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Before I actually show you the mechanics of what the curl of a

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vector field really is, let's try to get a little

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bit of intuition.

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So here I've drawn, I'm going to just draw a two-dimensional

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vector field.

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You can extrapolate to 3, but when we're getting

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the intuition, it's good to do it in 2.

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And so, let's see.

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I didn't even label the x and y axis.

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This is x, this is y.

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So when y is relatively low, our magnitude vector goes in

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the x direction, when it increases a little bit, it

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gets a little bit longer.

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So as we can see, as our change in the y-direction, as we go

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in the y-direction, the x-component of our vectors

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get larger and larger.

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And maybe in the x-direction they're constant, regardless

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of your level of x, the magnitude stays.

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So for given y, the magnitude of your x-component vector

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might stay the same.

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So I mean, this vector field might look something like this.

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I'm just making up numbers.

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Maybe it's just, I don't know, y squared i.

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So the magnitude of the x-direction is just a

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function of your y-value.

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And as your y-values get bigger and bigger, the magnitude in

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your x-direction will get bigger and bigger, proportional

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to the square of the magnitude of the y direction.

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But for any given x, it's always going to be the same.

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It's only dependent on y.

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So here, even if we make x larger, we still get

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the same magnitude.

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And remember, these are just sample points

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on our vector field.

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But anyway.

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That's enough of just getting the intuition behind

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that vector field.

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But let me ask you a question.

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If I were to, let's say that this vector field shows the

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velocity of a fluid at various points.

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And so you can view this, we're looking down on a river, maybe.

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If I were to take a little twig or something, and I were to

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place it in this fluid, so let me place the twig right here.

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Let me draw my twig.

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So let's say I place a twig, it's a funny-looking twig,

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but that's good enough.

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Let's say I place a twig right there.

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What's going to happen to the twig?

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Well, at this point on the twig, the water's moving to the

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right, so it'll push this part of the twig to the right.

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At the top of the twig, the water is also moving to the

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right, maybe with a faster velocity, but it's also going

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to push the top of the twig to the right.

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But the top of the twig is going to be being pushed to

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the right faster than the bottom of the twig, right?

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So what's going to happen?

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The twig's going to rotate, right?

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After, I don't know, some period of time, the

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twig's going to look something like this.

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The bottom will move a little bit to the right, but the

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top will move a lot more to the right.

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Right?

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And the whole thing would have been shifted to the right.

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But it's going to rotate a little bit.

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And maybe after a little bit further, maybe it looks

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something like this.

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So you can see that because the vectors increasing in a

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direction that is perpendicular to our direction

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of motion, right?

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This fairly simple example, all of the vectors point

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in the x-direction.

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But the magnitude of the vectors increase, they increase

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perpendicular, they increase in the y-dimension, right?

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And when this happens, when the flow is going in the same

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direction, but it's going at a different magnitude, you see

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that any object in it will rotate, right?

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So let's think about that.

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So if the derivative, the partial derivative, of this

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vector field with respect to y is increasing or decreasing, if

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it's just changing, that means as we increase in y, or as we

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decrease in y, the magnitude of the x-component of our vectors,

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right, the x-direction of our vectors changes.

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And so if you have a different speed for different levels of

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y, as something moves in the x-direction, it's going

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to be rotated, right?

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You could almost view it as if there's a net torque on an

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object that sits in the water here.

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And the ultimate would be, let me draw another vector field,

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the ultimate would be if I had this situation.

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Let me draw another vector field.

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If I had this situation, where maybe down here it's like this,

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then maybe it's like this, and then maybe it gets really

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small, then maybe it switches directions, up here, and then

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the vector field goes like this.

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So you could imagine up here that's going to the left, with

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a fairly large magnitude.

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So if you put a twig here, you would definitely hopefully see

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that the twig, not only will it not be shifted to the right,

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this side is going to be moved to the left, this side is going

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to be the right, it's going to be rotated.

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And you'll see that there's a net torque on the twig.

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So what's the intuition there?

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All of a sudden, we care about how much is the magnitude of a

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vector changing, not in its direction of motion, like in

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the divergence example, but we care how much is the magnitude

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of a vector changing as we go perpendicular to its

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direction of motion.

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So when we learned about dot and cross product,

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what did we learn?

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We learned that the dot product of 2 vectors tells you how much

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2 vectors move together, and the cross product tells you how

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much the perpendicular, it's kind of the multiplication

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of the perpendicular components of a vector.

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So this might give you a little intuition of what is the curl.

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Because the curl essentially measures what is the rotational

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effect, or I guess you could say, what is the curl of a

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vector field at a given point?

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And you can you can visualize it.

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You put a twig there, what would happen to the twig?

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If the twig rotates and there's some curl, if the magnitude

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of the rotation is larger, then the curl is larger.

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If it rotates in the other direction, you'll have the

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negative direction of curl.

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And so just like what we did with torque, we now care

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about the direction.

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Because we care whether it's going counterclockwise or

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clockwise, so we're going to end up with a vector

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quantity, right?

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So, and all of this should hopefully start fitting

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together at this point.

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We've been dealing with this Dell

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vector or this, you know, we could call this abusive

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notation, but it kind of is intuitive, although it really

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doesn't have any meaning when I describe it like this.

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You can kind of write it as a vector operator, and then it

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has a little bit more meeting.

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But this Dell

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operator, we use it a bunch of times.

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You know, if the partial derivative of something in the

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i-direction, plus the partial derivative, something with

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respect to y in the j-direction, plus the partial

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derivative, well, this is if we do it in three dimensions

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with respect to z in the k-direction.

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When we applied it to just a scalar or vector field, you

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know, like a three-dimensional function, we just multiplied

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this times that scalar function, we got the gradient.

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When we took the dot product of this with a vector field, we

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got the divergence of the vector field.

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And this should be a little bit intuitive

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to you, at this point.

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Because when we, you might want to review our original videos

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where we compared the dot product to the cross product.

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Because the dot product was, how much do two

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vectors move together?

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So when you're taking this Dell operator and dotting it with a

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vector field, you're saying, how much is the vector

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field changing, right?

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All a derivative is, a partial derivative or a normal

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derivative, it's just a rate of change.

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Partial derivative with respect to x is rate of change

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in the x-direction.

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So all you're saying is, when you're taking a dot product,

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how much is my rate of change increasing in my

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direction of movement?

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How much is my rate of change in the y-direction increasing

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in the y-direction?

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And so it makes sense that it helps us with divergence.

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Because remember, if this is a vector, and then as we increase

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this in the x-direction, the vectors increase, we took a

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little point, and we said, oh, at this point we're going to

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have more leaving than entering, so we have a

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positive divergence.

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But that makes sense, also, because as you go in the

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x-direction, the magnitudes of the vectors increase.

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Anyway, I don't want to confuse you too much.

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So now, the intuition, because now we don't care about the

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rate of change along with the direction of the vector.

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We care about the rate of change of the magnitudes of

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the vectors perpendicular the direction of the vector.

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So the curl, you might guess, is equal to the cross product

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of our Dell operator and the vector field.

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And if that was where your intuition led you, and that

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is what your guess is, you would be correct.

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That is the curl of the vector field.

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And it is a measure of how much is that field rotating, or

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maybe if you imagine an object in the field, how much is the

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field causing something to rotate because it's

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exerting a net torque?

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Because at different points in the object, you have a

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different magnitude of a field in the same direction.

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Anyway, I don't want to confuse you too much.

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Hopefully that example I just showed you will make

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a little bit of sense.

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Anyway, I realize I've already pushed 9 minutes.

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In the next video, I'll actually compute curl, and

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maybe we'll try to draw a couple more to hit

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the intuition home.

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See you in the next video.

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