Are all vector fields the gradient of a potential? ... and the Helmholtz Decomposition

[@amyers2141]

I realized in the first 2 minutes that I was watching a superb teacher. I had learned all these things about irrotational flow and the Helmholtz equation, but to wrap up the whole subject concisely on a single blackboard while writing backwards is an amazing feat. Thank you!

[@OloNadTrolo]

I don't think he writes backwards, the image is simply flipped.

[@Martin-iw1ll]

Yeah, he is left handed, just check his older videos that are presented with a conventional orientation

[@ishansinha1336]

@@Martin-iw1ll dude he is left handed in this vid too, flip of right would be left

[@TheJourneyWithKay]

You're breaking down and teaching whatever that is necessary that our professors actually have to tell us but they don't ! , In classes I just see a bunch of mathematical expressions without that of a meaning compared to your lectures. Thank You. I'm here to tell you Sir: Don't stop teaching!

[@marcosconceicao3223]

I am so amazed by your classes. Thank you very much!

[@binandakhungur]

You have explained so many fundamental things. Many likes tonyou

[@salvatoregiordano9050]

This could not have been explained better. Thank you so much!

[@edwincuevas9965]

Absolutely amazing vector calculus series! You are an amazing resource! Cheers!!!

[@dsaun777]

This is great! Keep up the amazing lectures they are gifts that we are grateful for!

[@ohault]

This episode could be the right time to introduce Geometric Algebra to move forward into deeper consolidation.

[@mMaximus56789]

Bump for the fundamental theorem of geometric calculus

[@NolanManteufel]

I love your videos. I love reconsidering these ideas that were developed by people thinking about electricity and magnetism. As a hobby I've been trying to apply them to knowledge and emotion. Thank you for making these videos, they are excellent.

[@justingreen8006]

Excellent teaching!! Thank you 🙂

[@shirsendumitra6044]

Excellent dear professor.

[@individuoenigmatico1990]

Wait, we can derive ∇×(∇f)=0 if f is a C² function, in which case Clairaut's theorem conditions hold (and so the second partial derivatives of f in ∇x(∇f) cancel each other out to give us 0). But in theory F could still be the gradient of a function f which is not C², in which case it may very well be the case that curl(F)=∇xF=∇×(∇f)≠0. In general what I'm saying is that a conservative field F can still have curl(F)≠0, if it is the gradient of a potential function f which is not C². Indeed the formal definition of a conservative field is that: a vector field F is conservative if and only if there exists a C¹ (countinuously differentiable) function f such that F=∇f. [But if f is not C², Clairaut's conditions don't hold, so we might have curl(F)≠0.] A better proof that not all vector fields are conservative, is just an exemplification.

[@NakulPatel-sd6uk]

A quick question. so before you said that potential flows are incompressible & irrotational while describing the set thing and for the generic flow you said that the first part is a potential flow that is irrotational. So is it also incompressible or not? or is that the potential flows can have either or both of the conditions?

[@owen7185]

Another awesome lecture Steve

[@NITESHKSAHU]

Suppose, F is a generic incompressible fluid velocity field. Then its divergence is always zero, div F=0. Using Helmholtz decomposition for this generic incompressible velocity field, div F= - div(grad phi) + div(curl A) = laplacian(phi) +div(curl A)= 0. But div (curl A) is always zero so, laplacian(phi) = 0 for any generic incompressible field. But, as far as I know, laplacian(phi) = 0 is only for potential flows, which is subsequently solved along with one N-S eqn to yield the fluid flow parameters. I will be very kind of you to put some light on this.

[@VIDDICACTUS]

Yep I got stumped there too... The better way to understand this is, Any generic field F can be decomposed as - "F = irrotational part + incompressible part" Irrotational part implies "curl(grad phi) = 0" Incompressible part implies "div(curl A)=0" Now lets say a generic field F is incompressible, that implies "div(F)=0"' therefore "div(grad phi)=0" i.e "laplacian(phi)=0" i.e conservative field Hence all incompressible fields are conservative and vice versa is not true...

[@hguy8449]

Thank you for making these videos, you are amazing

[@erockromulan9329]

One of the most impressive things about your videos is that you aren't using a mirror for the camera - you're writing is the mirror! Amazing.

[@RalphDratman]

It seems to me that one could create a vector field, say for force, in which an object would continually accelerate around a closed path. But such a force field could not correspond to a conservative system, ie., a time-independent potential.

[@Ruben-fk7zz]

at the end of the video with the generic flow equation, shouldn't the first irrotational element be called gradient flow instead of pot. flow? Because ist should be capable of doing the compression part, that the seccond can't right?

[@curtpiazza1688]

Great lecture! ❤ 😊

[@beebee_0136]

3:11 Thank you for revisiting these foundations. Another way I would like to think of the question posed, at least somewhat intuitively, is by considering natural vector fields as being composed of unit directional vectors (i, j, k) which are themselves vectors and form basic building blocks for their associated position vectorz (thus, xi + yj + zk). So, it follows that if at the most basic level we have unit vectors, then it seems to suggest there are *naturally existing* vectors distinct from scalar fields which can be turned into vector fields by applying the gradient (grad) operator.

[@lt4376]

I remember the various names like the Fundamental Theorem of Vector Calculus/Analysis or helmholtz decomp. But there is not an intuitive derivation.

[@joeyquiet4020]

thank you

[@adamb7088]

I agree with Olivier Hault. Would it be possible to see a covariant formulation of the Helmholtz Decomposition?

[@pritamroy3766]

Hi professor, @steve , my question is can a vector field be neither potential flow nor gradient flow? If YES then what will be mathmetical treatment for those kind of vector field? If NO, then why we can't have that kind of vetor field ???

[@Martin-iw1ll]

In the helmholtz decomp, should the irrotational part be called the gradient flow part and not potential flow part because potential flow is both irrotational and incompressible

[@AmentasOnIce]

I think you are right, he misspoke there.

[@lioneloddo]

So fascinating. I wonder if the decomposition of a field in terms of active and reactive field corresponds to a Helmholtz decomposition ? In complex space, the active and reactive fields are respectively given by the real and imaginary part of the field. I wonder if there is a link between the kind of field and the complex numbers...

[@AmentasOnIce]

Hmm, I know that if you form a vector field as a complex function (eg. V(z) is a complex function of the complex argument z, drawn as a vector at z), it's a potential vector field (curl-free and div-free).

[@kevincardenas6629]

Thanks a lot for these videos! How often are you publishing them?

[@anomalous4714]

Another awesome video in this series! I got a bit confused with the potential flow being defined as div- and curl-free. I always thought that potential flows are what you call gradient flows (curl-free), while flows that are div- and curl-free are harmonic flows. In the Helmholtz decomposition you call div(phi) a potential flow, so based on your definition of potential flow this must always be zero or am I misunderstanding this? I'm looking forward to the next videos.

[@kathrinskinder3891]

I was sort of asking myself the same thing. According to your definition of gradient flows and potential flows, shouldn't the Helmholtz decomposition be Vector field= gradient flow + solenoidal flow? Anyway, amazing video, your way of explaining this is so vivid. Thank you!

[@renwang870]

If the irrotational part of the generic flow is a potential flow (which by the last equation on the right half of the screen has divF = 0), and the solenoidal part also divergence-free, doesn’t it mean that the generic flow itself is also divergence-free?

[@AmentasOnIce]

I think you are right, and Prof. Brunton misspoke at around min 14 - grad phi is a grad flow, not a potential flow, according to the earlier definitions. I love these videos but there are occasional mistakes.

[@andreacomparini9381]

super

[@qwadratix]

Never mind the math. How are you writing on that (apparent) glass screen facing the camera without the result being reversed (mirror writing). The only scenario I can come up with is that the lecturer is left-handed and the image is flipped, either electronically or optically using a mirror. The alternative would have to be an ability to write right to left Leonardo da Vinci style. Somehow I doubt that.

[@nez9962]

Same question!

[@murillonetoo]

Great lecture, professor! I understood that all gradient flows are curl free. I'd like to ask about the other way around: are all curl free flows also gradient flows? How can we prove this?

[@VedJoshi..]

try proof by contradiction. Assume there exists a curl-free flow which is not a gradient flow. This should lead you to the conclusion.

[@blipblap614]

Check out "Notes on Differential Forms" (Lorenzo Sadun, UT Austin), page 6, Exercise 2. In this language, a gradient flow is an "exact" form (the derivative of something else), and therefore always curl-free ("closed" / no derivative of its own). Going the other direction is the Poincaré lemma -- in Euclidean space, yes; every curl-free flow is a gradient flow. It's not true if space has a "hole." arxiv.org/pdf/1604.07862.pdf

[@neiljf1089]

I'm not sure that potential flows are necessarily incompressible. I think they are simply defined as the gradient of a scalar potential. Incompressibility is an additional constraint.

[@wp4297]

Just definitions. Usually potential vector field means irrotational and thus can be written as the gradient of a scalar field (if the domain is regular enough - broadly speaking, simply connected - of course)

[@manuelsantosgutierrez7453]

Hi, is it analytically tractable to calculate the Helmholtz decomposition of a given 3D vector field? Is there a well-known method for that? Thanks!

[@agmessier]

In fluid dynamics, a field can be both irrotational (inviscid) and incompressible. Which term of the Helmholtz decomposition would capture these flows? Could it have both a scalar and vector potential field?

[@wp4297]

Yes, in incompressible irrotational flow you have both div(v)=0 and curl(v)=0. So, from the former you can write v=grad(phi), as a gradient of a scalar function, usually defined as the kinetic potential and indicated as phi, or from the latter as the curl of a vector field, v=curl(psi), being psi defined as the current function, nothing more than what you call a vector potential (usually used in 2D problem, but it's ok even in 3d). It really depends on what you need: as an example, if you deal with mass flow, it's usually easiest to use current function psi whose values (better, difference of values of psi evaluated in 2 points in space) is strictly related to the mass flow passing through a line connecting these two points. Please, remember that in the definition of the scalar and vector potential are not unique but are defined up to a gauge, usually this gauge has littele physical meaning, since the physical quantity of interest is the velocity and not the potentials. Please, when you deal with aerodynamics, also pay attention to the connection of the domain of the problem you're interested in. When you deal with a airfoil in 2d, the fluid flows around a object, so the space where fluid flows has a hole and is not simply connected, and: - mathematically: > the potential function phi becomes multi-valued > the mathematical problem is not fully defined and has an infinite number of solutions. Have you ever heard about Kutta condition? It's the mathematical condition that recover the most physical solution (the one the better suits experience and experimental results, at least when no separation occurs at high Reynolds number) among the infinite set of solutions - physically, this best solution is the one that has the right value of lift and the right distribution of pressure around the body

[@WarlordStarcraft2]

how is this video made? as in, how do we get a video of him facing us but writing backwards so that we can read what hes writing. He can't be actually writing backwards can he? I thought at first it was just a mirrored video, but he's facing us so that doesn't makes sense, and his hands actually properly follow the motions of what he's writing. Seems like this is pretty sophisticated production. I'm impressed.

[@whdaffer1]

I'd love to know the answer to your question too! I'm a wondering if the board and the software recording his writing is flipping it.

[@sreangsuacharyya5788]

Easily done. Just flip the direction of the vector normal to the writing surface. tongue in cheek

[@adiaphoros6842]

Can a solenoidal, that is an incompressible but rotational, field exist on its own, that is without being part of a Helmholtz or Hodge decomposition?

[@Ben777News]

Are you skilled at writing backwards on glass? Or is this some kind of image processing?

[@johnathancorgan3994]

Heh. He's writing normally behind the glass with the camera in front recording it backwards, then it is flipped left/right to make it look normal.

[@drslyone]

I think some people 'perform' in front of a mirror, and record the mirror.

[@timy8749]

I think he had been filmed behind the glass board, and then the image was flipped. Plus he’s a lefty. All these together seem to cause a lot of confusion😂

[@andreilucianlazer1871]

wait how on earth do you write backwards

[@anuardalhar6762]

Writing on transparent glass, and taking reverse video on other side.

[@12321dantheman]

is he writing all this mirrored??

[@takomamadashvili360]

🙌😍

[@abdjahdoiahdoai]

very interesting topic. I just looked up Hodge theory, and they seem to use it on understanding risk in finance. I wonder if we can combine Hodge with SINDY to gain knowledge on the underlying dynamics (flow/ f) in the financial world

[@georgesmith4639]

Is this some kind of video magic or is he really writing backwards at full speed with ease?

[@yizhang7027]

Are you left-handed?

[@ЗакировМарат-в5щ]

The answer is no.

[@siqi-47]

I think You are the ones who should do maths (genetically), not me…

[@xelktube]

The squeaky sound of the marker is *so* annoying, makes it impossible to watch (an otherwise great series of lectures)

[@p0k7lm]

better than excellent , thank - you ! ☺👍📚📐✏🔬