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Multivariable calculus
Course: Multivariable calculus > Unit 2
Lesson 10: Curl- 2d curl intuition
- Visual curl
- 2d curl formula
- 2d curl example
- Finding curl in 2D
- 2d curl nuance
- Describing rotation in 3d with a vector
- 3d curl intuition, part 1
- 3d curl intuition, part 2
- 3d curl formula, part 1
- 3d curl formula, part 2
- 3d curl computation example
- Finding curl in 3D
- Symbols practice: The gradient
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2d curl intuition
A description of how vector fields relate to fluid rotation, laying the intuition for what the operation of curl represents. Created by Grant Sanderson.
Want to join the conversation?
Why is positive curl anti-clockwise while negative curl's clockwise?
Is it more usefull in other fields of study that the positive curl is anti-clockwise, or is it just contrived convention?(9 votes)
Counterclockwise is defined as positive curl for the same reason the cross product is defined as it is (the right hand rule -- the cross product of i and j is k). For example, torque is the cross product of the arm and force (I'll just use x for cross product). So tau = r x F. If r points in the x direction and F points in the y direction, then tau is in the positive z direction, by the definition of cross product. But as we know, the torque is counterclockwise. So positive torque means counterclockwise.
So if one vector points forward, and another vector drags it to the left, the cross product is pointing up. Up means positive. Yet another way to say the cross product points up is to say the curl is counterclockwise. So counterclockwise is positive curl.
The convention of counterclockwise being positive and cross product using the right hand rule are dependent on each other. If you want to change one of them, you have to change the other.(13 votes)
Please could you tell me the formula for this particular vector field so I can simulate on mathematica?(3 votes)
I believe the formula is 3x^2-3y^2. Grant shows the formula in the 2D curl example video.
I know this answer is late, but I thought it might help future inquirers.(6 votes)
Why do I feel like Grant made a mistake with clockwise/counterclockwise rotations? The one that's clockwise rotation was called "counterclockwise", and vice versa! Could I be wrong?(3 votes)
He did indeed mix it up at the end. A popup box appears when you aren't in full screen.(3 votes)
- What I don't understand about this concept is that a vector is defined as having a length. So why are some of the molecules travelling half way along a vector then changing direction supposedly following another vector that is in the field but not in the image? Why doesn't the molecule follow the vector it is currently on for the entirety of its length and then change direction?(1 vote)
It can also depend on what the vectors are representing. In this case, unless I am wrong, the vectors represent the velocity of every particle, so each particle doesn't have to follow each vector from tail to point, as it just represent the direction to which particles travel on that instant, and the magnitude of the vector, aka the length, is just the speed that particles have exactly when passing by that point in space.
So actually when a particle changes its location by an infinitesimal amount in some direction, it's already subject to a new velocity represented by another vector.(5 votes)
in this case (2:36), where we have no rotation (just "molecules" converging/diverging) we are talking about divergent ?
div F ?
i mean, can we describe that origin point with a div ?(2 votes)- Aaaand why did someone decide to call the counter-clockwise rotation "positive"?? That doesn´t make any sense !(1 vote)
- Convention. We needed some standardized notation so confusion wouldn't arise. Plus, while going counter-clockwise, you go from the positive x axis to the positive y axis, so it is thought of as the simplest rotation one can come up with, and hence was standardized.(1 vote)
2:36Video transcript
- [Voiceover] Hello everyone. So I'm gonna start talking about curl. Curl is one of those very
cool vector calculus concepts, and you'll be pretty happy
that you've learned it once you have, if for no other reason because it's kind of
artistically pleasing. And, there's two different versions, there's a two-dimensional curl
and a three-dimensional curl. And naturally enough, I'll start talking about the two-dimensional version and kind of build our
way up to the 3D one. And in this particular video, I just want to lay down the intuition for what's visually going on. And, curl has to do with the
fluid flow interpretation of vector fields. Now this is something
that I've talked about in other videos, especially the ones on
divergents if you watch that, but just as a reminder,
you kind of imagine that each point in space is a particle, like an air molecule or a water molecule. And since what a vector field does is associate each point in
space with some kind of vector, now remember we don't always
draw every single vector, we just draw a small sub-sample, but in principle, every
single point in space has a vector attached to it. You can think of each particle, each one of these water
molecules or air molecules, as moving over time in such a
way that the velocity vector of its movement at any given point in time is the vector that it's attached to. So as it moves to a
different location in space and that velocity vector changes, it might be turning or
it might be accelerating, and that velocity might change. And you end up kind of a
trajectory for your point. And since every single
point is moving in this way, you can start thinking about a flow, kind of a global view of the vector field. And for this particular example, this particular vector
field that I have pictured, I'm gonna go ahead and put a blue dot at various points in space, and, each one of these you can think of as representing a water
molecule or something, and I'm just gonna let it play. And at any given moment,
if you look at the movement of one of these blue dots, it's moving along the
vector that it's attached to at that point, or if that
vector's not pictured, you know the vector that
would be attached to it at that point. And as we get kind of a
feel for what's going on in this entire flow, I want you to notice a
couple of particular regions. First, let's take a look at this region over here on the right. Kind of around here. And just kind of concentrate
on what's going on there. And I'll go ahead and start playing the animation over here. And what's most noticeable
about this region is that there's counterclockwise rotation. And this corresponds to an
idea that the vector field has a curl here, and
I'll go very specifically into what curl means, but just right now you should have the idea that in a region where there's counterclockwise rotation, we want to say the curl is positive. Whereas, if you look at a
region that also has rotation, but clockwise, going the other way, we think of that as being negative curl. Here I'll start it over here. And in contrast, if you look at a place where there's no rotation,
where like at the center here, you have some points coming
in from the top right and from the bottom
left, and then going out from the other corners. But there's no net rotation. If you were to just put like a
twig somewhere in this water, it wouldn't really be rotating. These are regions where you think of them as having zero curl. So with that as a general idea,
clockwise rotation regions correspond to positive curl. Counterclockwise rotation regions correspond to negative curl, and then no rotation
corresponds to zero curl. In the next video I'm
gonna start going through what this means in terms
of the underlying function defining the vector field
and how we can start looking at the partial differential
information of that function to quantify this intuition
of fluid rotation. And what's neat is that it's
not just about fluid rotation. If you have vector fields in other context and you just imagine that
they represent a fluid, even though they don't, this idea rotation and curling actually has certain importance in ways that you totally wouldn't expect. The gradient turns out
to relate to the curl, even though you wouldn't
necessarily think the grading has something to do with fluid rotation. In electromagnetism, this
idea of fluid rotation has a certain importance, even though fluids
aren't actually involved. So, it's more general than
just the representation that we have here, but
it's a very strong visual to have in your mind as
you study vector fields.