More on curl - Video Tutorials, Partial derivatives, gradient, divergence, curl Video Lecture - Engineering Mathematics | Best Video for Engineering Mathematics

Let's say I have the
vector field v.
The vector at any given point
is equal to minus-- or the
magnitude of my x direction
is actually dependent on y.
So where we are in the y
coordinate and the xy plane.
Plus-- and then my magnitude
in the y direction
is dependent on x.
Fair enough.
So first, let's just
chuck through this, and
figure out it's curl.
So the curl of v.
It is equal to our del
vector operator, cross v.
Which is nothing but this.
And even though this looks like
a two dimensional vector field,
we actually have to take the
cross product in 3 dimensions.
Because a curl is just like
torque, and when you-- like we
did the right hand rule when we
studied-- well, I hope you
watched some of the videos on
magnetism and torque-- but the
torque actually goes in a
direction that is perpendicular
to both of the vectors
in your cross product.
If both of these only have x
and y components, your actual
result is going to be
in the z direction.
It's going to be perpendicular
to both of these vectors.
So when you take the cross
product, you still have to
do it in 3 dimensions.
So i j k, partial with respect
to x, partial with respect to
y, partial with respect to z.
The x component is minus y.
The j component is x.
And we have no k component.
This one should be a little
bit cleaner to calculate
than the last example.
So this is going to be equal
to-- well, see the i
component-- let's cross out
its column and its row.
So it's going to be the partial
with respect to y of 0
minus the partial with
respect to z of x.
Minus that.
And all of that times i
minus-- and now it's going
to be the j component.
Remember, you do
plus minus plus.
So minus and this is
the j component.
Cross out its row and column,
the partial derivative
of x with respect to 0.
Or partial derivative of 0
with respect to x, actually.
Minus the partial
derivative of z.
Or the partial derivative
of the [? vector ?]
z of minus y.
That's its j component.
And finally, plus
its k component.
Row and column.
So the partial derivative with
respect to x minus the partial
derivative of y with
respect to minus y.
And I know you can't read it,
but that's that minus y there,
and that's the k component.
And now let's simplify it.
I'll simplify it above it.
So this term-- let's see,
partial derivative of 0.
Well, that's 0.
Partial derivative of
x with respect to z.
Well, as far as z's concerned,
x is a constant, so that's 0.
Partial derivative of 0
with respect to x, 0.
Partial derivative of minus
y with respect to z.
As far as z's concerned,
y is also constant.
So that's 0.
So all we're left is
with this last term.
That was pretty
straightforward.
Why don't we just cross
all of that out?
And that makes intuitive
sense, too, right?
Because at least from x's point
of view, the rotation is--
although, the rotation, if you
think about it, is going to be
in the direction perpendicular
to both the x direction and
perpendicular to the direction
in which it is changing.
You could kind of view it
orthogonally, to its direction
of motion, which is y.
So it would be the z direction.
If that confuses you,
don't worry about it.
But if it doesn't, then
you could apply the same
argument to the j vector.
But anyway, lets simplify this.
So this is equal to the
partial derivative of
x with respect to x.
Well, that's just 1.
Minus the partial derivative
of y-- of minus y
with respect to y.
Well, that's just minus 1.
So it equals 2.
So the curl, at any point of
this vector field, is 2.
Let's see what this vector
field looks like, and let's
see if that gives us-- if
our intuition holds
in this example.
And let me try to make
it a little bit bigger.
Make the window bigger.
There you go.
Well, I think it's clear, you
know, right when you look at
it, that this vector field
looks like it's spinning.
If you were to stick something,
especially in the middle, it's
very clear that it would spin.
But what might be a little
unintuitive-- you might think,
wow, well, wouldn't something
spin faster near the center
than it would here?
Why is the-- you know,
the curl we got is 2.
It's a constant.
The curl is the same throughout
this entire vector field.
So you'll be like, whoa, that's
kind of implying that the field
is making something spin
equally, no matter where
you are in the field.
Let's see if that makes sense.
Well, in the middle, it
definitely makes sense that
something is spinning.
If I had a little stick here,
I'd be pushing in this
direction, with not that much
of a magnitude in
that direction.
And then I'd be pushing down to
the right in this direction,
so it would cause it to spin.
But what if I had that
same stick here?
You'd say, well, on the
top right I'm pushing
up, up and to the left.
And the bottom left, I'm also
pushing up and to the left.
You know, it wouldn't
spin as much.
But it would.
Because the difference in
magnitudes of these 2-- you
could almost view them as the
torque producing forces-- the
difference in magnitude is
enough that you'd still have
the same counterclockwise
rotation here as you
would have here.
So because the curl is a
constant positive number, when
we look at the xy plane like
this, if you put a twig-- that
same twig anywhere where you
put it on this plane--
it'll have the same
counterclockwise rotation.
I think that's pretty neat.
Now let's do a
little experiment.
What would have happened
if this was plus y.
So let's just do
that experiment.
If this was plus y then this
would have been plus y, then
this final term-- we would have
taken the partial derivative
with respect to y of plus y--
and so this would have
been 1 minus plus 1.
And then our curl
would have been 0.
Which would have meant that we
would have had no rotation.
And what's the
intuition of that?
Well, if we just look at it
mathematically, if we have 0
curl, somehow, the rotation in
our x direction must be being
offset by a rotation
in the y direction.
That the torques must
be just perfectly
offsetting each other.
Let's see what happens
if I were to change.
So this was our old graph.
Let me actually change it
to my new vector field.
So that's our new vector field.
This is our vector
field plus yi plus xj.
And now the curl
is 0 everywhere.
Which implies, or which means,
that I could put a twig
anywhere here and I'm not going
to get any kind of rotation.
Let's see if that makes sense.
If I put a twig in the center,
or some kind of stick in
the center, let's see.
I would have the forces or the
fluids pushing in in this
direction, but they're not
helping to rotate, and pushing
out in that direction.
Well, that's not going
to help me either.
So I'm definitely not
going to rotate there.
And actually, you could
put a twig anywhere.
And maybe a twig might be
pushed in a direction.
For example, if I put a twig
here, it's going to be pushed
outward by the flow of the
water, by the velocity
of the water.
But it's not going to rotate.
So the [? ink ?]
kind of holds.
That's even though I kind of
have curl in the x direction,
or I have curl in the
y direction, they're
offsetting each other.
So that in 2 dimensions,
I actually end up
having no rotation.
And actually, this is called an
irrotational-- I think that's
the word-- vector field.
Where you're not going to
have any rotation here.
All of the-- if you think of it
as force or velocity of the
vector field-- is going to be
applying translation to
objects in that field.
And actually, just for fun,
let's think about the
divergence of that field.
Just to, I don't know.
Just because I have 2 minutes.
So if I were to think about the
divergence of that field, it's
fairly easy to calculate.
Let me erase this real fast.
It's always fun to just
interpret a vector
field to death.
Let's do the divergence
of this one.
So the divergence of that
vector field, is just a
partial derivative of
this with respect to x.
So div of v is the same thing
as our del operator dot
our vector field, v.
And that's the partial
derivative of this
with respect to x.
Well, the partial derivative of
y with respect to x is just 0.
Plus the partial derivative
of x with respect to y.
So this is 0.
So the divergence of
this field is 0.
And even in the other case,
when this was a minus, it
still would have been 0.
And does that make sense?
Well, if we take a circle
anywhere here-- let's take it
in the center, because the
center's the most interesting.
Let's take a circle.
So we do have some fluid or
particles coming in at a
certain velocity from the
bottom right and the top left.
But just as much is coming
out through the top right
and the bottom left.
Whatever's coming in through
here and here is leaving
through there and there.
So if I had an infinitesimally
small circle, or sphere, in
this vector field, I would have
no net density increasing.
Or nothing would be entering
into that circle, or in any
given amount of time the
concentration of that
circle wouldn't change.
And that's true pretty
much anywhere.
If I were to draw a circle
here, because the divergence
is 0, it's telling us that
whatever's coming in in 1
direction is coming out
in the other directions.
So I'm not getting any
denser, or any less dense.
I have no divergence
or convergence.
So that's just interesting.
Well, now I've run of time.
So we're done analyzing
this vector field.
See you in the next video.