Both grad and div involve finding fields using partial derivatives. We’ll look at yet another useful field. Once again it involves partial derivatives. Water can flow in many different and often complex ways. Let’s look at a relatively simple case. What’s the velocity field that

describes the flow on the surface of a river? We can explore this field by watching floating objects. What kind of motion can you see here? There appear to be two types of motion. Downstream and rotation about a vertical axis. In fact if we’re travelling along with the disc all we see is the rotation. This rotation is related

to the velocity field. But how? To answer that question, we need to define the velocity field on the surface of the river. We’ll define the surface velocity vector at any fixed point on the river, fixed, that is, relative to the river bank, as the velocity of any floating object as it passes through that point. To model the surface velocity we’ll need to make some assumptions, both about the shape of the river and about the way that the water flows. We’ll assume that there are no rocks affecting the flow. Let’s also assume that the river is straight and of uniform width. And the flow is steady and streamline; it all flows downstream. So this is our model river. To model the velocity field we first need some axes. And a function v(x,y) that describes

the unique velocity vector at each point (x,y) on the surface. We’ll assume that for this short stretch of the river v(x,y) doesn’t depend on y. In other words the velocity at any point

only depends on its distance x from the left bank. This implies a model surface velocity field of the form v(x,y)=u(x) j. But what sort of function is u(x)? Let’s go back to the real river for a moment. The velocity of the water varies as

we go across the river. And we can assume that at each of the banks the surface speed is zero. That is u(0) and u(d) are both zero

– where d is the width of the river. Somewhere in between the surface speed will be a maximum. Let’s assume that there’s some symmetry and so that the maximum flow is midstream. What does this imply for u(x)? Well, the simplest possibility is parabolic; u(x)=Cx(d -x), where C is a constant. And so our model for the surface velocity field v is this. But how can a velocity field cause rotation? Let’s go back to the model. Can you see what’s happening? Let’s subtract the downstream velocity of the centre of the disc. While the outer edge is being tugged downstream by the flow, the nearer edge is being tugged upstream. So the disc rotates. This indicates the presence of another vector field, one that describes the magnitude and direction

of the rotation at any point in the river. Here the rotation is anticlockwise. So using the right hand rule the direction is vertically upwards, that is in the direction of k, the Cartesian unit vector in the z direction. This new vector field is called the curl of v, or just curl v. How does this vector curl v vary over the river? As we’ve seen, here the rotation is anticlockwise. But here it’s clockwise. On the left-hand side of the river the velocity difference across the disc causes rotation in the positive k direction. While on the right hand side of the river the velocity difference causes rotation in the negative k direction. So what’s the rotation midstream? Well midstream there’s no rotation; in other words curl v=0. The vector field showing

curl v for the whole river looks like this. Everywhere curl v is perpendicular to the velocity field v. Any vector field can have a curl field associated with it. And later in this unit you’ll see how curl v can be found from partial derivatives. For our model river the rotation was localised – there was rotation about each point. And it’s local rotation that curl describes. Not the bulk rotation of water you see in river bends, or in swirling water. So what do you think is happening here? There’s plenty of water movement – bulk rotation. But what about any local rotation? What would curl be? Let’s model the problem by looking at the two-dimensional surface velocity. All the water’s going round and round in a circle. But whilst that nearer the centre is flowing in a vortex, further out it gets slower and slower. So there are two different kinds of flow to model. Let’s start with the water inside the vortex. At the centre the velocity is zero. It’s like a hurricane or a tornado, where the wind velocity is zero at its centre, its eye. As we move further out, the speed increases. If you ignore the bulk rotation causing it to flow around in a circular path you can see that the disc is also rotating about its centre. So with this velocity field there is local rotation. But what about the flow outside the vortex? Clearly there’s still bulk rotation of the water. But what is the local rotation? This time the disc isn’t rotating at all about its own centre so there’s no local rotation. In other words this velocity field has a curl of zero. But why should anybody want to know about the curl of a vector field? Professor Eric Priest: (St. Andrews University, Scotland) In weather patterns you often find concentrations of the vorticity where the curl of the velocity is extremely large. These are called cyclones. A cyclone is a region where the pressure is lower than normal and so naturally the air tends to flow in towards the centre of the cyclone. But as it flows in, it doesn’t just come

in radially, it acquires a twist. In cyclones you get a build up of the vorticity and when the vorticity gets extremely intense, of course you get

hurricanes and tornadoes and these can do a great deal of damage. So it’s very important that we understand how the vorticity is built up and how tornadoes and hurricanes behave so that we can predict them better in future. I’m particularly interested in the sun. There are very strong magnetic fields and the curl of the magnetic field is in fact the electric current. In most of the sun the current is zero, there’s no curl. But in small regions where the curl is very

large, you find dynamic phenomena produced. As you go away from the surface of the sun you might expect going away from a hot body that the temperature would get lower. In fact, the opposite occurs, it actually increases, and in the atmosphere of the sun the temperature is several million degrees, five million degrees, really hot. There are giant tubes of magnetic flux in the atmosphere. When these have no curl they just sit there in a quiet state for months at a time. But when the curl builds up, when the electric current builds up, eventually they could reach a stage where they go unstable and they erupt outwards from the sun and produce enormous ejections of mass. Knowing about the curl enables us to understand how and why these ejections occur.

wonderful explanation !

Excellent. Thanks for the video !

Excellent !!!!

I had no idea Melisandre, the red priestess, was tutoring physics on youtube!

For the night is dark and full of terrors.

Hello!

My name is Alfredo, and it's a big pleasure to watch your nice videos about teaching in pleasurable way. Hence I resort to you by reason of learning the right way to work with some differential opertator as gradient, del, divergence, del dot, quabla, and quabla dot operators, by virtue of learning much better the relativistic Maxwell's equations. I assert that I've found a difference between them when they are applied to tensor fields. So I underscore the following summary which was composed by myself, wishing you can absolve my query.

http://s28.postimg.org/6f7g8abx8/tensor_calculus_q.jpg

Thus I'll be looking forward to your reply as soon as possible.

Regards from Peru!

extremely useful, thank you

PLEASE REFER LANGARAGING MECHANICS — WHAT do u mean by generalised co ordinates ?

madam the lecture is excellent– but volume is very very low– pray raise volume of u r voice– thanking u

The exquisite piece of explanation one coud ever ask for 🙂 Thanks !!

thanks

One of the best videos I have seen on this topic. Kudos for the effort

Great video, thank you!

"The simplest possibility is parabolic" – why not linear (i.e., here, absolute value, to make a triangle)?

perfect , epic , wonderful these are the god damn physical explanation that i need ffs

+masterxilo the velocity profile is parabolic because at the edges, the velocity is 0. As you get closer to the center of the river, the flow velocity increases to a maximum in the center and thus giving you a parabolic profile. Its called the no-slip condition in fluid dynamics. Look it up

thanks ………..amazing . 🙂

Great video. Thanks

Now I should never forget what the curl vector means physically. Great!

beautiful !

v=Cx(d-x) parabola projected in xy plane proof please

but why's the curl of magnetic field give the electric current ?

first time i got it thnks 🙂

I love the dude at the end! You can tell that he gets a lot of satisfaction from studying these phenomena.

amazing xplntn!!

An excellent short talk. I wonder whether for completeness she could have shown a film of flowing lava – I have one which shows the drag effect of the 'riverbank' very clearly.

Really interesting way of looking at things. Thank you for sharing

Thanks a lot! A new intuition opens with a new perspective!

This is a beautiful presentation of gradient,divergence and curl. These kind of ideas should be exchanged in the classrooms so that the students can identify better with the course instead of just writing abstract equations on the board and expecting the student to understand them.

very useful

THANK YOU

nice 👍👍👍

the explanation is beautiful and pragmatic just like your voice. i love you

thank you!!

Such videos invoke passion for learning. Thanks teacher.

so is the curl why protoplanets aquire rotation when they form?

Thank you very much!

Brilliant

i ve been trying to understand curl and div for a long long time now i think finaly the 'aha' moment is here

really superb work….

how do you know that the host is a British? you just see how he/she pronounces 'water'. They tend to say it like 'woo-tuh'. Nice explanation though.

Awesome

really awesome presentation

tremendous outstanding presentation

to the point approach

I liked this sort of documentary type class. Great!

It takes a peculiar kind of mind to give different titles to three videos in a series, so they don't sort together out of a longer list and don't arrange themselves in numerical order.

Wouldn't want that, would we, now? Well done, Open University.

I learned what to do when I see "curl v" on my exam, but never learned WHY we learn about it. This video and the others in the series are so useful to show students the practical application to what we learn about. Thank you for these videos.

What is j

thanks sir

What is the geometric interpretation of gradient of dot product of two vectors becoming zero?

How you made the animation ? which software ?

superb……. thank you…..you are great…..

Great video! Question: At 4:30, the ccw rotation of the disc is described as being caused by the downstream flow on the right & the relatively lesser flow on the left of the disc. In reality, wouldn't this give rise to the magnus effect l, causing the disc to float toward the center of the stream?

Thanks for providing some knowledge👍👍

Vortex lines (curl of V) moves with the flow by Helmholtz II

what program is used to make this animation????

اجمل شرح عن الدوران

What's the difference between grad and div?

Awesome physical interpretation . We should bring them to text books ,rather studying waste stuff . Pls consider

Great physics .

👌👌👌

at 7:14, all the arrows except the one at center should point upwards right?? the direction of rotation is same at every point

great efforts, and explanation as well

your efforts make me to subscribe

wonderful

Amazingly understandable

I hope an OU administrator reads these comments at some point because there's huge enthusiasm for the way these ideas are presented here. I would have loved to watch these videos as part of my electromagnetism module.

does that mean theres chance life exist inside the sun🤔🤔

That was great.

awesome

This science is real science and weather scientist brought on is a major physesist .

These videos are very helpful. Thank you 😍 ♥

Now I understand the physical meaning of curl and divergence. Well presented! 👍

true beauty of maths

Good!

Greens therom ????🤔🤔🤔🤔

The "correct" order should be: grad, curl, and div.

that is superb, what a fabulous explanation. thanks a lot

Thank you…

for more than 1 year I was in a in a state of confusion about the physical significance of curl, divergence…

thank you again

Very nicely explained!

Thank you!

Especially the last part!!

Great !

A big thanks!

My teachers never told the physical significance if these values. It took around 4 years for me to learn the real meaning.

I can't imagine that such a great video can exist. It was a couple of years that I studied about curl, but i didn't know what it exactly was, but now I recognize it very greatly.

Thank you very much for this undescribable video