building blocks. 1. Cauchy Riemann relations 2. The Grad operator. 3. The curl operator. 4. the Divergence operator. 5 . Green's Curl theorems of circulation 6. Green's Divergent theorem of flux 7. Stoke's Curl theorem involving circulation 8. Divergence theorem involving divergences through volumes, I always thought that students should see the close links there are in how these derivatives are combined to produce their " engineered" activity. dU/dx dU/dy dU/dz dV/dx dV/dy dV/dz dZ/dx dZ/dy dZ/dz and reduced to two dimensions dU/dx dU/dy dV/dx dV/dy

[@nippletonuniversity8464]

Now this is the stuff I should be covering. I approve

[@swayamkumarpatro776]

Thank you for this lecture series sir. I have my end semester exams now on Integration in Vector Fields and Multivariable Calculus. You have helped a lot!!!

[@DrTrefor]

Good luck!

[@structuralanalysis6885]

This is a brilliant content for visualization. Thank you so much for uploading these in youtube. God bless you. Keep up the good work.

[@aidanbaxter204]

your diagrams are great. understanding this so intuitively after your 7 minute video is incredible. what an awesome theorem!

[@ravikant_kumar_i.b.c3433]

Thank you sir.. I am a college student at IIT Bombay INDIA and your lectures are helping a lot ..😀😀

[@sethbeckett8481]

Yo thank you for this video (and the Stoke's Theorem one), super duper helpful!!

[@DrTrefor]

You’re most welcome!

[@davehumphreys1725]

Its been many years now, but I seem to recall that there are two 'things' about vector fields that, if known, tell us all there is to know about the field. One is where are the sources and sinks , if they exist, located within the field, and the other is where in the field are the spots that might cause a rotation of the field, located. Given the equation of the vector field, if you then run the divergence on it, using the dot product of nabla with the field equation, you will end up with a formula that contains 3 functions of x,y and z [ assuming Cartesian co-ordinates], that are then added algebraically. If you then enter any values of x, y and z into this formula the result with be a scaler number that is either +ve, -ve or zero. If its +ve, its telling you that at that xyz location, the field lines are diverging away from the point and that there is a source of the field there. If its -ve, its telling you that, at that location, the field lines are converging toward the point and that there is a sink of the field there. I find this idea easy to visualize by thinking about a static charge distribution and considering the electric field. If my divergence equation gives me a positive answer, then its telling me that at that xyz location there is a +ve charge ie a source of the field. This idea also explains why the divergence of the magnetic field is zero, since the field lines form closed loops and have no source or sink. The curl of a vector field is, in my opinion, very similar to the angular momentum vector. The curl vectors are just lines you can draw to represent a rotation of the field. They also have no sources or sinks, which, again, explains why the divergence of the curl is zero.

[@schmetterling4477]

That only applies to vector fields in three dimensions. The situation is more complicated in higher dimensions.

[@Matthias27182]

I love your videos. They inspire me to seek out explanations for all kinds of math. Thank you for being such a great teacher.

[@zexisun1243]

Thank you so much, my classes are lacking of these geometric interpretations, now I am a lot clear about the topic

[@nasimhossain2328]

I love every second of your explanation

[@latestjobsupdates4453]

@2:41 correction : it takes a vector function and spits out a scaler function.

[@DrTrefor]

Quite right, thank you!

[@latestjobsupdates4453]

@@DrTrefor You are doing an amazing job teacher. God bless you. Which series is coming up next in mathematics ? Thoroughly enjoying your videos.

[@j.o.5957]

So if I got this right, we find the flux which is how much something tends to pass normally past a surface. A pretty formula. Question to self: What would be the boundary and parameterization? We could use spherical coordinates, r*dr*dtheta*dphi and find the boundaries that way.

[@dhruv0x0x0]

it was so easy, but even tho our prof is great i didn't got the concept, vector calculus is best with all these animations, really thanks for all these efforts!!!!!

[@haushofer100]

Fantastic explanation. Many thanks!

[@DrTrefor]

Glad it was helpful!

[@zethayn]

You are amazing sir, thank you very much!

[@sdsa007]

thanks! Having studied both flux forms of the 2D green theorem, I was wondering why there was something missing in the 3D Kelvin-Stokes Theorem! it only has a 3D curly form , but finally, now found the 3D divergence form (its in the name duh!)

[@nrcarl00]

This video saved my life

[@JB-ji4yq]

Very interesting, all so liked what was behind you as well.

[@tanvirfarhan5585]

best channel

[@robmarks6800]

Hey, amazing videos! Will you make the last videos public too? Just wondering:)

[@DrTrefor]

Yup, everything will be public by the end of this week, just release to members ~1 week in advance:)

[@pjpaulpiti]

Very good video! One small misprint: The text in small letters entering at 6:23 in the bottom left corresponds to Stokes' Thm., rather than to Divergence Thm.

[@ahmadawlagi6481]

shouldn't the left-hand side of the divergence theorem have d sigma instead of ds? 4:40

[@aidealczar6075]

Superb!

[@stephend.4342]

Masterful graphics and presentation, but there is something that has been gnawing at me in the 1D, 2D and 3D cases: what about flux of a field which does not cross a curve, does not cross a surface, and does not cross an enclosed volume in the normal direction (positive, going out) to the the curve , to the surface or to the enclosed volume? There are an infinite number of directions by which the flux can proceed outwards, of which only one is proceeding outwards (crossing) in the normal direction. How therefore is this total flux, in all directions outwards, calculated?

[@tihaelbou2384]

Thank you so much sir this is what I was looking for. Could you please make a video explaining " la matrice jacobienne" and " le jacobien " 😁.

[@DrTrefor]

Thank you! I do have a video on Jacobean in my multicariable calculus playlist

[@imonwani9322]

@@DrTrefor please I have a question I failed to answer am begging you if you can help me answer it

[@张鑫-e9o]

ds in the divergence theorem should be capital dS?

[@fernandojackson7207]

Excellent lecture. what is the dot product of operators, though. I'm aware of dotting points/vectors, though not operators. And, just curious, what kind of Mathematical Object is the Divergence of a Vector Field?

[@manirarebajeanpaul9312]

Then clearly using words. What is the statement of the fundamental theorem of divergence?

[@forrestkennedy5458]

can someone explain why we get interior cancellation in for the divergence in the volume integral? I don't think I am understanding that.

[@DrTrefor]

Imagine a vertical boundary. The flux from left to right is the negative of the flux from right to left. So if we add up both fluxes it would be zero. This is true in the interior. But for a boundary, you only get the one side, not the other.

[@mayukhmalidas]

Thanks a lot 👍

[@wilurbean]

why does the divergence cancel out within a volume? Why *_must_* they cancel?

[@shmkrar1153]

Thank you! It helped me a lot!

[@saiankitsahoo7663]

Awesome content

[@AbhijatBhat]

I don't understand why the divergences in the interior volume cancel out one another ... can someone please explain?

[@adilmazer123]

Man your Videos are awesome, thank you so much

[@DrTrefor]

I appreciate that!

[@cristianmeraz4181]

Thanks professor!

[@sergiolucas38]

very good video, as always :)

[@TheDeluxeBacon]

Can you make a video on a Divergence Theorem example pls?

[@DrTrefor]

Check out the playlist, it's the very next video!

[@leoads]

You mean.... spits out a scalar function.... right?

[@carultch]

Yes, divergence is a scalar function, as is the volumetric integral of divergence.

[@mahirpokar1528]

I believe the divergence operator will spit out a scalar function not a vector function.

[@thiagonadimmartinho6837]

thanks!

[@andrespenafiel4408]

I just have one question. What is M and what is N here? What is the difference between dM/dx and just d/dx, or dN/dy and just d/dy?

[@carultch]

He's using the M/N/P trio of letters to name the three component functions that define the vector field. It is very common to avoid O as a variable name, so P is what follows. Some books/instructors call them P/Q/R, which was the trio of letters my instructor chose to use. The vector field is given as: F(x, y, z) = Or in another notation F(x, y, z) = M(x, y, z)*ihat + N(x,y,z)*jhat + P(x, y, z)*khat

[@carultch]

d/dx is a verb. dM/dx is a noun. d/dx says "take the derivative of the following, with respect to x" dM/dx says "the derivative of M with respect to x" In this case, the derivatives are really partial derivatives, so these would be those funky d's, rather than ordinary d's. It still means approximately the same calculus action as a derivative in general.

[@johnanderson7840]

I thought taking the divergence of a vector changed it into a scalar and the gradient of a scale function turns it into a vector. If you’re taking a vector and multiplying it by the del operator wouldn’t that technically be taking the gradient

[@carultch]

If you take a vector field and "multiply" it by the del operator, that could be divergence, or that could be curl, depending on which kind of multiplication it is. In the case of divergence, it is an operation analogous to the dot product, where you distribute each differential operator within the del, to each corresponding term of the vector field, and then add them up. Not really multiplication by strict definition, but I get what you mean. In the case of curl, it is an operation analogous to the cross product. Each differential operator operates on a non-corresponding term, in a pattern spelled out by the determinant of the unit vector row, the differential operator row, and the vector field row. The order matters, and it produces a vector.

[@carultch]

The gradient is formed when you start with a scalar field, and use the del operator to take its derivatives, and generate a vector with them as its components. This is an operation analogous to multiplying a vector by a scalar.

[@JoeMac123a]

Why can't the portion along the boundary cancel while what's bounded (inside) can? at ~6:00? Also, would this work if the divergence was negative and there was contraction? Or would it contradict the unit normal vector???

[@aspiredifferent8085]

You look like the young Jack Dorsey😄😄.

[@ashwaniagrawal5770]

I am tired of adding your videos to my favourite playlist

[@DrTrefor]

Ha it’s hard work but someone’s got to satisfy the KZbin algorithm!

[@ashwaniagrawal5770]

@@DrTrefor My pleasure to get a reply from u sir. Much respect for u. U have made mathematics a fun which was boring to me.

[@kudzai63]

加油大家！

[@adamm150a]

is green theorem and gasuss divergence theorem are the same? i am conffused

[@DrTrefor]

Greens theorem is 2d, divergence is 3D, but they are very similar

[@stasdolinsek9460]

Great stuff

[@DrTrefor]

Thank you!

[@f1tech249]

Here I am to remember Gaussian Théorème to use for convective heat transfer lol

[@amirhosseindaraie5622]

Can you start teaching neural networks math ?

[@DrTrefor]

ooh, that would be a fun one. To be honest, I know almost nothing about this, but I'd be excited to learn it:)

[@amirhosseindaraie5622]

@@DrTrefor It would be a game changer course;)

[@RahulSharma-oc2qd]

There is a book on tensor and there is one equation written in it about the divergence of a vector field in such way...... div(S)=LimV->0 (1/V) { Sn dA..... Where curly brackets are nothing but integration sign over the domain (or surface area). I am unable to understand the limit part and from where this V (volume) comes into the equation. Help me understanding this please!

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Please make a playlist on Complex Analysis

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stroke theorem

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