3d curl intuition, part 2

WEBVTT
Kind: captions
Language: en

00:00:00.496 --> 00:00:01.329
- [Voiceover] So where we left off,

00:00:01.329 --> 00:00:03.770
I had this two-dimensional vector field V,

00:00:03.770 --> 00:00:06.703
and I have it pictured here as
kind of a yellow vector field

00:00:06.703 --> 00:00:08.423
and I just stuck it in three dimensions

00:00:08.423 --> 00:00:10.495
in kind of an awkward way
where I put it on the XY plane

00:00:10.495 --> 00:00:12.727
and said pretend this
is in three dimensions.

00:00:12.727 --> 00:00:14.550
And then when you describe the rotation,

00:00:14.550 --> 00:00:17.356
around each point what we
were familiar with is 2D curl,

00:00:17.356 --> 00:00:20.772
that's where you get this
vector field, it's not quite

00:00:20.772 --> 00:00:22.676
a 3D vector field because
you're only assigning

00:00:22.676 --> 00:00:25.308
points on the XY plane to
three dimensional vectors,

00:00:25.308 --> 00:00:27.404
rather than every point
in space to a vector,

00:00:27.404 --> 00:00:28.757
but we're getting there.

00:00:28.757 --> 00:00:30.061
So here let's actually
extend this to a fully

00:00:30.061 --> 00:00:32.380
three dimensional vector field,

00:00:32.380 --> 00:00:34.882
and first of all let me just kind of

00:00:34.882 --> 00:00:37.308
clear up the board from the computations

00:00:37.308 --> 00:00:39.308
we did in the last part.

00:00:40.846 --> 00:00:42.270
And as I do that kind
of start thinking about

00:00:42.270 --> 00:00:45.527
how you might want to
extend the vector field

00:00:45.527 --> 00:00:47.518
that I have here that's
pretty much two dimensional

00:00:47.518 --> 00:00:49.303
into three dimensions.

00:00:49.303 --> 00:00:52.880
And one idea you might, we'll
kind of get rid of the circles

00:00:52.880 --> 00:00:56.693
and the plane, is to
take this vector field

00:00:56.693 --> 00:01:00.028
and then just kind of copy
it to different slices.

00:01:00.028 --> 00:01:03.071
So, you might get something
kind of like this.

00:01:03.071 --> 00:01:05.290
And then, I've drawn each
slice a little bit sparser

00:01:05.290 --> 00:01:07.431
than the original one, so
technically that original

00:01:07.431 --> 00:01:08.685
one if you look on the XY plane

00:01:08.685 --> 00:01:10.541
I've pictured many more vectors,

00:01:10.541 --> 00:01:12.381
but it's really the same vector field,

00:01:12.381 --> 00:01:15.519
and all I've done here is
said at every slice in space,

00:01:15.519 --> 00:01:18.007
just copy that same vector field.

00:01:18.007 --> 00:01:20.317
So if you look from above,
you can maybe see how really

00:01:20.317 --> 00:01:22.654
it's just the same vector
field kind of copied a bunch,

00:01:22.654 --> 00:01:25.661
and if you look at it each
slice, you know in the same way

00:01:25.661 --> 00:01:28.379
that in the XY plane,
you've got this vector field

00:01:28.379 --> 00:01:30.762
sitting on the slice,
every other part of space

00:01:30.762 --> 00:01:32.298
will have that.

00:01:32.298 --> 00:01:35.459
And even though there's only
what, like six or seven slices

00:01:35.459 --> 00:01:37.379
displayed here, in principal
you're thinking that

00:01:37.379 --> 00:01:39.885
every one of those infinitely
many slices of space

00:01:39.885 --> 00:01:42.288
has a copy of this vector field.

00:01:42.288 --> 00:01:44.731
And in a formula, what does that mean?

00:01:44.731 --> 00:01:46.691
Well what it means is
what it means is that

00:01:46.691 --> 00:01:48.790
we're taking not just X
and Y as input points,

00:01:48.790 --> 00:01:51.235
But we're gonna start taking Z in as well.

00:01:51.235 --> 00:01:55.891
So if I go, I'm gonna say that
Z is an input point as well.

00:01:55.891 --> 00:01:57.011
And I want to be considering these

00:01:57.011 --> 00:01:58.460
as vectors in three dimensions,

00:01:58.460 --> 00:02:00.677
so rather than just saying
that that it's got X and Y

00:02:00.677 --> 00:02:05.008
components, I'm gonna pretend
like it has a Z component

00:02:05.008 --> 00:02:08.823
that just happens to
be zero for this case.

00:02:08.823 --> 00:02:12.032
And the fact that you
have a Z in the input,

00:02:12.032 --> 00:02:14.104
but the output doesn't depend on the Z,

00:02:14.104 --> 00:02:16.743
corresponds to the fact that
all the slices are the same,

00:02:16.743 --> 00:02:18.263
as you change the Z direction

00:02:18.263 --> 00:02:19.504
the vectors won't change at all,

00:02:19.504 --> 00:02:21.487
they're just carbon copies of each other.

00:02:21.487 --> 00:02:24.047
And the fact this output
has a Z component,

00:02:24.047 --> 00:02:26.393
but it just happens to be
zero is what corresponds

00:02:26.393 --> 00:02:28.222
to the fact that it's very flat looking,

00:02:28.222 --> 00:02:30.569
you know, none of them point
up or down in the Z direction,

00:02:30.569 --> 00:02:32.167
they're all purely X and Y.

00:02:32.167 --> 00:02:34.621
So, as three dimensional vector fields go,

00:02:34.621 --> 00:02:37.559
this one is only barely a
three dimensional vector field,

00:02:37.559 --> 00:02:39.701
it's kind of phoning it in
as far as three dimensional

00:02:39.701 --> 00:02:40.808
vector fields are concerned,

00:02:40.808 --> 00:02:42.809
but it'll be quite good
for our example here,

00:02:42.809 --> 00:02:46.148
because now if we start
thinking of this as representing

00:02:46.148 --> 00:02:49.527
a three dimensional fluid
flow, so now rather than

00:02:49.527 --> 00:02:51.561
just kind of the fluid
flow like the one I have

00:02:51.561 --> 00:02:54.185
pictured over here, where
you've got water molecules

00:02:54.185 --> 00:02:56.224
moving in two dimensions
and it's very easy to

00:02:56.224 --> 00:02:59.184
understand clockwise rotation,
counter clockwise rotation,

00:02:59.184 --> 00:03:01.880
things like that, whereas
over here it's a very

00:03:01.880 --> 00:03:04.409
kind of chaotic three
dimensional fluid flow,

00:03:04.409 --> 00:03:06.523
but because it's so flat
if you view it from above

00:03:06.523 --> 00:03:08.993
it's still loosely the same just kind of

00:03:08.993 --> 00:03:10.799
counter clockwise over here on the right,

00:03:10.799 --> 00:03:12.766
and clockwise like there above,

00:03:12.766 --> 00:03:13.775
so if I were to draw like a column,

00:03:13.775 --> 00:03:15.941
you could think of this column as being,

00:03:15.941 --> 00:03:18.232
having a tornado of fluid flow, right,

00:03:18.232 --> 00:03:20.319
where it's, everything is
kind of rotating together

00:03:20.319 --> 00:03:21.855
in that same direction.

00:03:21.855 --> 00:03:24.760
So if you were to assign a
vector to each point in space

00:03:24.760 --> 00:03:26.992
to describe the kind of rotation happening

00:03:26.992 --> 00:03:29.122
around each one of those points in space,

00:03:29.122 --> 00:03:31.104
you would expect that
those inside this column,

00:03:31.104 --> 00:03:34.552
inside this sort of counter
clockwise rotating tornado,

00:03:34.552 --> 00:03:36.276
and I say counter
clockwise, but if we viewed

00:03:36.276 --> 00:03:37.746
it from below it would look clockwise,

00:03:37.746 --> 00:03:39.581
so that's the tricky part
about three dimensions

00:03:39.581 --> 00:03:41.329
and why we need to
describe it with vectors,

00:03:41.329 --> 00:03:43.616
but you expect these using
your right hand rule,

00:03:43.616 --> 00:03:45.504
where you curl the
fingers of your right hand

00:03:45.504 --> 00:03:47.536
around the direction of rotation here,

00:03:47.536 --> 00:03:50.708
you would expect vectors that
point up in the Z direction,

00:03:50.708 --> 00:03:53.024
the positive Z direction,
and if I do that,

00:03:53.024 --> 00:03:56.540
if I show what all of the
rotation vectors look like,

00:03:56.540 --> 00:03:58.888
you'll get this, and maybe
this is kind of a mess

00:03:58.888 --> 00:04:01.560
because there's a lot of things
on the screen at this point.

00:04:01.560 --> 00:04:03.144
So for the moment I'll kind of

00:04:03.144 --> 00:04:05.566
remove that original
vector field and remove

00:04:05.566 --> 00:04:07.688
the XY plane, and just
kind of focus on this

00:04:07.688 --> 00:04:09.875
new vector field that
I have pictured here.

00:04:09.875 --> 00:04:13.500
Inside that column where we
have that tornado rotation

00:04:13.500 --> 00:04:15.224
I was describing, all of the vectors

00:04:15.224 --> 00:04:17.464
point in the positive Z direction,

00:04:17.464 --> 00:04:19.314
but if we were to view it elsewhere,

00:04:19.314 --> 00:04:20.504
like over in this region,

00:04:20.504 --> 00:04:22.560
those are pointing in
the negative Z direction,

00:04:22.560 --> 00:04:24.581
and if you stick your
thumb in the direction

00:04:24.581 --> 00:04:26.719
of all of these vectors in
the negative Z direction,

00:04:26.719 --> 00:04:30.570
that tells you the direction
of, that tells you how

00:04:30.570 --> 00:04:32.526
the fluid, maybe you're
thinking of it as air

00:04:32.526 --> 00:04:33.930
kind of rushing about the room,

00:04:33.930 --> 00:04:37.166
how that fluid rotates
it in three dimensions.

00:04:37.166 --> 00:04:38.993
So what curl is gonna
do, here I'll kind of

00:04:38.993 --> 00:04:41.581
clear things up, I have
the formula from last time,

00:04:41.581 --> 00:04:43.069
that hopefully hasn't looked too

00:04:43.069 --> 00:04:44.621
in the way while I've been doing this,

00:04:44.621 --> 00:04:47.525
that described curl for a
two dimensional vector field

00:04:47.525 --> 00:04:50.165
if we imagine that's
not just taking X and Y

00:04:50.165 --> 00:04:53.005
as its inputs because it's a
three dimensional vector field,

00:04:53.005 --> 00:04:56.577
but if we imagine it taking X, Y and Z,

00:04:56.577 --> 00:04:59.442
so it's a proper three
dimensional vector field,

00:04:59.442 --> 00:05:02.107
the output is gonna tell
you at every point in space

00:05:02.107 --> 00:05:06.807
what the rotation that
corresponds to that point is.

00:05:06.807 --> 00:05:09.176
And in the next video I'm
gonna give you the formula

00:05:09.176 --> 00:05:11.472
and tell you how you
actually compute this curl

00:05:11.472 --> 00:05:13.143
given an arbitrary function,

00:05:13.143 --> 00:05:15.231
but for right now we're just
getting the visual intuition,

00:05:15.231 --> 00:05:16.847
we're just trying to understand

00:05:16.847 --> 00:05:19.482
what it is that curl
is going to represent.

00:05:19.482 --> 00:05:21.307
And in this vector field,
this one that was just kind of

00:05:21.307 --> 00:05:24.095
copies of a 2D put above,
it's almost contrived

00:05:24.095 --> 00:05:27.026
because all of the rotation
happens in these perfect

00:05:27.026 --> 00:05:29.558
tornado-like patters that
doesn't really change

00:05:29.558 --> 00:05:32.413
as you move up and down
in the XY direction,

00:05:32.413 --> 00:05:34.469
but more generally you might
have a more complicated

00:05:34.469 --> 00:05:36.026
looking vector field, so I'll go ahead and

00:05:36.026 --> 00:05:39.338
finally erase this since it's
been a little bit in the way

00:05:39.338 --> 00:05:41.171
for a while, and erase this guy too,

00:05:41.171 --> 00:05:42.690
and if you think about just arbitrary

00:05:42.690 --> 00:05:46.034
three dimensional vector
fields, like let's say

00:05:46.034 --> 00:05:49.906
this one that I have here,
so I don't know about you,

00:05:49.906 --> 00:05:53.226
but for me it's really hard
to think about the fluid flow

00:05:53.226 --> 00:05:55.741
associated with this, I have
a vague notion in my mind

00:05:55.741 --> 00:05:58.322
that okay, like fluid is
flowing out from this corner

00:05:58.322 --> 00:05:59.769
and kind of flowing in here,

00:05:59.769 --> 00:06:01.857
but it's very hard to
think about it all at once,

00:06:01.857 --> 00:06:04.017
and certainly if you start
talking about rotation,

00:06:04.017 --> 00:06:05.617
it's hard to look at
a given point and say,

00:06:05.617 --> 00:06:07.721
oh yeah there's gonna be
a general fluid rotation

00:06:07.721 --> 00:06:10.601
in some certain way and I can
give you the vector for that.

00:06:10.601 --> 00:06:14.337
But as a more loose and
vague idea, I can say,

00:06:14.337 --> 00:06:16.941
okay, given that there's
some kind of crazy

00:06:16.941 --> 00:06:19.567
air current fluid flow
happening around here,

00:06:19.567 --> 00:06:21.761
I can maybe understand
that at a specific point,

00:06:21.761 --> 00:06:23.545
you're gonna have some kind of rotation,

00:06:23.545 --> 00:06:25.582
and here I'll picture it
as if there's like a ball

00:06:25.582 --> 00:06:28.135
or a globe sitting there
in space, and maybe you're

00:06:28.135 --> 00:06:30.517
imagining your new vector field and saying

00:06:30.517 --> 00:06:32.482
what kind of rotation is it kind of induce

00:06:32.482 --> 00:06:34.669
in that ball that's just
floating there in space?

00:06:34.669 --> 00:06:37.178
Maybe you're imagining this
as like a tennis ball that

00:06:37.178 --> 00:06:39.785
you're sort of holding in
place in space using magnets

00:06:39.785 --> 00:06:42.593
or magic or something like that,
and you're letting the wind

00:06:42.593 --> 00:06:44.694
sort of freely rotate
it, and you're wondering

00:06:44.694 --> 00:06:46.953
what direction it tends to rotate,

00:06:46.953 --> 00:06:49.082
and then when it does and
when you have this rotation,

00:06:49.082 --> 00:06:52.533
you can describe that 3D rotation
with some kind of vector,

00:06:52.533 --> 00:06:55.041
and in this case it would
be a vector that points out

00:06:55.041 --> 00:06:57.345
in that direction because we're
kind of curling our fingers,

00:06:57.345 --> 00:07:00.105
curling our right hand fingers
over in that direction,

00:07:00.105 --> 00:07:03.281
and if you don't understand
how we describe 3D rotation

00:07:03.281 --> 00:07:06.033
with a vector, I have a
video on that, maybe go back

00:07:06.033 --> 00:07:08.436
and check out that video,
but the idea here is that

00:07:08.436 --> 00:07:10.760
when you have some sort
of crazy fluid flow

00:07:10.760 --> 00:07:12.752
that's induced by some
sort of vector field,

00:07:12.752 --> 00:07:14.923
and you do this at every point and say,

00:07:14.923 --> 00:07:16.584
hey what's the rotation
at every single point,

00:07:16.584 --> 00:07:19.094
that's gonna give you the
curl, that is what the curl

00:07:19.094 --> 00:07:23.120
of a three dimensional vector
field is trying to represent.

00:07:23.120 --> 00:07:24.890
And if this feels confusing,
if this feels like something

00:07:24.890 --> 00:07:27.746
that's hard to wrap your
mind around, don't worry

00:07:27.746 --> 00:07:31.422
we've been there, 3D curl is
one of the most complicated

00:07:31.422 --> 00:07:34.315
things in multivariable calculus
that we have to describe.

00:07:34.315 --> 00:07:37.238
But I think the key to
understanding it is to just kind of

00:07:37.238 --> 00:07:40.479
patiently think through and
take the time to think about

00:07:40.479 --> 00:07:43.402
what 2D curl is instead of
thinking about how you extend

00:07:43.402 --> 00:07:45.678
that to three dimensions
and slowly say, okay, okay,

00:07:45.678 --> 00:07:47.807
I kind of get it, tornados of rotation,

00:07:47.807 --> 00:07:49.590
that sort of makes sense,
and if you understand

00:07:49.590 --> 00:07:51.614
how to represent three
dimensional rotation around

00:07:51.614 --> 00:07:54.949
a single point with a
vector, then understanding

00:07:54.949 --> 00:07:57.055
three dimensional curl
comes down to thinking about

00:07:57.055 --> 00:07:58.943
doing that at every single point in space

00:07:58.943 --> 00:08:00.750
according to whatever rotation

00:08:00.750 --> 00:08:03.654
the wind flow around
that point would induce.

00:08:03.654 --> 00:08:05.918
Like I said, it is complicated,
and it's okay if it

00:08:05.918 --> 00:08:08.314
doesn't sink the first
time, it certainly took me

00:08:08.314 --> 00:08:12.698
awhile to really wrap my head
around this 3D curl idea,

00:08:12.698 --> 00:08:15.865
and with that I'll see you next video.


3d curl formula, part 2

WEBVTT
Kind: captions
Language: en

00:00:00.154 --> 00:00:02.331
- [Voiceover] So I'm
explaining the formula for

00:00:02.331 --> 00:00:04.221
three-dimensional curl
and where we left off,

00:00:04.221 --> 00:00:06.801
we have this determinant
of a three-by-three matrix,

00:00:06.801 --> 00:00:07.955
which looks absurd because

00:00:07.955 --> 00:00:09.215
none of the individual components

00:00:09.215 --> 00:00:11.165
are actual numbers, but nevertheless,

00:00:11.165 --> 00:00:12.324
I'm about show how when you kind of

00:00:12.324 --> 00:00:14.415
go through the motions
of taking a determinant,

00:00:14.415 --> 00:00:16.104
you get a vector-valued function

00:00:16.104 --> 00:00:18.663
that corresponds to the curl.

00:00:18.663 --> 00:00:20.736
So let me show you what I mean by that.

00:00:20.736 --> 00:00:22.303
If you're computing the determinant,

00:00:22.303 --> 00:00:23.825
of the guy that we have pictured

00:00:23.825 --> 00:00:25.977
there in the upper-right,

00:00:25.977 --> 00:00:28.743
you start by taking this
upper-left component,

00:00:28.743 --> 00:00:31.037
and then multiplying it by the determinant

00:00:31.037 --> 00:00:33.678
of the sub-matrix, the sub-matrix whose

00:00:33.678 --> 00:00:35.446
rows are not the row of I and

00:00:35.446 --> 00:00:37.864
whose columns are not the row of I.

00:00:37.864 --> 00:00:39.346
So what that looks like over here,

00:00:39.346 --> 00:00:41.397
is we're gonna take that unit vector I,

00:00:41.397 --> 00:00:44.446
and then multiple it by a
certain little determinant,

00:00:44.446 --> 00:00:46.537
and what this sub-determinant involves

00:00:46.537 --> 00:00:49.626
is multiplying this
partial-partial Y by R,

00:00:49.626 --> 00:00:51.577
which means taking the partial derivative

00:00:51.577 --> 00:00:54.933
with respect to Y of the
multi-variable function R

00:00:54.933 --> 00:00:57.024
and then subtracting off
the partial derivative

00:00:57.024 --> 00:00:59.136
with respect to Z of Q.

00:00:59.136 --> 00:01:01.716
So we're subtracting
off partial derivative

00:01:01.716 --> 00:01:06.477
with respect to Z of the
multi-variable function Q,

00:01:06.477 --> 00:01:09.437
and then that,

00:01:09.437 --> 00:01:11.020
so that's the first thing that we do,

00:01:11.020 --> 00:01:12.260
and then as a second part,

00:01:12.260 --> 00:01:14.799
we take this J and we're
gonna subtract off,

00:01:14.799 --> 00:01:17.485
so you're kind of thinking
plus, minus, plus,

00:01:17.485 --> 00:01:19.128
for the elements in this top row,

00:01:19.128 --> 00:01:21.648
so we're gonna subtract off J,

00:01:21.648 --> 00:01:24.167
multiplied by another sub-determinant,

00:01:24.167 --> 00:01:26.607
and then this one is gonna involve,

00:01:26.607 --> 00:01:28.395
you know, this column
that it's not part of

00:01:28.395 --> 00:01:29.917
and this column that it's not part of,

00:01:29.917 --> 00:01:32.719
and you imagine those guys
as a two-by-two matrix,

00:01:32.719 --> 00:01:34.125
and its determinant involves taking the

00:01:34.125 --> 00:01:36.927
partial derivative with respect to X of R,

00:01:36.927 --> 00:01:38.898
so that's kind of the diagonal,

00:01:38.898 --> 00:01:42.919
partial-partial X of R,

00:01:42.919 --> 00:01:44.995
and then subtracting off
the partial derivative

00:01:44.995 --> 00:01:47.066
with respect to Z of P,

00:01:47.066 --> 00:01:51.350
so partial-partial Z of P,

00:01:52.840 --> 00:01:55.091
and then that's just two out of three

00:01:55.091 --> 00:01:57.344
of the things we need to do
for our overall determinant,

00:01:57.344 --> 00:01:58.931
because the last part we're gonna add,

00:01:58.931 --> 00:02:03.114
we're gonna add that
top-right component, K,

00:02:03.114 --> 00:02:06.752
multiplied by the sub-matrix whose

00:02:06.752 --> 00:02:08.970
columns involve the
column it's not part of

00:02:08.970 --> 00:02:12.623
and whose rows involve the
rows that it's not part of,

00:02:12.623 --> 00:02:15.874
so K multiplied by the
determinant of this guy

00:02:15.874 --> 00:02:17.844
is going to be, let's see,

00:02:17.844 --> 00:02:19.774
partial-partial X of Q,

00:02:19.774 --> 00:02:24.774
so that's partial-partial X of Q

00:02:26.154 --> 00:02:28.796
minus partial-partial Y of P,

00:02:28.796 --> 00:02:32.394
so partial derivative with respect to Y

00:02:32.394 --> 00:02:35.277
of the multi-variable function P,

00:02:35.277 --> 00:02:39.344
and that entire expression is the

00:02:39.344 --> 00:02:41.617
three-dimensional curl of the function

00:02:41.617 --> 00:02:44.948
whose components are P, Q, and R.

00:02:44.948 --> 00:02:46.575
So here we have our vector function,

00:02:46.575 --> 00:02:48.404
vector-valued function V whose

00:02:48.404 --> 00:02:50.193
components are P, Q, and R,

00:02:50.193 --> 00:02:51.574
and when you go through this whole process

00:02:51.574 --> 00:02:53.524
of imagining the cross-product between

00:02:53.524 --> 00:02:55.921
the Del operator, this Nabla symbol,

00:02:55.921 --> 00:03:00.471
and the vector output P, Q, and R,

00:03:00.471 --> 00:03:03.092
what you get is this whole expression,

00:03:03.092 --> 00:03:06.342
and, you know, here we're
writing with I, J, K notation,

00:03:06.342 --> 00:03:09.492
if you're writing it as a column vector,

00:03:09.492 --> 00:03:11.442
I guess I didn't erase some of these guys,

00:03:11.442 --> 00:03:13.478
but if you're writing
this as a column vector,

00:03:13.478 --> 00:03:16.219
it would look like saying the curl

00:03:16.219 --> 00:03:18.331
of your vector valued function V

00:03:18.331 --> 00:03:20.689
as a function of X, Y, and Z

00:03:20.689 --> 00:03:24.681
is equal to,

00:03:24.681 --> 00:03:26.237
and then what I'd put in
for this first component

00:03:26.237 --> 00:03:27.270
would be what's up there,

00:03:27.270 --> 00:03:32.007
so that would be your partial
with respect to Y of R

00:03:32.007 --> 00:03:34.200
minus partial of Q with respect to Z,

00:03:34.200 --> 00:03:36.740
so partial of Q with respect to Z,

00:03:36.740 --> 00:03:38.912
and I won't copy it down
for all of the other ones

00:03:38.912 --> 00:03:39.890
but in principle, you know,

00:03:39.890 --> 00:03:42.203
you'd kind of, whatever
this J component is,

00:03:42.203 --> 00:03:43.342
and I guess we're subtracting it

00:03:43.342 --> 00:03:45.030
so you'd subtract there,

00:03:45.030 --> 00:03:47.791
you'd copy that as the next
component and then over here.

00:03:47.791 --> 00:03:49.621
But often times when
you're computing curl,

00:03:49.621 --> 00:03:52.302
you kinda switch to
using this IJK notation.

00:03:52.302 --> 00:03:54.091
My personal preference,
I typically default

00:03:54.091 --> 00:03:55.920
to column vectors and other people

00:03:55.920 --> 00:03:57.744
will write in terms of I, J, and K,

00:03:57.744 --> 00:03:59.044
it doesn't really matter as long as

00:03:59.044 --> 00:04:01.872
you know how to go back
and forth between the two.

00:04:01.872 --> 00:04:03.373
One really quick thing
that I wanna highlight

00:04:03.373 --> 00:04:05.021
before doing an example of this

00:04:05.021 --> 00:04:08.595
is that the K-component here,

00:04:08.595 --> 00:04:10.544
the Z-component of the output,

00:04:10.544 --> 00:04:12.943
is exactly the
two-dimensional curl formula.

00:04:12.943 --> 00:04:15.422
If you kind of look back
to the videos on 2-D curl

00:04:15.422 --> 00:04:16.581
and what its formula is,

00:04:16.581 --> 00:04:18.043
that is what we have here.

00:04:18.043 --> 00:04:19.242
And in fact all the other components

00:04:19.242 --> 00:04:20.830
kind of look like mirrors of that

00:04:20.830 --> 00:04:22.760
but you're using slightly
different operators

00:04:22.760 --> 00:04:24.463
and slightly different functions

00:04:24.463 --> 00:04:25.642
but if you think about rotation

00:04:25.642 --> 00:04:27.553
that happens purely in the XY plane,

00:04:27.553 --> 00:04:29.160
just two-dimensional rotation,

00:04:29.160 --> 00:04:30.315
and how in three dimensions that's

00:04:30.315 --> 00:04:32.834
described with a vector in the K direction

00:04:32.834 --> 00:04:34.421
and again, if that
doesn't quite seem clear,

00:04:34.421 --> 00:04:36.124
maybe look back at the video on

00:04:36.124 --> 00:04:38.341
describing rotation with
a three-dimensional vector

00:04:38.341 --> 00:04:39.521
and the right-hand rule,

00:04:39.521 --> 00:04:42.322
but vector is pointing
in the pure Z direction,

00:04:42.322 --> 00:04:44.560
describe rotation in the XY plane,

00:04:44.560 --> 00:04:46.283
and what's happening with these other guys

00:04:46.283 --> 00:04:47.684
is kind of similar, right?

00:04:47.684 --> 00:04:50.714
Rotation that happens
purely in the XZ plane

00:04:50.714 --> 00:04:52.704
is gonna correspond with a rotation vector

00:04:52.704 --> 00:04:54.170
in the Y direction, the direction

00:04:54.170 --> 00:04:56.730
perpendicular to the X, let's see, so

00:04:56.730 --> 00:04:58.721
the XZ plane over here.

00:04:58.721 --> 00:05:00.711
And then similarly this first component

00:05:00.711 --> 00:05:02.294
kind of tells you all
the rotation happening

00:05:02.294 --> 00:05:04.773
in the YZ plane and the vectors

00:05:04.773 --> 00:05:08.104
in the I direction, the X
direction of the output,

00:05:08.104 --> 00:05:11.213
kind of corresponds to
rotation in that plane.

00:05:11.213 --> 00:05:12.372
Now when you compute it,

00:05:12.372 --> 00:05:13.471
you're not always thinking about

00:05:13.471 --> 00:05:14.912
oh, you know, this corresponds to

00:05:14.912 --> 00:05:15.991
rotation in that plane and this

00:05:15.991 --> 00:05:17.492
corresponds to rotation in that plane,

00:05:17.492 --> 00:05:19.035
you're just kind of computing it to

00:05:19.035 --> 00:05:20.012
get a formula out,

00:05:20.012 --> 00:05:20.965
but I think it's kind of nice to

00:05:20.965 --> 00:05:22.633
recognize that all the intuition

00:05:22.633 --> 00:05:24.134
that we put into the two-dimensional curl

00:05:24.134 --> 00:05:25.722
does show up here

00:05:25.722 --> 00:05:27.673
and another thing I wanna emphasize

00:05:27.673 --> 00:05:30.393
is this is not a formula to be memorized.

00:05:30.393 --> 00:05:32.162
I would not, if I were you, try to

00:05:32.162 --> 00:05:34.641
sit down and memorize
this long expression.

00:05:34.641 --> 00:05:36.632
The only thing that you need to remember,

00:05:36.632 --> 00:05:38.723
the only thing, is that curl is

00:05:38.723 --> 00:05:41.404
represented as this Del cross V,

00:05:41.404 --> 00:05:43.823
this Nabla symbol cross product with

00:05:43.823 --> 00:05:45.451
a vector-valued function V,

00:05:45.451 --> 00:05:47.300
because from there, whatever
your components are,

00:05:47.300 --> 00:05:48.862
you can kind of go through the process

00:05:48.862 --> 00:05:50.384
that I just did and the more you do it,

00:05:50.384 --> 00:05:51.644
the quicker it becomes,

00:05:51.644 --> 00:05:53.902
it's kind of long but it
doesn't take that long,

00:05:53.902 --> 00:05:55.892
and it's certainly much
more fault-tolerant

00:05:55.892 --> 00:05:57.459
than trying to remember something that

00:05:57.459 --> 00:06:00.161
has as many moving parts as the formula

00:06:00.161 --> 00:06:01.582
that you see here,

00:06:01.582 --> 00:06:02.942
and in the next video I'll go through

00:06:02.942 --> 00:06:04.041
an actual example of that.

00:06:04.041 --> 00:06:06.374
I'll have functions for P, Q, and R

00:06:06.374 --> 00:06:08.430
and walk through that process in

00:06:08.430 --> 00:06:10.034
a more concrete context.

00:06:10.034 --> 00:06:11.160
I'll see you then!


Bond enthalpy and enthalpy of reaction

WEBVTT
Kind: captions
Language: en

00:00:00.404 --> 00:00:02.911
- [Voiceover] We're gonna be
talking about bond enthalpy

00:00:02.911 --> 00:00:04.769
and how you can use it to calculate

00:00:04.769 --> 00:00:07.416
the enthalpy of reaction.

00:00:07.416 --> 00:00:10.642
Bond enthalpy is the energy that it takes

00:00:10.642 --> 00:00:12.975
to break one mole of a bond.

00:00:16.425 --> 00:00:18.258
So one mole of a bond.

00:00:20.350 --> 00:00:22.021
So different types of bonds will have

00:00:22.021 --> 00:00:24.188
different bond enthalpies.

00:00:25.040 --> 00:00:27.969
So as an example, we can talk about

00:00:27.969 --> 00:00:32.136
a carbon hydrogen bond, or a
carbon hydrogen single bond.

00:00:33.655 --> 00:00:36.859
So this carbon is probably
attached to some other stuff,

00:00:36.859 --> 00:00:40.574
because carbons usually have
more than one single bond.

00:00:40.574 --> 00:00:42.409
But we're gonna ignore everything else

00:00:42.409 --> 00:00:43.964
attached to the carbon,

00:00:43.964 --> 00:00:47.656
we're just gonna represent it
as a big blob, like popcorn,

00:00:47.656 --> 00:00:49.978
maybe it's a protein, it could be,

00:00:49.978 --> 00:00:52.950
it could be a sugar molecule,
it could be a lot of things.

00:00:52.950 --> 00:00:55.926
But we're ignoring that blob.

00:00:55.926 --> 00:00:59.057
And one other thing I
forgot to say earlier

00:00:59.057 --> 00:01:01.031
is that this is the energy it takes

00:01:01.031 --> 00:01:04.769
to break one mole of a
bond in the gas phase.

00:01:04.769 --> 00:01:07.852
So it's a pretty specific definition.

00:01:10.969 --> 00:01:14.165
So in the case of our
carbon hydrogen bond,

00:01:14.165 --> 00:01:16.728
the bond enthalpy of this bond,

00:01:16.728 --> 00:01:19.746
so if we break this bond...

00:01:19.746 --> 00:01:21.790
Let's do a sort of dotted line.

00:01:21.790 --> 00:01:23.790
If we break this bond...

00:01:25.319 --> 00:01:28.753
We have to add energy, and what we'll get

00:01:28.753 --> 00:01:32.169
as our products is we'll get our popcorn,

00:01:32.169 --> 00:01:35.652
and what happens is,
when we break this bond,

00:01:35.652 --> 00:01:39.112
the two electrons that
originally made up the bond,

00:01:39.112 --> 00:01:42.037
one of the electrons
will go to the carbon,

00:01:42.037 --> 00:01:45.451
and the other electron'll
go to the hydrogen.

00:01:45.451 --> 00:01:48.678
And we usually represent single electrons

00:01:48.678 --> 00:01:50.652
like that using a single dot,

00:01:50.652 --> 00:01:52.719
sort of like when you
write Lewis structures,

00:01:52.719 --> 00:01:55.087
you can write lone pairs with two dots.

00:01:55.087 --> 00:01:58.965
So here's our carbon with
one dot, or one electron,

00:01:58.965 --> 00:02:01.287
and our hydrogen with one electron,

00:02:01.287 --> 00:02:04.787
and these are both still in the gas phase.

00:02:06.916 --> 00:02:10.761
So the delta H of this
reaction is the bond enthalpy,

00:02:10.761 --> 00:02:13.261
which I will abbreviate as BE.

00:02:14.174 --> 00:02:16.960
So some important things to
remember about bond enthalpy

00:02:16.960 --> 00:02:20.460
are that bond enthalpy is always positive.

00:02:22.463 --> 00:02:24.893
So it's always going to take energy,

00:02:24.893 --> 00:02:29.060
you're always gonna have to
add energy to break a bond.

00:02:30.070 --> 00:02:32.985
If we take the reverse of the bond,

00:02:32.985 --> 00:02:37.394
if we take the reverse
of the bond enthalpy,

00:02:37.394 --> 00:02:41.561
so another way to think about
this is to flip this reaction,

00:02:42.433 --> 00:02:44.778
so if we take the
reverse of this reaction,

00:02:44.778 --> 00:02:47.361
that means we're making a bond.

00:02:51.628 --> 00:02:55.180
And since we know that breaking
a bond always takes energy,

00:02:55.180 --> 00:02:59.824
that means making a bond
always releases energy.

00:02:59.824 --> 00:03:03.574
So it will always be
negative to make a bond.

00:03:05.676 --> 00:03:07.603
And that's another way of saying,

00:03:07.603 --> 00:03:10.103
it will always release energy.

00:03:13.036 --> 00:03:15.405
And then the third thing
that we're gonna discuss

00:03:15.405 --> 00:03:18.354
about bond enthalpy is
that you can use it,

00:03:18.354 --> 00:03:22.521
you can use bond enthalpy to
estimate delta H of reaction.

00:03:27.247 --> 00:03:31.009
And delta H of reaction is,
or the enthalpy of reaction,

00:03:31.009 --> 00:03:33.354
is something that chemists
are often interested in.

00:03:33.354 --> 00:03:35.908
We wanna know if it's
exothermic or endothermic.

00:03:35.908 --> 00:03:37.487
You might know that
there's lots of other ways

00:03:37.487 --> 00:03:40.289
of calculating delta H of reaction,

00:03:40.289 --> 00:03:42.539
such as using Hess's law...

00:03:44.801 --> 00:03:49.186
Or another way is using
delta H of formation.

00:03:49.186 --> 00:03:51.187
And then there are other ways too.

00:03:51.187 --> 00:03:53.184
So this is just another
way that we can use

00:03:53.184 --> 00:03:57.247
to calculate delta H of
reaction using bond enthalpies.

00:03:57.247 --> 00:04:00.557
So we're gonna go through
an example of that next.

00:04:00.557 --> 00:04:04.027
So the example reaction is taking propyne,

00:04:04.027 --> 00:04:05.694
which is C3H4 gas...

00:04:08.300 --> 00:04:12.196
And reacting it with
hydrogen, so hydrogen gas,

00:04:12.196 --> 00:04:14.279
to get propane, C3H8 gas.

00:04:19.422 --> 00:04:21.373
And I don't know about you,

00:04:21.373 --> 00:04:24.925
I'm pretty bad at looking at
a chemical formula like this

00:04:24.925 --> 00:04:27.085
and knowing exactly what
the molecule looks like,

00:04:27.085 --> 00:04:30.429
so I'm gonna draw out
the Lewis structures.

00:04:30.429 --> 00:04:32.727
So the Lewis structure for propyne,

00:04:32.727 --> 00:04:36.227
propyne has three carbons, and one triple,

00:04:37.557 --> 00:04:40.057
one carbon carbon triple bond,

00:04:41.133 --> 00:04:43.716
and then it has four hydrogens.

00:04:46.497 --> 00:04:47.997
So that's propyne.

00:04:48.981 --> 00:04:51.481
And we also have hydrogen gas.

00:04:52.534 --> 00:04:56.701
And our product is propane, so
propane has all single bonds.

00:04:59.198 --> 00:05:02.115
So three carbons with single bonds,

00:05:03.036 --> 00:05:06.453
and eight hydrogens bound to the carbons.

00:05:09.902 --> 00:05:12.898
So that's the reaction
we are interested in,

00:05:12.898 --> 00:05:17.785
and what we wanna know here is
what is delta H of reaction?

00:05:17.785 --> 00:05:21.489
And how can we calculate
it using bond enthalpies?

00:05:21.489 --> 00:05:23.695
We said earlier that bond enthalpies,

00:05:23.695 --> 00:05:26.412
a bond enthalpy is the energy
it takes to break a bond.

00:05:26.412 --> 00:05:29.430
So what we're gonna do next
is look at our reaction

00:05:29.430 --> 00:05:33.331
in terms of what bonds are
broken and what bonds are formed.

00:05:33.331 --> 00:05:36.559
And this is a lot easier to
do using the Lewis structures.

00:05:36.559 --> 00:05:40.188
First let's talk about
which bonds are broken.

00:05:40.188 --> 00:05:43.943
We, if we compare our
reactants and our products,

00:05:43.943 --> 00:05:47.519
we're breaking this
carbon carbon triple bond.

00:05:47.519 --> 00:05:49.910
If we're breaking this
carbon carbon triple bond,

00:05:49.910 --> 00:05:52.859
and we're also gonna break
this hydrogen hydrogen bond,

00:05:52.859 --> 00:05:54.438
and one thing we forgot to do earlier

00:05:54.438 --> 00:05:55.762
which is super important,

00:05:55.762 --> 00:05:58.804
is we actually need to make
sure our reaction is balanced.

00:05:58.804 --> 00:06:00.685
And we have four plus two,

00:06:00.685 --> 00:06:03.216
six hydrogens on our reactant side,

00:06:03.216 --> 00:06:05.746
and we have eight hydrogens
on our product side.

00:06:05.746 --> 00:06:06.861
That's not balanced.

00:06:06.861 --> 00:06:09.253
So we actually need two hydrogen molecules

00:06:09.253 --> 00:06:10.530
on the reactant side.

00:06:10.530 --> 00:06:12.736
So let's draw one more in.

00:06:12.736 --> 00:06:15.847
So yes, we said we are breaking
a hydrogen hydrogen bond,

00:06:15.847 --> 00:06:19.377
we're actually breaking two
hydrogen hydrogen bonds.

00:06:19.377 --> 00:06:20.979
It's important to keep track of

00:06:20.979 --> 00:06:22.976
how many of each type
of bond we're breaking

00:06:22.976 --> 00:06:25.182
because the bond enthalpy is per mole,

00:06:25.182 --> 00:06:27.109
so if you have twice as many moles

00:06:27.109 --> 00:06:31.637
it'll take twice as much energy
to break all of those bonds.

00:06:31.637 --> 00:06:34.423
And then we can look at
the bonds that are formed.

00:06:34.423 --> 00:06:36.374
So we have, not, since we broke

00:06:36.374 --> 00:06:38.931
this carbon carbon triple bond,

00:06:38.931 --> 00:06:42.264
that means we needed to make a new bond,

00:06:43.154 --> 00:06:45.917
and the new bond we made
in our product molecule

00:06:45.917 --> 00:06:48.750
is this carbon carbon single bond.

00:06:51.197 --> 00:06:53.417
Not only did we form a new single bond

00:06:53.417 --> 00:06:55.785
between these two carbons,
but now these carbons

00:06:55.785 --> 00:06:57.504
are attached to a bunch of hydrogens,

00:06:57.504 --> 00:07:01.004
so we made four new carbon hydrogen bonds.

00:07:03.564 --> 00:07:05.840
So let's write that out so
that we can keep track of them

00:07:05.840 --> 00:07:09.741
when we do our final calculation
of delta H of reaction.

00:07:09.741 --> 00:07:13.158
So if we just look at the bonds broken...

00:07:14.849 --> 00:07:16.599
The bonds we broke...

00:07:18.704 --> 00:07:21.467
We have a carbon carbon triple bond,

00:07:21.467 --> 00:07:25.228
and we have a couple
hydrogen hydrogen bonds.

00:07:25.228 --> 00:07:29.431
Let's also just write down
how many of each we have,

00:07:29.431 --> 00:07:32.334
because we'll need that
for our calculation.

00:07:32.334 --> 00:07:35.059
So we have one carbon carbon triple bond,

00:07:35.059 --> 00:07:39.300
and we have two hydrogen
hydrogen bonds that are broken.

00:07:39.300 --> 00:07:42.968
And then we can also look
up their bond enthalpies,

00:07:42.968 --> 00:07:45.718
which are in kilojoules per mole.

00:07:48.927 --> 00:07:51.258
Bond enthalpies you can typically look up

00:07:51.258 --> 00:07:53.185
in your textbook or online,

00:07:53.185 --> 00:07:56.227
and they usually come in a
table of bond enthalpies.

00:07:56.227 --> 00:07:58.735
And so the units can
be kilojoules per mole,

00:07:58.735 --> 00:08:02.311
sometimes you'll also see
calories or kilocalories per mole.

00:08:02.311 --> 00:08:04.865
I already looked up these
bond enthalpy values.

00:08:04.865 --> 00:08:07.605
So carbon carbon triple
bonds have a bond enthalpy

00:08:07.605 --> 00:08:10.461
of 835 kilojoules per mole,

00:08:10.461 --> 00:08:13.503
and hydrogen hydrogen
bonds have a bond enthalpy

00:08:13.503 --> 00:08:16.753
of 800, sorry, 436 kilojoules per mole.

00:08:19.592 --> 00:08:23.719
And then next, if we look at
the bonds that are broken,

00:08:23.719 --> 00:08:26.719
we have a carbon carbon single bond.

00:08:29.719 --> 00:08:32.969
And we have one of those bonds forming.

00:08:34.517 --> 00:08:36.885
And the bond enthalpy for that,

00:08:36.885 --> 00:08:41.052
which is also in terms of
kilojoules per mole, is 346.

00:08:44.176 --> 00:08:46.916
And last but not least, we have the

00:08:46.916 --> 00:08:49.494
carbon hydrogen bonds that we're forming,

00:08:49.494 --> 00:08:51.932
and we have four of
those, and each of those,

00:08:51.932 --> 00:08:55.682
the bond enthalpy is
413 kilojoules per mole.

00:08:59.446 --> 00:09:02.148
So now we can take all of this information

00:09:02.148 --> 00:09:05.144
and put it together to
calculate delta H of reaction.

00:09:05.144 --> 00:09:08.371
So delta H of reaction,
if we're thinking about it

00:09:08.371 --> 00:09:10.693
in terms of bonds made and broken,

00:09:10.693 --> 00:09:14.443
it's a total energy
change during a reaction.

00:09:16.057 --> 00:09:18.751
And so it's just the energy it takes

00:09:18.751 --> 00:09:22.501
to break all of our
bonds in the reactants...

00:09:24.307 --> 00:09:27.040
So to break this carbon carbon triple bond

00:09:27.040 --> 00:09:30.040
and the two hydrogen hydrogen bonds,

00:09:31.691 --> 00:09:35.274
plus the energy it
takes to make the bonds,

00:09:37.071 --> 00:09:39.238
to make new product bonds.

00:09:43.108 --> 00:09:44.664
We said earlier that you always

00:09:44.664 --> 00:09:46.545
have to add energy to break bonds,

00:09:46.545 --> 00:09:48.565
so bond enthalpy is always positive,

00:09:48.565 --> 00:09:50.701
so we know this part of our calculation

00:09:50.701 --> 00:09:52.838
should always be a positive number.

00:09:52.838 --> 00:09:54.440
What that means is that it always

00:09:54.440 --> 00:09:56.506
releases energy to make new bonds,

00:09:56.506 --> 00:10:00.407
and when energy is released,
delta H becomes more negative.

00:10:00.407 --> 00:10:02.729
So this number here,
when we're talking about

00:10:02.729 --> 00:10:05.353
adding up the energy it
takes to make new bonds,

00:10:05.353 --> 00:10:07.304
these should be negative numbers.

00:10:07.304 --> 00:10:09.968
So now let's plug in the values
we have for bond enthalpy

00:10:09.968 --> 00:10:11.878
for all of these bonds that are

00:10:11.878 --> 00:10:14.804
made and broken in our reaction.

00:10:14.804 --> 00:10:17.753
Let's start with the
bonds that are broken.

00:10:17.753 --> 00:10:19.866
So we have our carbon carbon single bond,

00:10:19.866 --> 00:10:23.976
that will require 835 kilojoules per mole,

00:10:23.976 --> 00:10:27.203
and we have only one of them.

00:10:27.203 --> 00:10:30.384
And we also have to break
two hydrogen hydrogen bonds,

00:10:30.384 --> 00:10:33.467
so two times 436 kilojoules per mole,

00:10:34.982 --> 00:10:38.558
which is the bond enthalpy of that bond.

00:10:38.558 --> 00:10:42.087
So that's all of the bonds we break.

00:10:42.087 --> 00:10:45.686
Now we have to add up the
energy that's released

00:10:45.686 --> 00:10:47.474
when we make the new bonds.

00:10:47.474 --> 00:10:50.121
So we have this carbon carbon single bond.

00:10:50.121 --> 00:10:54.301
So that is 346 kilojoules per
mole, and that's negative,

00:10:54.301 --> 00:10:56.968
because that energy is released.

00:10:57.854 --> 00:11:01.244
And then, the last bond is
the carbon hydrogen bond,

00:11:01.244 --> 00:11:03.682
also negative because
the energy is released,

00:11:03.682 --> 00:11:06.282
and we have four of them, and each of them

00:11:06.282 --> 00:11:09.365
will release 413 kilojoules per mole.

00:11:10.950 --> 00:11:13.852
So if we still all of
this into our calculator

00:11:13.852 --> 00:11:15.733
to get our final answer,

00:11:15.733 --> 00:11:18.937
what I got was that the
delta H of reaction for this,

00:11:18.937 --> 00:11:20.864
for this hydrogenation reaction

00:11:20.864 --> 00:11:23.001
between propine and hydrogen gas,

00:11:23.001 --> 00:11:25.334
is -291 kilojoules per mole.

00:11:30.269 --> 00:11:34.053
We can see that this overall
reaction releases energy,

00:11:34.053 --> 00:11:38.053
because delta H is negative,
so it's exothermic.

00:11:41.336 --> 00:11:43.690
And that's how you can use bond enthalpies

00:11:43.690 --> 00:11:46.277
to calculate delta H of reaction.


Beginning sentences with conjunctions

WEBVTT
Kind: captions
Language: en

00:00:00.206 --> 00:00:01.733
- [Voiceover] Hey grammarians.

00:00:01.733 --> 00:00:03.804
Some of you may have been raised like me,

00:00:03.804 --> 00:00:05.030
with the superstition

00:00:05.030 --> 00:00:08.559
that it wasn't okay to start
a sentence with a conjunction.

00:00:08.559 --> 00:00:11.315
Like for, or and, or nor, or but.

00:00:11.315 --> 00:00:13.018
But I'm here to tell you,

00:00:13.018 --> 00:00:15.014
not to bury the lead, totally fine.

00:00:15.014 --> 00:00:16.629
Like, you may do this.

00:00:16.629 --> 00:00:20.169
There is nothing ungrammatical
about this construction.

00:00:20.169 --> 00:00:22.445
Let me show you what I mean

00:00:22.445 --> 00:00:26.253
and what sort of sentences have
historically been regarded,

00:00:26.253 --> 00:00:29.109
by some people, as unacceptable.

00:00:29.109 --> 00:00:32.778
So let's start by looking at
this two sentence paragraph.

00:00:32.778 --> 00:00:35.750
"Ginny looked at the
painting suspiciously.

00:00:35.750 --> 00:00:39.419
But, as she turned away, she
didn't see it look at her."

00:00:39.419 --> 00:00:42.159
Or, just starting something on its own,

00:00:42.159 --> 00:00:44.666
without connecting it to, you know,

00:00:44.666 --> 00:00:46.547
you can start a paragraph with it like,

00:00:46.547 --> 00:00:50.007
"But the question remains, what is art?"

00:00:50.007 --> 00:00:52.399
There's this, there's this superstition

00:00:52.399 --> 00:00:55.092
that says that you can't begin
sentences with conjunctions,

00:00:55.092 --> 00:00:57.391
that it's ungrammatical or weak writing,

00:00:57.391 --> 00:00:59.272
and I don't think either
of these things are true.

00:00:59.272 --> 00:01:01.942
In fact, I think is a conflation,

00:01:01.942 --> 00:01:05.727
or a confusion of a couple
separate issues in writing.

00:01:05.727 --> 00:01:08.513
But sometimes you wanna
punctuate a sentence

00:01:08.513 --> 00:01:09.999
by beginning with a conjunction.

00:01:09.999 --> 00:01:12.902
There's a kind of dramatic
tension you can access

00:01:12.902 --> 00:01:14.179
by beginning a sentence like that,

00:01:14.179 --> 00:01:16.617
kind of unexpectedly leaping into action.

00:01:16.617 --> 00:01:19.659
What I like about sentence
initial "But" here

00:01:19.659 --> 00:01:23.117
is that it kind of, you
think the sentence is over,

00:01:23.117 --> 00:01:24.033
and it is,

00:01:24.033 --> 00:01:26.323
but then this other thing happens,

00:01:26.323 --> 00:01:29.504
and putting it after a piece
of terminal punctuation

00:01:29.504 --> 00:01:32.918
like a period really serves to bring into

00:01:32.918 --> 00:01:36.089
sharp relief whatever it
is you're trying to say

00:01:36.089 --> 00:01:37.608
after the "But."

00:01:37.608 --> 00:01:40.232
I think this an awfully useful technique,

00:01:40.232 --> 00:01:41.439
and I'm not alone.

00:01:41.439 --> 00:01:44.318
Brian Garner, the author
of the usage manual

00:01:44.318 --> 00:01:45.549
that I use most of the time,

00:01:45.549 --> 00:01:47.779
Garner's Modern American Usage,

00:01:47.779 --> 00:01:50.402
says that about 9% of sentences

00:01:50.402 --> 00:01:52.515
by, what he calls, first-rate writers

00:01:52.515 --> 00:01:55.394
begin with "and," "but"
and "so" or other words

00:01:55.394 --> 00:01:59.644
in the FANBOYS mnenomic,
if you remember FANBOYS.

00:01:59.644 --> 00:02:01.402
And that's "for,"

00:02:01.402 --> 00:02:02.485
"and," "nor,"

00:02:03.358 --> 00:02:04.358
"but," "or,"

00:02:05.691 --> 00:02:06.691
"yet," "so."

00:02:07.654 --> 00:02:10.216
And then those are the
coordinating conjunctions.

00:02:10.216 --> 00:02:11.899
And it's not just Garner,

00:02:11.899 --> 00:02:13.779
'cause that's a relatively
recent publication,

00:02:13.779 --> 00:02:16.394
but we're talking about
language authorities

00:02:16.394 --> 00:02:18.447
going back to like, Anglo-Saxon times?!

00:02:18.447 --> 00:02:19.422
I've never had a problem

00:02:19.422 --> 00:02:22.957
with starting sentences with conjunctions.

00:02:22.957 --> 00:02:26.643
I think where the prohibition
has traditionally come from,

00:02:26.643 --> 00:02:29.685
where this language
superstition comes from,

00:02:29.685 --> 00:02:32.680
is a conflation with another problem.

00:02:32.680 --> 00:02:35.004
Let's get to that on the next screen.

00:02:35.004 --> 00:02:37.812
Alright, consider the following utterance.

00:02:37.812 --> 00:02:39.391
"Because I said so."

00:02:39.391 --> 00:02:40.691
Is this a sentence?

00:02:40.691 --> 00:02:44.383
No, actually this is what's
called a sentence fragment.

00:02:44.383 --> 00:02:47.472
"Because I said so."
is a dependent clause,

00:02:47.472 --> 00:02:49.910
it can't stand on its own in a sentence.

00:02:49.910 --> 00:02:53.602
"Because" is what we'd call
a subordinating conjunction.

00:02:53.602 --> 00:02:56.481
So dependent clauses follow
subordinating conjunctions,

00:02:56.481 --> 00:02:59.267
which means that they grant context,

00:02:59.267 --> 00:03:01.729
but they can't stand on
their own as sentences.

00:03:01.729 --> 00:03:04.283
If you remember the
analogy I used previously,

00:03:04.283 --> 00:03:09.066
an independent clause,
you know, is like a tree.

00:03:09.066 --> 00:03:12.572
And, a subordinating conjunction
marking a dependent clause

00:03:12.572 --> 00:03:14.825
is like a ladder leaning
up against that tree.

00:03:14.825 --> 00:03:16.706
You can have the tree without the ladder,

00:03:16.706 --> 00:03:18.331
but without the tree to lean on,

00:03:18.331 --> 00:03:21.048
the ladder's not gonna stand up.

00:03:21.048 --> 00:03:24.600
Now it's totally fine to begin
a sentence with "because,"

00:03:24.600 --> 00:03:28.223
as long as it's attached
to an independent clause.

00:03:28.223 --> 00:03:29.717
So, you know, you could say,

00:03:29.717 --> 00:03:31.800
"Because I told them to,"

00:03:36.175 --> 00:03:37.526
comma,

00:03:37.526 --> 00:03:40.529
the goblins built me a sandcastle."

00:03:40.529 --> 00:03:43.501
Very nice, so the little goblins do that.

00:03:43.501 --> 00:03:46.168
So, I think because of that fear

00:03:47.124 --> 00:03:50.607
of just ending the sentence
as "Because I told them to,"

00:03:50.607 --> 00:03:52.720
of creating the sentence fragment,

00:03:52.720 --> 00:03:57.157
I think it's pretty easy
to simplify all that down

00:03:57.157 --> 00:03:59.523
into just saying, oh,
don't start sentences

00:03:59.523 --> 00:04:02.356
with conjunctions, just generally.

00:04:03.650 --> 00:04:07.000
So this is really less about
how you start an utterance,

00:04:07.000 --> 00:04:08.765
and more about how you end it.

00:04:08.765 --> 00:04:10.111
If you're gonna start a sentence

00:04:10.111 --> 00:04:12.457
with a conjunction of any kind,

00:04:12.457 --> 00:04:13.487
you have to make sure

00:04:13.487 --> 00:04:15.986
that you're actually producing a sentence.

00:04:15.986 --> 00:04:17.379
So think about your follow through,

00:04:17.379 --> 00:04:20.328
is ultimately the take away here.

00:04:20.328 --> 00:04:23.045
If you start a sentence
with a conjunction,

00:04:23.045 --> 00:04:25.503
make sure you're building towards

00:04:25.503 --> 00:04:28.177
some kind of independent clause.

00:04:28.177 --> 00:04:29.198
Now don't get me wrong,

00:04:29.198 --> 00:04:32.078
sentence fragments
definitely have their place,

00:04:32.078 --> 00:04:34.794
but, not in formal writing.

00:04:34.794 --> 00:04:37.186
You would use this maybe
for rhetorical effect,

00:04:37.186 --> 00:04:41.661
or, to approach a kind
of realism in dialogue,

00:04:41.661 --> 00:04:45.870
but not in essay writing,
not for the newspaper.

00:04:45.870 --> 00:04:48.355
I just wanna repeat,
there's not really a rule

00:04:48.355 --> 00:04:51.884
against beginning sentences
with a conjunction.

00:04:51.884 --> 00:04:54.206
It's a superstition, frankly.

00:04:54.206 --> 00:04:57.155
And in practice, the rule
is generally ignored.

00:04:57.155 --> 00:04:59.872
So I read this paper
from like 1994, or so,

00:04:59.872 --> 00:05:02.249
that analyzed the frequency

00:05:02.249 --> 00:05:04.400
of the word "but" in newspapers,

00:05:04.400 --> 00:05:06.838
and found that 60% of the time

00:05:06.838 --> 00:05:09.601
it was used at the
beginnings of sentences.

00:05:09.601 --> 00:05:13.223
Which is way more often
than certainly I expected!

00:05:13.223 --> 00:05:14.616
But it makes sense for that medium.

00:05:14.616 --> 00:05:16.846
It's punchy, it's dramatic.

00:05:16.846 --> 00:05:19.096
But, it gets a little stale

00:05:20.398 --> 00:05:22.953
if you use it too often, I think.

00:05:22.953 --> 00:05:25.274
But, again, you can absolutely begin

00:05:25.274 --> 00:05:27.039
sentences with conjunctions.

00:05:27.039 --> 00:05:29.849
It's just that if you start a
sentence with a conjunction,

00:05:29.849 --> 00:05:31.010
you have to follow through,

00:05:31.010 --> 00:05:33.239
and actually make it a sentence.

00:05:33.239 --> 00:05:35.723
If you begin a sentence with
a subordinating conjunction,

00:05:35.723 --> 00:05:37.302
you're writing a dependent clause

00:05:37.302 --> 00:05:38.417
that needs to be followed up

00:05:38.417 --> 00:05:40.321
by an independent clause, right?

00:05:40.321 --> 00:05:42.202
So if you're beginning with
a subordinating clause,

00:05:42.202 --> 00:05:43.711
you're making the ladder

00:05:43.711 --> 00:05:47.104
and you need to follow it
up with a tree to lean on.

00:05:47.104 --> 00:05:49.470
But if you begin a sentence
with a coordinating conjunction,

00:05:49.470 --> 00:05:50.933
like one of the FANBOYS conjunctions,

00:05:50.933 --> 00:05:52.372
you're in good shape.

00:05:52.372 --> 00:05:54.253
Just try not to overuse it,

00:05:54.253 --> 00:05:56.877
as you'd avoid overusing anything.

00:05:56.877 --> 00:05:58.827
Moderation in all things!

00:05:58.827 --> 00:06:00.081
So, take away?

00:06:00.081 --> 00:06:02.659
Write full sentences,

00:06:02.659 --> 00:06:04.888
and put your best "but" forward.

00:06:04.888 --> 00:06:06.420
You can learn anything.

00:06:06.420 --> 00:06:07.253
David out.


Coordinating conjunctions

WEBVTT
Kind: captions
Language: en

00:00:00.427 --> 00:00:01.331
- [Voiceover] Hello grammarians.

00:00:01.331 --> 00:00:04.048
Alright, today I want to start
talking about conjunctions,

00:00:04.048 --> 00:00:06.870
and conjunctions are
this part of speech that

00:00:06.870 --> 00:00:09.958
has a very particular function
in English and what it does

00:00:09.958 --> 00:00:13.801
conjunctions unite words,
phrases, and clauses.

00:00:13.801 --> 00:00:15.681
Let me, let me show you an example.

00:00:15.681 --> 00:00:17.121
So if you want to talk about two things,

00:00:17.121 --> 00:00:21.545
you could say it's a
peanut butter and jelly.

00:00:21.545 --> 00:00:24.087
Or if I were going to
paraphrase Eddie Izzard,

00:00:24.087 --> 00:00:27.920
I might offer you the
choice of cake or death.

00:00:28.905 --> 00:00:31.866
You could also describe
something as being sad but true.

00:00:31.866 --> 00:00:34.327
And these three are the most
commonly used conjunctions

00:00:34.327 --> 00:00:36.672
because I would be remiss
if I didn't mention the

00:00:36.672 --> 00:00:40.411
seminal conjunction song that got me into

00:00:40.411 --> 00:00:43.004
the grammar game in the first
place, Conjunction Junction,

00:00:43.004 --> 00:00:45.241
which I think was written by Jack Sheldon

00:00:45.241 --> 00:00:48.434
or performed by Jack
Sheldon in like, 1973.

00:00:48.434 --> 00:00:51.069
Schoolhouse Rock. It's great. Look it up.

00:00:51.069 --> 00:00:53.148
But what I'm going to talk about today

00:00:53.148 --> 00:00:57.640
is a mnemonic, or a memory
aide, called FANBOYS.

00:00:57.640 --> 00:01:00.624
You may have heard this before, FANBOYS.

00:01:00.624 --> 00:01:03.585
And this is how we remember
the coordinating conjunctions.

00:01:03.585 --> 00:01:05.001
And you don't need to worry about the name

00:01:05.001 --> 00:01:07.962
coordinating conjunctions,
we'll get to that later.

00:01:07.962 --> 00:01:10.545
For now, just remember FANBOYS.

00:01:12.320 --> 00:01:16.070
For, And, Nor, But, Or,
Yet, and So. FANBOYS.

00:01:22.589 --> 00:01:24.691
We'll go through how each
one of these are used

00:01:24.691 --> 00:01:28.274
over the next screen.
Follow me downstairs!

00:01:31.472 --> 00:01:35.555
So we've got For, And,
Nor, But, Or, Yet, and So,

00:01:36.638 --> 00:01:39.517
and each of them have a
different basic function.

00:01:39.517 --> 00:01:41.201
So let's, let's review them.

00:01:41.201 --> 00:01:43.813
For has this connotation.

00:01:43.813 --> 00:01:45.461
You can use it the way you'd use a word

00:01:45.461 --> 00:01:48.044
like since or because, as in...

00:01:49.688 --> 00:01:52.752
I do not eat buttons,
for they are not food.

00:01:52.752 --> 00:01:56.476
You can see I'm combining
these two little sentences

00:01:56.476 --> 00:01:58.059
using the word for.

00:01:58.894 --> 00:02:02.679
We're using this for to
explain my reasoning.

00:02:02.679 --> 00:02:05.245
I do not eat buttons
because they are not food,

00:02:05.245 --> 00:02:08.662
and for is a simpler way to express that.

00:02:10.233 --> 00:02:14.133
The conjunction and? Very
elemental, very important

00:02:14.133 --> 00:02:18.881
conjunction combines one
thing with another as in...

00:02:18.881 --> 00:02:22.323
The kangaroo robbed the
bank and torched the saloon.

00:02:22.323 --> 00:02:24.157
A rather criminal kangaroo.

00:02:24.157 --> 00:02:27.675
And you can see we're using
and to combine these two ideas.

00:02:27.675 --> 00:02:30.798
We're saying, this thing happened,
this thing also happened,

00:02:30.798 --> 00:02:32.463
they happened together.

00:02:32.463 --> 00:02:34.172
Nor is similar to and,

00:02:34.172 --> 00:02:36.634
but we use it to combine untrue things.

00:02:36.634 --> 00:02:38.711
We use it to express negation.

00:02:38.711 --> 00:02:41.254
So if I were going to talk
about an angry tree spirit,

00:02:41.254 --> 00:02:45.743
for example, just to pull
an example out of a hat,

00:02:45.743 --> 00:02:46.862
I could say,

00:02:46.862 --> 00:02:51.029
she won't leave her tree, nor
will she speak with humans.

00:02:52.214 --> 00:02:56.498
We use but to express exceptions, as in...

00:02:56.498 --> 00:03:01.002
We used every building
material but chewing gum.

00:03:01.002 --> 00:03:05.054
Chewing gum is the exception
to every building material.

00:03:05.054 --> 00:03:09.838
We or as a conjunction to
choose between options...

00:03:09.838 --> 00:03:12.043
As in, would you rather have a pet bear

00:03:12.043 --> 00:03:13.599
or a pet giraffe?

00:03:13.599 --> 00:03:15.980
We ask the hard questions on Khan Academy.

00:03:15.980 --> 00:03:18.573
Yet is kind of like but,

00:03:18.573 --> 00:03:23.143
except that we use it to
express unexpected things

00:03:23.143 --> 00:03:25.895
so I'm just going to
write, to the contrary!

00:03:25.895 --> 00:03:29.644
As if I were Sherlock Holmes
hollering at a confused Watson.

00:03:29.644 --> 00:03:31.862
And to the contrary, my good man Watson!

00:03:31.862 --> 00:03:34.323
This is how we use yet. I'll show you.

00:03:34.323 --> 00:03:37.296
I want to leave, yet I cannot.

00:03:37.296 --> 00:03:39.942
So it sets up this kind of push and pull

00:03:39.942 --> 00:03:42.265
dynamic in a sentence, is what yet does.

00:03:42.265 --> 00:03:44.796
We set the expectation, I want to leave,

00:03:44.796 --> 00:03:48.337
and yet here's the unexpected,
what I'd say an ironic part,

00:03:48.337 --> 00:03:50.543
yet I cannot. I want to leave...

00:03:50.543 --> 00:03:54.114
But unfortunately, or but unexpectedly.

00:03:54.114 --> 00:03:58.903
When you would say but
unexpectedly, just use yet.

00:03:58.903 --> 00:04:02.129
And finally, the last part of FANBOYS,

00:04:02.129 --> 00:04:03.962
so shows consequences.

00:04:05.300 --> 00:04:07.099
Dougal was allergic to sheep,

00:04:07.099 --> 00:04:10.489
so he skipped the wool festival.

00:04:10.489 --> 00:04:12.881
Right so, so we're trying to set up that

00:04:12.881 --> 00:04:15.888
as a consequence of the first clause,

00:04:15.888 --> 00:04:17.025
Dougal being allergic to sheep,

00:04:17.025 --> 00:04:19.784
the thing that follows
is, because of that,

00:04:19.784 --> 00:04:22.710
he decided to skip the wool festival.

00:04:22.710 --> 00:04:26.793
This is FANBOYS, For, And,
Nor, But, Or, Yet, So.

00:04:28.074 --> 00:04:29.933
These are all covered in the exercises.

00:04:29.933 --> 00:04:31.232
You can learn anything.

00:04:31.232 --> 00:04:32.065
David out.


Correlative conjunctions

WEBVTT
Kind: captions
Language: en

00:00:00.437 --> 00:00:01.757
- [Voiceover] Hey,
grammarians, today we're going

00:00:01.757 --> 00:00:04.313
to be talking about
correlative conjunctions and

00:00:04.313 --> 00:00:07.585
I know this this looks like
pretty ugly word, correlative,

00:00:07.585 --> 00:00:09.725
like, it's kind of
complicated looking but,

00:00:09.725 --> 00:00:11.117
let's break it down.

00:00:11.117 --> 00:00:12.881
This co part comes from the Latin,

00:00:12.881 --> 00:00:15.631
cum, meaning with or, together...

00:00:17.361 --> 00:00:20.383
And relative, well, we know
that the word relative is, like,

00:00:20.383 --> 00:00:23.075
your aunt or, your uncle or,
your cousin is a relative.

00:00:23.075 --> 00:00:24.583
So, we're just going to say, that

00:00:24.583 --> 00:00:27.250
just means relative or, related.

00:00:28.462 --> 00:00:30.343
So, correlative conjunctions, there

00:00:30.343 --> 00:00:32.875
we see it, definitionally
or, etymologically,

00:00:32.875 --> 00:00:35.219
are related together in some way.

00:00:35.219 --> 00:00:38.515
They're matched pairs;
they're a matching set.

00:00:38.515 --> 00:00:41.395
And this just means that,
when you see one, it's

00:00:41.395 --> 00:00:43.299
probably time to use the other.

00:00:43.299 --> 00:00:47.131
In this video, we'll go
through five of my favorites.

00:00:47.131 --> 00:00:50.427
So, either/or, is a good
pair to start off with.

00:00:50.427 --> 00:00:52.891
And when you start a sentence
with either or, either,

00:00:52.891 --> 00:00:54.541
I don't know how you say it.

00:00:54.541 --> 00:00:56.537
It sets up the expectation
that you're going to have to

00:00:56.537 --> 00:00:59.229
be choosing between two things,
so, we're going to say, or,

00:00:59.229 --> 00:01:02.201
later in the sentence,
as in, what is reputed to

00:01:02.201 --> 00:01:04.317
be Oscar Wilde's last words, either the

00:01:04.317 --> 00:01:06.317
wallpaper goes, or I do.

00:01:08.285 --> 00:01:12.452
The opposite of either/or is neither/nor.

00:01:13.533 --> 00:01:17.591
So, either/or sets up this
choice between two options and

00:01:17.591 --> 00:01:20.197
neither/nor rejects both options, so,

00:01:20.197 --> 00:01:23.114
neither Jia nor Becca liked Howard.

00:01:23.981 --> 00:01:25.677
And I recognize that I say neither and

00:01:25.677 --> 00:01:27.137
some of you say neither and there's a

00:01:27.137 --> 00:01:28.344
whole George Gershwin song about that but,

00:01:28.344 --> 00:01:29.775
let's just chalk it up to

00:01:29.775 --> 00:01:33.303
my Midwestern American accent, how 'bout?

00:01:33.303 --> 00:01:36.241
♫ Either, either, neither, neither

00:01:36.241 --> 00:01:39.889
♫ Let's learn about some conjunctions

00:01:39.889 --> 00:01:44.056
♫ But oh, if we talk about the
both and you use and, and oh,

00:01:47.065 --> 00:01:51.232
♫ If you learn to use conjunctions,
that is grand, oh yes ♫

00:01:53.217 --> 00:01:57.001
Both Bigby and LaDoux
were career criminals.

00:01:57.001 --> 00:01:58.977
Why we're using both to indicate a

00:01:58.977 --> 00:02:02.041
connection between two things
and then we use and later in

00:02:02.041 --> 00:02:05.041
the sentence to really firm that up.

00:02:06.801 --> 00:02:09.749
As/so, is kind of an interesting one.

00:02:09.749 --> 00:02:12.349
It's a little formal and it kind of

00:02:12.349 --> 00:02:14.717
allows you to set up this relationship of

00:02:14.717 --> 00:02:18.505
consequence, right, to say, if
one thing is happening then,

00:02:18.505 --> 00:02:22.700
another thing happens
as a consequence, so...

00:02:22.700 --> 00:02:26.174
As goes Kansas, so goes the nation.

00:02:26.174 --> 00:02:28.634
So, you know, as one thing happens,

00:02:28.634 --> 00:02:30.422
so must another thing happen, is

00:02:30.422 --> 00:02:33.302
what this is trying to set up.

00:02:33.302 --> 00:02:37.318
Whether and or, is
similar to either and or,

00:02:37.318 --> 00:02:41.485
except it has a, kind of,
whiff of possibility about it.

00:02:43.078 --> 00:02:45.910
Whether you like shrimp or
chocolate, there's certain to

00:02:45.910 --> 00:02:50.438
be something for you at
the Veracruz food fair.

00:02:50.438 --> 00:02:51.582
Right, because whether is, kind of,

00:02:51.582 --> 00:02:54.090
setting up this possibility
between these two options and

00:02:54.090 --> 00:02:56.758
it doesn't really matter
which one you choose.

00:02:56.758 --> 00:02:59.962
It's just offering up possibilities.

00:02:59.962 --> 00:03:01.358
And I would like to offer you the

00:03:01.358 --> 00:03:03.238
possibility of checking out more of these

00:03:03.238 --> 00:03:06.394
correlative conjunctions in our exercises.

00:03:06.394 --> 00:03:09.554
So, stick around; you can learn anything.

00:03:09.554 --> 00:03:10.387
David out.


Subordinating conjunctions

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:01.543
- [Voiceover] Hey, grammarians.

00:00:01.543 --> 00:00:04.686
Today let's start talking about
subordinating conjunctions.

00:00:04.686 --> 00:00:08.734
Words like, although
and after and because.

00:00:08.734 --> 00:00:11.028
It's a pretty complicated topic,

00:00:11.028 --> 00:00:13.676
because in order to understand
subordinating conjunctions,

00:00:13.676 --> 00:00:15.835
you have to understand the difference

00:00:15.835 --> 00:00:19.505
between an independent,
and a dependent clause.

00:00:19.505 --> 00:00:21.460
Because that's what
subordinating conjunctions do,

00:00:21.460 --> 00:00:25.958
is they unite independent
and dependent clauses.

00:00:25.958 --> 00:00:27.818
That's what subordinating conjunctions do.

00:00:27.818 --> 00:00:29.485
But, what are these?

00:00:30.600 --> 00:00:33.800
You know, what is an independent
and a dependent clause?

00:00:33.800 --> 00:00:36.777
Well okay so first of
all, let's back up again.

00:00:36.777 --> 00:00:37.610
What is a clause?

00:00:37.610 --> 00:00:39.268
A clause is just a language chunk

00:00:39.268 --> 00:00:42.601
that has a subject and a verb.

00:00:42.601 --> 00:00:44.942
That's what a clause does.

00:00:44.942 --> 00:00:46.939
All sentences are clauses,

00:00:46.939 --> 00:00:49.285
but not all clauses are sentences.

00:00:49.285 --> 00:00:50.340
I'll write that down.

00:00:50.340 --> 00:00:52.872
So, all sentences are clauses,

00:00:52.872 --> 00:00:55.913
but not all clauses are sentences.

00:00:55.913 --> 00:00:57.453
It's possible to have
something that has both

00:00:57.453 --> 00:01:00.575
a subject and a verb, that
doesn't stand on its own,

00:01:00.575 --> 00:01:01.867
and that's a dependent clause.

00:01:01.867 --> 00:01:03.421
We'd call that a sentence fragment.

00:01:03.421 --> 00:01:05.838
What a dependent clause does,

00:01:07.428 --> 00:01:11.495
is provide extra information
that isn't necessary

00:01:11.495 --> 00:01:14.535
to the understanding
of the sentence, right.

00:01:14.535 --> 00:01:16.451
But an independent clause has

00:01:16.451 --> 00:01:18.807
to be able to stand on its own.

00:01:18.807 --> 00:01:20.353
So, let's take a look
at the sentence here.

00:01:20.353 --> 00:01:23.044
Herbert performed his
irresponsible experiments

00:01:23.044 --> 00:01:25.296
because he was curious.

00:01:25.296 --> 00:01:28.379
And this sentence is
composed of two clauses.

00:01:28.379 --> 00:01:30.087
In red, we've got this independent clause,

00:01:30.087 --> 00:01:33.111
Herbert performed his
irresponsible experiments.

00:01:33.111 --> 00:01:36.143
And in green, we have
this dependent clause,

00:01:36.143 --> 00:01:37.975
because he was curious.

00:01:37.975 --> 00:01:39.915
Now, we know that because he was curious,

00:01:39.915 --> 00:01:41.422
is the dependent clause.

00:01:41.422 --> 00:01:45.589
Because, it explains more of
why Herbert did what he did.

00:01:46.787 --> 00:01:48.915
Right, it is explaining the reason

00:01:48.915 --> 00:01:52.694
for his performing
irresponsible experiments.

00:01:52.694 --> 00:01:56.250
But, because he was curious on its own,

00:01:56.250 --> 00:02:00.180
doesn't work as a sentence,
he was curious does.

00:02:00.180 --> 00:02:01.013
That's a sentence.

00:02:01.013 --> 00:02:02.830
But, because he was curious

00:02:02.830 --> 00:02:05.923
asks more questions than it answers.

00:02:05.923 --> 00:02:08.326
Because he was curious, what?

00:02:08.326 --> 00:02:11.848
You know, if a clause causes you to ask.

00:02:11.848 --> 00:02:13.388
Yeah, so what?

00:02:13.388 --> 00:02:16.757
Then it's probably a dependent
clause, write that down.

00:02:16.757 --> 00:02:18.926
Put that in the doctrine.

00:02:18.926 --> 00:02:20.582
So, a dependent clause should

00:02:20.582 --> 00:02:23.559
cause you to ask, yeah so what?

00:02:23.559 --> 00:02:25.260
Because a dependent clause leans

00:02:25.260 --> 00:02:27.049
against an independent clause.

00:02:27.049 --> 00:02:29.712
It's like a ladder laid
up against the tree.

00:02:29.712 --> 00:02:31.046
The tree is still gonna be standing there

00:02:31.046 --> 00:02:32.208
whether or not the ladder is there or not.

00:02:32.208 --> 00:02:35.212
But, the ladder is useful
because it allows you to get up

00:02:35.212 --> 00:02:38.238
further into the tree
and explore its leaves,

00:02:38.238 --> 00:02:39.071
if that makes sense.

00:02:39.071 --> 00:02:40.078
It allows you to get further context

00:02:40.078 --> 00:02:42.838
and further understanding.

00:02:42.838 --> 00:02:44.515
But the tree could just as easily,

00:02:44.515 --> 00:02:45.823
it's still a tree without the ladder.

00:02:45.823 --> 00:02:46.856
Right, you don't have to have

00:02:46.856 --> 00:02:49.439
a ladder in order to be a tree.

00:02:55.400 --> 00:02:57.784
All right so, another
thing about the difference

00:02:57.784 --> 00:03:01.894
between a coordinating and
a subordinating conjunction,

00:03:01.894 --> 00:03:03.561
is that coordinating conjunctions

00:03:03.561 --> 00:03:07.509
only unite independent
clauses with each other.

00:03:07.509 --> 00:03:09.793
Which means, any sentence that has

00:03:09.793 --> 00:03:11.015
a coordinating conjunction like,

00:03:11.015 --> 00:03:14.344
but or anything else in fanboys,.

00:03:14.344 --> 00:03:18.443
Which is for and nor,
but, or yet so, right.

00:03:18.443 --> 00:03:22.617
Can be separated into two
sentences, two separate sentences.

00:03:22.617 --> 00:03:25.838
Look at this, so Jimothy
liked to wash the dishes,

00:03:25.838 --> 00:03:28.254
but Beckany preferred to sweep the floor.

00:03:28.254 --> 00:03:30.429
Now these can each be their own sentences.

00:03:30.429 --> 00:03:31.840
Jimothy liked to wash the dishes.

00:03:31.840 --> 00:03:33.954
Beckany preferred to sweep the floor.

00:03:33.954 --> 00:03:37.982
We can combine these if
we want to and, we have.

00:03:37.982 --> 00:03:41.269
Whereas, this subordinating
conjunction sentence

00:03:41.269 --> 00:03:43.436
that begins with although,

00:03:44.978 --> 00:03:48.546
cannot be separated into two sentences.

00:03:48.546 --> 00:03:50.211
Although she loved Sir Reginald,

00:03:50.211 --> 00:03:52.101
Lady Penelope hated his pranks.

00:03:52.101 --> 00:03:55.020
So, Lady Penelope hated his
pranks, that's a sentence.

00:03:55.020 --> 00:03:57.784
Although she loved Sir
Reginald, that's not a sentence.

00:03:57.784 --> 00:04:02.079
This is a dependent clause
so, this can be split.

00:04:02.079 --> 00:04:03.871
(mouth sound effects)

00:04:03.871 --> 00:04:05.384
This cannot be split,

00:04:05.384 --> 00:04:06.932
because although she loved Sir Reginald,

00:04:06.932 --> 00:04:10.634
doesn't stand on its own, it
just grants additional context

00:04:10.634 --> 00:04:12.090
and detail to the fact that Lady Penelope

00:04:12.090 --> 00:04:14.423
hated Sir Reginald's pranks.

00:04:15.940 --> 00:04:16.897
So, with that in mind,

00:04:16.897 --> 00:04:18.260
here are some of the most

00:04:18.260 --> 00:04:21.728
common subordinating
conjunctions in English.

00:04:21.728 --> 00:04:24.834
And some of these you might
recognize as being adverbs

00:04:24.834 --> 00:04:27.868
or prepositions in other contexts,

00:04:27.868 --> 00:04:29.800
but you could also use them

00:04:29.800 --> 00:04:32.911
to unite dependent and
independent clauses.

00:04:32.911 --> 00:04:34.161
So, here we go.

00:04:35.700 --> 00:04:38.700
After, although, as, as if, because,

00:04:39.898 --> 00:04:42.315
before, even if, even though,

00:04:43.893 --> 00:04:46.110
if, if only, like,

00:04:46.110 --> 00:04:48.193
rather than, since, that,

00:04:49.580 --> 00:04:52.149
though, unless, until,

00:04:52.149 --> 00:04:53.899
when, where, whereas,

00:04:56.389 --> 00:04:59.472
wherever, whether, which, and, while.

00:05:02.155 --> 00:05:03.389
All of these words have the power

00:05:03.389 --> 00:05:06.972
to unite independent
and dependent clauses.

00:05:08.092 --> 00:05:09.374
You may notice some of them like,

00:05:09.374 --> 00:05:12.308
after and before, are prepositions.

00:05:12.308 --> 00:05:14.084
They can be used in many ways.

00:05:14.084 --> 00:05:16.399
English is complicated, but I am confident

00:05:16.399 --> 00:05:17.900
that you can figure this out

00:05:17.900 --> 00:05:20.525
because you can learn anything.

00:05:20.525 --> 00:05:23.889
These are some of the subordinating
conjunctions of English.

00:05:23.889 --> 00:05:24.722
David out.


Node voltage method (step 5)

WEBVTT
Kind: captions
Language: en

00:00:01.675 --> 00:00:03.742
- [Voiceover] And now we're
down to solving this circuit.

00:00:03.742 --> 00:00:06.737
What I want to do now is put
in the con pointed values

00:00:06.737 --> 00:00:09.404
and solve this specific circuit.

00:00:10.475 --> 00:00:13.142
Let me move the screen up again.

00:00:16.745 --> 00:00:20.530
We'll leave the list of steps
up there so we can see them.

00:00:20.530 --> 00:00:23.850
Let's go to work on this equation now.

00:00:23.850 --> 00:00:25.243
We have a little bit of algebra,

00:00:25.243 --> 00:00:28.215
and we can plug in
values where we need to.

00:00:28.215 --> 00:00:30.798
We can plug in 15 volts for V1.

00:00:32.767 --> 00:00:35.850
And for R1, we can plug in 4000 ohms.

00:00:38.038 --> 00:00:39.779
We can put in, for V2.

00:00:39.779 --> 00:00:41.362
V2's still unknown.

00:00:43.425 --> 00:00:45.508
And that's divided by 4K.

00:00:47.163 --> 00:00:51.330
In this expression here we
still have V2 unknown over 2K

00:00:56.172 --> 00:00:59.505
And let's put IS over on the other side.

00:01:03.297 --> 00:01:05.985
IS was three milliamps.

00:01:05.985 --> 00:01:08.377
Let's just keep working at this now.

00:01:08.377 --> 00:01:10.377
V2 times 1/4K

00:01:14.205 --> 00:01:15.872
plus 1/2K

00:01:19.198 --> 00:01:21.115
equals three milliamps.

00:01:25.769 --> 00:01:27.789
Oh, get my minus signs right.

00:01:27.789 --> 00:01:30.227
Minus sign over here.

00:01:30.227 --> 00:01:32.526
Let's bring the constant
term over to this side,

00:01:32.526 --> 00:01:35.359
so this is 15 volts divided by 4K.

00:01:38.842 --> 00:01:40.978
And continuing over here,

00:01:40.978 --> 00:01:43.137
we have minus V2.

00:01:43.137 --> 00:01:46.272
Let's combine those two resistor terms.

00:01:46.272 --> 00:01:49.355
So it's going to be 1+2 over 4K,

00:01:58.811 --> 00:02:00.728
equals three milliamps.

00:02:02.926 --> 00:02:05.759
And 15 volts divided by 4K is 3.75

00:02:10.810 --> 00:02:12.560
minus 3.75 milliamps.

00:02:17.052 --> 00:02:18.635
Moving on, minus V2

00:02:21.255 --> 00:02:23.172
equals -0.75 milliamps,

00:02:32.284 --> 00:02:33.951
times 4K over three.

00:02:40.556 --> 00:02:42.065
Let's get rid of the two minus signs.

00:02:42.065 --> 00:02:44.248
We don't need those anymore.

00:02:44.248 --> 00:02:46.165
And V2 equals one volt.

00:02:55.277 --> 00:02:56.763
That's good; we solved it.

00:02:56.763 --> 00:02:59.527
We solved our prior two voltages.

00:02:59.527 --> 00:03:03.110
We have one here and
we have one over here.

00:03:04.194 --> 00:03:07.111
And we can check off the last step.

00:03:08.582 --> 00:03:12.692
So that's our first application
of the node voltage method.

00:03:12.692 --> 00:03:14.573
I want to show you one more thing

00:03:14.573 --> 00:03:18.265
that is a powerful part of this technique.

00:03:18.265 --> 00:03:21.765
Let me quickly sketch the schematic again.

00:03:30.231 --> 00:03:32.460
So this was our schematic,

00:03:32.460 --> 00:03:35.270
and we assigned node voltages.

00:03:35.270 --> 00:03:36.965
We assigned node voltages here.

00:03:36.965 --> 00:03:40.965
V1, and V2, and we made
this our reference node.

00:03:43.885 --> 00:03:45.905
And one of the things we did not do

00:03:45.905 --> 00:03:47.646
as part of the node voltage method,

00:03:47.646 --> 00:03:52.546
we did not us KVL to write
equations around these loops.

00:03:52.546 --> 00:03:54.589
And one of the features
of the node voltage method

00:03:54.589 --> 00:03:56.679
is that the KVL equations,

00:03:56.679 --> 00:03:58.397
because we're using node voltages,

00:03:58.397 --> 00:04:02.739
the KVL equations are
automatically satisfied.

00:04:02.739 --> 00:04:03.970
And I'll show you why.

00:04:03.970 --> 00:04:06.292
I want to put one more label on here,

00:04:06.292 --> 00:04:08.709
which is the element voltage.

00:04:09.775 --> 00:04:11.191
We'll call this VR1.

00:04:11.191 --> 00:04:14.326
That's the element voltage across here.

00:04:14.326 --> 00:04:17.326
The element voltage here is just V2.

00:04:19.550 --> 00:04:21.733
So in this case for R2,

00:04:21.733 --> 00:04:23.660
V2 is the element voltage

00:04:23.660 --> 00:04:25.634
and the node voltage at the same time.

00:04:25.634 --> 00:04:27.801
VR1 is an element voltage.

00:04:29.047 --> 00:04:31.625
And now we're going to write KVL,

00:04:31.625 --> 00:04:32.948
starting from this point,

00:04:32.948 --> 00:04:36.615
and going around the
loop in this direction.

00:04:38.359 --> 00:04:42.887
And what we have is... let's
get all our labels on here.

00:04:42.887 --> 00:04:47.054
The loop voltages, we
start with a rise of +VS,

00:04:48.784 --> 00:04:50.617
then we take away VR1,

00:04:52.360 --> 00:04:54.443
and then we take away V2,

00:04:55.750 --> 00:04:58.258
and that equals zero.

00:04:58.258 --> 00:05:02.298
So that would be the KVL
equation for this circuit.

00:05:02.298 --> 00:05:04.806
Now I'm going to plug in,

00:05:04.806 --> 00:05:06.826
using node voltages,
I'm going to write VR1.

00:05:06.826 --> 00:05:08.826
So I get +VS minus,

00:05:11.424 --> 00:05:15.174
VR1 is node voltage V1
minus node voltage V2.

00:05:18.808 --> 00:05:19.808
V1 minus V2,

00:05:24.056 --> 00:05:25.806
minus V2 equals zero.

00:05:26.796 --> 00:05:29.234
And we'll just do one more substitution.

00:05:29.234 --> 00:05:31.532
But I forgot VS and V1
are the same voltage.

00:05:31.532 --> 00:05:34.199
So this is actually V1 minus V1,

00:05:36.594 --> 00:05:39.177
minus V2, minus V2 equals zero.

00:05:41.587 --> 00:05:44.025
And if we look at this equation,

00:05:44.025 --> 00:05:45.775
that goes, that goes,

00:05:46.811 --> 00:05:48.311
plus V2, minus V2.

00:05:49.737 --> 00:05:51.780
This equation is automatically true,

00:05:51.780 --> 00:05:54.697
if we write Kirchhoff's Current Law

00:05:55.913 --> 00:05:57.933
in terms of node voltages.

00:05:57.933 --> 00:06:00.279
That always turns out to be the case.

00:06:00.279 --> 00:06:02.717
That's why we don't bother to do it.

00:06:02.717 --> 00:06:04.133
We know it's going to be true.

00:06:04.133 --> 00:06:06.339
So that's a nice feature
of the node voltage method.

00:06:06.339 --> 00:06:08.893
It's a really efficient
way to write equations.

00:06:08.893 --> 00:06:11.099
You only write KCL equations.

00:06:11.099 --> 00:06:13.746
And this is such a good method in fact,

00:06:13.746 --> 00:06:15.511
that circuit simulators, like,

00:06:15.511 --> 00:06:19.644
you may come across a circuit
simulator called Spice.

00:06:19.644 --> 00:06:22.965
Almost every circuit simulator
uses this node voltage method

00:06:22.965 --> 00:06:24.635
to do its computations.


Divergence formula, part 1

WEBVTT
Kind: captions
Language: en

00:00:01.054 --> 00:00:01.887
- [Voiceover] Hello everyone.

00:00:01.887 --> 00:00:02.979
So now that we have an intuition

00:00:02.979 --> 00:00:05.044
for what divergence is
trying to represent,

00:00:05.044 --> 00:00:07.454
let's start actually
drilling in on a formula.

00:00:07.454 --> 00:00:08.713
The first thing I want to do

00:00:08.713 --> 00:00:11.665
is just limit our perspective to functions

00:00:11.665 --> 00:00:13.927
that only have an X component, or rather,

00:00:13.927 --> 00:00:18.010
where the Y component of
the output is just zero.

00:00:19.453 --> 00:00:21.004
So this is some kind of vector field

00:00:21.004 --> 00:00:22.341
and if there's only an X component

00:00:22.341 --> 00:00:23.838
what this means it's going to look like

00:00:23.838 --> 00:00:26.294
is all of the vectors
only go left or right,

00:00:26.294 --> 00:00:30.595
and there's kind of no up or
down involved in any of them.

00:00:30.595 --> 00:00:32.935
So in this case, let's
start thinking about what

00:00:32.935 --> 00:00:35.800
positive divergence of your vector fields

00:00:35.800 --> 00:00:38.800
might look like near some point X-Y.

00:00:40.032 --> 00:00:41.323
So if you have your point,

00:00:41.323 --> 00:00:44.847
this is that point X-Y sitting
off somewhere in space,

00:00:44.847 --> 00:00:48.159
two cases where the divergence
of this might look positive

00:00:48.159 --> 00:00:51.052
are, one, where nothing
happens at the point.

00:00:51.052 --> 00:00:55.219
So in this case P would be
equal to zero at our point.

00:00:56.182 --> 00:00:59.095
But then to left of it things
are moving to the left,

00:00:59.095 --> 00:01:02.773
meaning P, the X component of
our vector valued function,

00:01:02.773 --> 00:01:03.773
is negative.

00:01:04.934 --> 00:01:07.863
That's why the X component
of this vector is negative.

00:01:07.863 --> 00:01:10.148
But then to the right,

00:01:10.148 --> 00:01:11.982
vectors would be moving off to the right.

00:01:11.982 --> 00:01:14.679
So over here, P would be positive.

00:01:14.679 --> 00:01:16.682
So this would be an example of a

00:01:16.682 --> 00:01:18.432
positive divergence circumstance

00:01:18.432 --> 00:01:20.922
where only the X component is responsible.

00:01:20.922 --> 00:01:21.755
And what you'll notice here,

00:01:21.755 --> 00:01:23.290
this would be P starts negative,

00:01:23.290 --> 00:01:25.515
goes zero, then becomes positive.

00:01:25.515 --> 00:01:27.669
So as you're changing in the X direction,

00:01:27.669 --> 00:01:29.591
P should be increasing.

00:01:29.591 --> 00:01:33.655
So a positive divergence
here seems to correspond to

00:01:33.655 --> 00:01:37.988
a positive partial derivative
of P with respect to X.

00:01:37.988 --> 00:01:39.363
And if that seems a little unfamiliar,

00:01:39.363 --> 00:01:41.746
if you're not sure how to
think about partial derivatives

00:01:41.746 --> 00:01:44.563
of a component of a vector field,

00:01:44.563 --> 00:01:46.904
I have a video on that
and you can take a look

00:01:46.904 --> 00:01:48.735
and refresh yourself, how
you might think about this

00:01:48.735 --> 00:01:51.011
partial derivative of P with respect to X.

00:01:51.011 --> 00:01:52.436
And once you do, hopefully it makes sense

00:01:52.436 --> 00:01:55.078
why this specific positive
divergence example

00:01:55.078 --> 00:01:58.515
corresponds with the positive
partial derivative of P.

00:01:58.515 --> 00:01:59.646
But remember, this isn't the only way

00:01:59.646 --> 00:02:01.184
that a positive divergence might look.

00:02:01.184 --> 00:02:05.351
You're gonna have another
circumstance where, let's say,

00:02:07.851 --> 00:02:09.731
your point, X-Y,

00:02:09.731 --> 00:02:11.208
actually has a vector attached to it.

00:02:11.208 --> 00:02:14.875
So this here again
represents our point X-Y.

00:02:15.800 --> 00:02:19.794
And in this specific example
this would be P is positive.

00:02:19.794 --> 00:02:22.223
P of X-Y is positive at your point there.

00:02:22.223 --> 00:02:24.452
But another way that positive
divergence might look

00:02:24.452 --> 00:02:27.433
is that you have things
coming in towards that point

00:02:27.433 --> 00:02:30.034
and things going away,
but the things going away

00:02:30.034 --> 00:02:31.729
are bigger than the ones coming in.

00:02:31.729 --> 00:02:35.189
But again, this exhibits the
idea of P increasing in value.

00:02:35.189 --> 00:02:38.434
P starts off small, it's a
positive but small component,

00:02:38.434 --> 00:02:40.822
and then it gets bigger and
then it gets even bigger.

00:02:40.822 --> 00:02:42.944
So once again we have this idea of

00:02:42.944 --> 00:02:45.661
positive partial derivative
of P with respect to X,

00:02:45.661 --> 00:02:47.054
because changes in X,

00:02:47.054 --> 00:02:49.443
as you increase X it
causes an increase in P,

00:02:49.443 --> 00:02:52.066
seems to correspond to
positive divergence.

00:02:52.066 --> 00:02:54.226
And you can even look at
it if you go the other way

00:02:54.226 --> 00:02:58.336
where you have a little bit of
negative component to P here.

00:02:58.336 --> 00:03:00.495
So P is a little bit negative,

00:03:00.495 --> 00:03:03.375
but to the left of your
point it's really negative,

00:03:03.375 --> 00:03:06.207
and then to the right it's
not nearly as negative.

00:03:06.207 --> 00:03:07.670
And in this case it's kind of like

00:03:07.670 --> 00:03:10.364
as you're moving to the
right, as X is increasing,

00:03:10.364 --> 00:03:12.105
you start off very negative,

00:03:12.105 --> 00:03:13.490
and then only kind of negative,

00:03:13.490 --> 00:03:15.054
and then barely negative.

00:03:15.054 --> 00:03:17.341
And once again that
corresponds to an increase

00:03:17.341 --> 00:03:20.491
in the value of P as X increases.

00:03:20.491 --> 00:03:23.501
So what you'd expect is that
a partial derivative of P,

00:03:23.501 --> 00:03:26.684
that X component of the
output, with respect to X,

00:03:26.684 --> 00:03:28.785
is gonna be somewhere
involved in the formula

00:03:28.785 --> 00:03:33.000
for the divergence of our
vector field at a point X-Y.

00:03:33.000 --> 00:03:35.420
In the next video I'm gonna
go a similar line of reasoning

00:03:35.420 --> 00:03:37.846
to see what should go on
with that Y component.


Divergence intuition, part 1

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.524
- [Voiceover] Alright everyone.

00:00:01.524 --> 00:00:03.259
We've gotten to one of
my all-time favorite

00:00:03.259 --> 00:00:06.267
multi-variable calculus
topics, divergence.

00:00:06.267 --> 00:00:08.376
In the next few videos I'm gonna describe

00:00:08.376 --> 00:00:10.502
what it is mathematically
and how you compute it

00:00:10.502 --> 00:00:12.504
and all of that but
here I just want to give

00:00:12.504 --> 00:00:15.000
a very visual understanding of what it is

00:00:15.000 --> 00:00:17.380
that it's trying to represent.

00:00:17.380 --> 00:00:20.782
So I've got a picture in
front of us, a vector field.

00:00:20.782 --> 00:00:22.516
I've said before that a pretty neat way

00:00:22.516 --> 00:00:24.499
to understand vector
fields is to think of them

00:00:24.499 --> 00:00:26.406
as representing a fluid flow.

00:00:26.406 --> 00:00:28.398
What I mean by that is you can think of

00:00:28.398 --> 00:00:31.174
every single point in space as a particle,

00:00:31.174 --> 00:00:33.559
maybe an air particle or a water particle.

00:00:34.499 --> 00:00:35.947
Something to that effect.

00:00:35.947 --> 00:00:37.751
Since what a vector field does

00:00:37.751 --> 00:00:39.775
is it associates each point in space

00:00:39.775 --> 00:00:41.646
with some kind of vector

00:00:41.646 --> 00:00:43.932
and remember I mean, whenever
we represent vector fields,

00:00:43.932 --> 00:00:46.381
we only show a small subset
of all of those vectors,

00:00:46.381 --> 00:00:48.763
but in principle you should be thinking of

00:00:48.763 --> 00:00:51.290
every one of those infinitely
many points in space

00:00:51.290 --> 00:00:53.671
being associated with
one of these vectors.

00:00:53.671 --> 00:00:56.003
The fact that they're
kind of smoothly changing

00:00:56.003 --> 00:00:57.607
as you traverse across space

00:00:57.607 --> 00:01:00.793
,means that showing this
very small finite subsample

00:01:00.793 --> 00:01:02.675
of those infinitely many vectors

00:01:02.675 --> 00:01:04.998
still gives a pretty good
feel for what's going on.

00:01:04.998 --> 00:01:07.002
So if we have these fluid particles

00:01:07.002 --> 00:01:09.874
and you have a vector
assigned to each one,

00:01:09.874 --> 00:01:12.257
kind of a natural thought
you might have is to say

00:01:12.257 --> 00:01:15.096
what would happen if you let
things progress over time

00:01:15.096 --> 00:01:18.702
where at any given instant
the velocity of one of

00:01:18.702 --> 00:01:21.822
these particles is given by
that vector connected to it.

00:01:21.822 --> 00:01:24.040
As it moves it will be
touching a different vector.

00:01:24.040 --> 00:01:25.436
So it's velocity might turn.

00:01:25.436 --> 00:01:26.983
It might go in a different direction.

00:01:26.983 --> 00:01:29.243
For each one it will kind
of traverse some path

00:01:29.243 --> 00:01:32.243
as determined by the vectors
that it's touching as it goes

00:01:32.243 --> 00:01:34.537
and when you think of all
of them doing this at once,

00:01:34.537 --> 00:01:37.503
it will feel like a certain fluid flow.

00:01:37.503 --> 00:01:39.409
For this you don't
actually have to imagine.

00:01:39.409 --> 00:01:42.227
I went ahead and put together
an animation for you.

00:01:42.227 --> 00:01:44.762
So we'll put some water molecules or dots

00:01:44.762 --> 00:01:47.130
to represent a small sample
of the water molecules

00:01:47.130 --> 00:01:49.765
throughout space here and then
I'm just gonna let it play

00:01:49.765 --> 00:01:54.248
where each one moves along the
vector that it's closest to.

00:01:54.248 --> 00:01:56.496
I'll just let it play forward here

00:01:56.496 --> 00:01:58.923
where each one is flowing along the vector

00:01:58.923 --> 00:02:02.012
that's touching the point
where it is in that moment.

00:02:04.742 --> 00:02:07.289
So for example, if we were to go back

00:02:07.289 --> 00:02:09.668
and maybe focus our
attention on just one vector

00:02:09.668 --> 00:02:12.252
like this guy, one particle, excuse me,

00:02:12.252 --> 00:02:14.291
he's attached to this
vector so he'll be moving

00:02:14.291 --> 00:02:16.006
in that direction but just for an instant

00:02:16.006 --> 00:02:17.252
because after he moves a little

00:02:17.252 --> 00:02:18.873
he'll be attached to a different vector.

00:02:18.873 --> 00:02:22.991
So if you kind of let it play
and follow that particular dot

00:02:22.991 --> 00:02:26.533
after a little bit you'll
find him elsewhere.

00:02:26.533 --> 00:02:28.419
I think this is the one, right.

00:02:28.419 --> 00:02:30.248
Now he's gonna be moving along this vector

00:02:30.248 --> 00:02:32.769
or whatever one is really attached to him.

00:02:32.769 --> 00:02:35.510
Thinking about all of the
particles all at once doing this

00:02:35.510 --> 00:02:38.761
gives a sort of global
view of the vector field.

00:02:38.761 --> 00:02:40.841
If you're studying math,
you might start to ask

00:02:40.841 --> 00:02:43.991
some natural questions about
the nature of that fluid flow.

00:02:43.991 --> 00:02:46.559
For example, you might
wonder if you were to just

00:02:46.559 --> 00:02:49.044
look in a certain region
and count the number

00:02:49.044 --> 00:02:51.323
of water molecules that
are inside that region,

00:02:51.323 --> 00:02:53.534
does that count of yours change

00:02:53.534 --> 00:02:56.254
as you play this animation,

00:02:56.254 --> 00:02:58.282
as you let this flow over time?

00:02:58.282 --> 00:02:59.989
In this particular example you can look

00:02:59.989 --> 00:03:02.228
and it doesn't look
like the count changes.

00:03:02.228 --> 00:03:03.273
Certainly not that much.

00:03:03.273 --> 00:03:06.248
It's not increasing over
time or decreasing over time

00:03:06.248 --> 00:03:08.183
In a little bit, if I
gave you the function

00:03:08.183 --> 00:03:09.751
that determines this vector field,

00:03:09.751 --> 00:03:12.034
you will be able to tell
me why it's the case

00:03:12.034 --> 00:03:14.383
that the number of
molecules in that region

00:03:14.383 --> 00:03:15.993
doesn't tend to change

00:03:15.993 --> 00:03:17.892
but if you were to look
at another example,

00:03:17.892 --> 00:03:20.983
like a guy that looks like this

00:03:20.983 --> 00:03:22.510
and if I were to say I want you to focus

00:03:22.510 --> 00:03:24.474
on what happens around the origin,

00:03:24.474 --> 00:03:26.219
in that little region around the origin,

00:03:26.219 --> 00:03:29.269
you can probably predict
how once I start playing it,

00:03:29.269 --> 00:03:31.245
once I put some water molecules in there

00:03:31.245 --> 00:03:34.264
and let them flow along the
vectors that they flow along,

00:03:34.264 --> 00:03:38.510
the density inside that region
around the origin decreases.

00:03:38.510 --> 00:03:40.035
So we put a whole bunch of vectors there

00:03:40.035 --> 00:03:41.257
and I'll just play it for a quick instant.

00:03:41.257 --> 00:03:43.742
Just kind of let it jump for an instant.

00:03:43.742 --> 00:03:46.264
One thing that characterizes
this field around the origin

00:03:46.264 --> 00:03:48.520
is that decrease in density.

00:03:48.520 --> 00:03:51.264
What you might say if you
wanted to be suggestive

00:03:51.264 --> 00:03:53.410
of the operation that I'm leading to here

00:03:53.410 --> 00:03:55.505
is that the water
molecules tend to diverge

00:03:55.505 --> 00:03:57.001
away from the origin.

00:03:57.001 --> 00:03:59.325
So the kind of divergence
of the vector field

00:03:59.325 --> 00:04:01.338
near that origin is positive.

00:04:01.338 --> 00:04:03.077
You'll see what I mean
mathematically by that

00:04:03.077 --> 00:04:04.745
in the next couple videos,

00:04:04.745 --> 00:04:06.954
but if we were to flip
over these vectors, right,

00:04:06.954 --> 00:04:08.282
if we were to flip them around,

00:04:08.282 --> 00:04:10.250
now if I were to ask about the density

00:04:10.250 --> 00:04:12.188
in that same region around the origin,

00:04:12.188 --> 00:04:14.159
we can probably see how
it's gonna increase.

00:04:14.159 --> 00:04:18.247
When I play that fluid flow
over just a short spurt of time,

00:04:18.247 --> 00:04:20.763
the density in the region increases.

00:04:20.763 --> 00:04:23.243
So these don't diverge away, they converge

00:04:23.243 --> 00:04:26.012
towards the origin.

00:04:26.012 --> 00:04:28.769
That fact actually has some
mathematical significance

00:04:28.769 --> 00:04:31.503
for the function representing
this vector field

00:04:31.503 --> 00:04:33.036
around that point.

00:04:33.036 --> 00:04:35.220
Even if the vector field
doesn't represent fluid flow,

00:04:35.220 --> 00:04:36.846
if it represents a magnetic field

00:04:36.846 --> 00:04:40.886
or an electric field or things like that,

00:04:40.886 --> 00:04:43.352
there's a certain meaning to this idea

00:04:43.352 --> 00:04:46.262
of diverging away from a point
or converging to a point.

00:04:46.262 --> 00:04:49.267
Another way that people
sometimes think about this,

00:04:49.267 --> 00:04:50.739
if you look at that same kind of

00:04:50.739 --> 00:04:53.258
outward-flowing vector field

00:04:53.258 --> 00:04:55.512
as rather than thinking
of a decrease in density,

00:04:55.512 --> 00:04:57.763
imagining that the fluid
would have to constantly

00:04:57.763 --> 00:04:59.834
be repopulated around that point.

00:04:59.834 --> 00:05:02.285
So you're really thinking of
the origin as a source of fluid

00:05:02.285 --> 00:05:03.920
and if I had animated this better,

00:05:03.920 --> 00:05:06.004
a whole bunch of other points
should be sources of fluid

00:05:06.004 --> 00:05:08.085
so that the density doesn't
decrease everywhere,

00:05:08.085 --> 00:05:11.136
but the idea is that points
of positive divergence

00:05:11.136 --> 00:05:12.994
where things are diverging away

00:05:12.994 --> 00:05:15.068
would have to have a source of that fluid

00:05:15.068 --> 00:05:17.154
in order to kind of
keep things sustaining.

00:05:17.154 --> 00:05:19.928
Conversely, if you were to look
at that kind of inward flow

00:05:19.928 --> 00:05:23.249
or what you might call
negative divergence example

00:05:23.249 --> 00:05:25.763
and you were to play it but
it were to go continuously,

00:05:25.763 --> 00:05:27.740
you'd have to think of
that center point as a sink

00:05:27.740 --> 00:05:29.930
where all the fluid kind
of just sort of flows away.

00:05:29.930 --> 00:05:31.459
That's actually a technical term.

00:05:31.459 --> 00:05:33.305
People will say the
vector field has a sink

00:05:33.305 --> 00:05:36.366
at such and such point or
the electromagnetic field

00:05:36.366 --> 00:05:37.756
has a sink at such and such point

00:05:37.756 --> 00:05:40.451
and that often has a certain significance.

00:05:40.451 --> 00:05:43.767
If we go back to that
original example here

00:05:43.767 --> 00:05:46.517
where there is no change
in fluid density, right,

00:05:46.517 --> 00:05:48.756
what you might notice,
this feels a lot more like

00:05:48.756 --> 00:05:50.497
actual water than the other ones

00:05:50.497 --> 00:05:53.128
because there is no
change in density there.

00:05:53.128 --> 00:05:55.267
If you can find a way to
mathematically describe

00:05:55.267 --> 00:05:58.500
that lack of a change in density,

00:05:58.500 --> 00:06:01.159
it's a pretty good way
to model water flow.

00:06:01.159 --> 00:06:03.005
Again, even if it's not water flow

00:06:03.005 --> 00:06:04.874
but it's something like
the electromagnetic field,

00:06:04.874 --> 00:06:06.475
there's often a significance to this

00:06:06.475 --> 00:06:09.005
no changing in density idea.

00:06:09.005 --> 00:06:10.952
So with that I think
I've jabbered on enough

00:06:10.952 --> 00:06:12.928
about the visuals of it
and in the next video

00:06:12.928 --> 00:06:15.552
I'll tell you what
divergence is mathematically,

00:06:15.552 --> 00:06:17.899
how you compute it, go
through a couple examples,

00:06:17.899 --> 00:06:18.848
things like that.

00:06:18.848 --> 00:06:19.760
See you next video.