History and prehistory

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:02.379
- [Instructor] Anatomically
modern human beings

00:00:02.379 --> 00:00:07.322
have been on this planet
for roughly 200,000 years.

00:00:07.322 --> 00:00:09.202
And even though that's a small fraction

00:00:09.202 --> 00:00:11.344
of the amount of time the
Earth has been around,

00:00:11.344 --> 00:00:13.545
which is over four billion years,

00:00:13.545 --> 00:00:16.843
on a human scale it's an
incredibly long amount of time.

00:00:16.843 --> 00:00:20.383
Just to put it in perspective,
if this is present time,

00:00:20.383 --> 00:00:23.183
if you wanted to put,
when was the Roman Empire?

00:00:23.183 --> 00:00:25.542
Well if we're talking
about 2000 years ago,

00:00:25.542 --> 00:00:30.045
it would show up on our
timeline right about there.

00:00:30.045 --> 00:00:31.090
If you wanted to talk about

00:00:31.090 --> 00:00:33.098
when the pyramids were constructed,

00:00:33.098 --> 00:00:35.133
it would be right about there.

00:00:35.133 --> 00:00:37.092
You can hardly see the time difference

00:00:37.092 --> 00:00:40.973
between now and several
thousand years ago.

00:00:40.973 --> 00:00:43.280
If you want to see how
long we've had writing,

00:00:43.280 --> 00:00:46.591
about 5000 years is our
current best estimate.

00:00:46.591 --> 00:00:49.146
Once again, it barely
shows up on this timeline.

00:00:49.146 --> 00:00:50.922
How long have we had agriculture?

00:00:50.922 --> 00:00:52.442
Well, 10 to 15,000 years.

00:00:52.442 --> 00:00:55.191
Once again, it's a small fraction of this.

00:00:55.191 --> 00:00:56.650
Another way to think about it,

00:00:56.650 --> 00:00:58.974
think about all of our ancestors,

00:00:58.974 --> 00:01:01.315
the various generations that have passed

00:01:01.315 --> 00:01:03.482
since the first appearance

00:01:05.990 --> 00:01:09.284
of anatomically modern human beings.

00:01:09.284 --> 00:01:13.451
This is over six or 7,000
generations into the past.

00:01:14.998 --> 00:01:18.483
Think about all of the stories
that must have happened.

00:01:18.483 --> 00:01:19.885
A lot of simple things.

00:01:19.885 --> 00:01:22.885
It might be a founding of a village,

00:01:23.856 --> 00:01:28.460
a killing of an animal,
a very simple courtship,

00:01:28.460 --> 00:01:31.215
a tenderness between a
mother and their child.

00:01:31.215 --> 00:01:32.705
And think about the big things,

00:01:32.705 --> 00:01:36.115
the wars, the battles,
the natural disasters.

00:01:36.115 --> 00:01:38.301
It's hard for us to imagine
how much has occurred

00:01:38.301 --> 00:01:41.387
even in the last 100 or
200 or thousand years,

00:01:41.387 --> 00:01:43.387
much less 200,000 years.

00:01:44.864 --> 00:01:48.410
But we seek to understand regardless.

00:01:48.410 --> 00:01:52.113
And that is what history is all about.

00:01:52.113 --> 00:01:54.192
And as we'll see, history is in general

00:01:54.192 --> 00:01:56.990
trying to understand
the stories of our past.

00:01:56.990 --> 00:01:58.850
And if we want to get a
little bit more technical,

00:01:58.850 --> 00:02:03.177
we can also think about
pre-history, which is, technically,

00:02:03.177 --> 00:02:06.678
the things that happened
before we had writing.

00:02:06.678 --> 00:02:10.428
Because writing is our
main tool for history.

00:02:11.777 --> 00:02:13.773
What I have right over here, this picture,

00:02:13.773 --> 00:02:15.999
this is Egyptian hieroglyphs.

00:02:15.999 --> 00:02:20.236
And Egyptian hieroglyphs
are over 5,000 years old.

00:02:20.236 --> 00:02:24.549
I could write 5,000
years before the present.

00:02:24.549 --> 00:02:26.611
But even when you have
writing, it's not enough

00:02:26.611 --> 00:02:28.259
as we'll see not only in this video,

00:02:28.259 --> 00:02:31.600
but in many videos as we study
history and world history.

00:02:31.600 --> 00:02:34.937
It's really a lot of science
and a lot of detective work

00:02:34.937 --> 00:02:36.898
to make sense of what has happened.

00:02:36.898 --> 00:02:39.647
And that understanding
will constantly evolve.

00:02:39.647 --> 00:02:42.700
For example, we didn't know
what these hieroglyphs said

00:02:42.700 --> 00:02:46.283
until 1799 when we
found the Rosetta Stone.

00:02:48.700 --> 00:02:51.002
And what was useful
about the Rosetta Stone

00:02:51.002 --> 00:02:55.584
is that they had some text
written in the hieroglyphs

00:02:55.584 --> 00:02:58.237
and they had the same text
then written in a Greek

00:02:58.237 --> 00:02:59.978
that we were able to understand

00:02:59.978 --> 00:03:03.900
and that started to help unlock
what these hieroglyphs said.

00:03:03.900 --> 00:03:06.755
But even once you have
a sense of what they say

00:03:06.755 --> 00:03:08.528
or even if you understand it quite well,

00:03:08.528 --> 00:03:10.683
you still have to do a
lot of detective work

00:03:10.683 --> 00:03:13.482
and take everything with a grain of salt.

00:03:13.482 --> 00:03:14.732
You can imagine

00:03:16.815 --> 00:03:19.766
if there's a bunch of
groups of people here

00:03:19.766 --> 00:03:23.661
and we get the history from this group,

00:03:23.661 --> 00:03:26.578
for some reason we were able
to find what they wrote,

00:03:26.578 --> 00:03:28.848
well it might not be completely unbiased.

00:03:28.848 --> 00:03:32.672
They might have a negative view
of this group or that group.

00:03:32.672 --> 00:03:34.553
And so you have to take it
all with a grain of salt.

00:03:34.553 --> 00:03:37.393
At the extreme form, they
might have eliminated

00:03:37.393 --> 00:03:40.509
some of these groups and
then only they were around

00:03:40.509 --> 00:03:42.982
to say what actually happened.

00:03:42.982 --> 00:03:45.195
You also have to be skeptical
because you don't know

00:03:45.195 --> 00:03:47.372
whether these stories are actual accounts

00:03:47.372 --> 00:03:50.813
or whether someone just made
it up to fit a worldview.

00:03:50.813 --> 00:03:52.623
You also have to keep in
mind that these stories,

00:03:52.623 --> 00:03:55.867
whether they were transmitted
written or orally,

00:03:55.867 --> 00:03:57.852
they're oftentimes retransmitted

00:03:57.852 --> 00:04:01.489
from generation to generation,
especially in the oral case.

00:04:01.489 --> 00:04:03.503
But even in the written,
you've gotta wonder

00:04:03.503 --> 00:04:06.330
what's added over each generation.

00:04:06.330 --> 00:04:07.759
You can imagine people embellishing,

00:04:07.759 --> 00:04:09.384
making the story a little bit better.

00:04:09.384 --> 00:04:11.798
Or taking out things of the story

00:04:11.798 --> 00:04:16.138
that really doesn't fit
in with their worldview.

00:04:16.138 --> 00:04:18.801
So even when we have the writing,

00:04:18.801 --> 00:04:21.878
and this is once again, some
images of early writing.

00:04:21.878 --> 00:04:26.625
This is the famous Sumerian
cuneiform tablet here.

00:04:26.625 --> 00:04:27.727
Even when we have it, obviously,

00:04:27.727 --> 00:04:29.343
we have much much more writing

00:04:29.343 --> 00:04:31.409
as we get to the more recent past,

00:04:31.409 --> 00:04:33.665
we have to be very very very skeptical.

00:04:33.665 --> 00:04:34.641
We know, today,

00:04:34.641 --> 00:04:37.974
even if two observers
observe something yesterday,

00:04:37.974 --> 00:04:40.088
something that just
happened, they might have

00:04:40.088 --> 00:04:42.439
very very different
perceptions of what happened.

00:04:42.439 --> 00:04:45.158
So even though there's writing,
we have to be skeptical.

00:04:45.158 --> 00:04:48.098
But things get, in some
ways, even more interesting

00:04:48.098 --> 00:04:52.612
before we have writing,
when we go into pre-history.

00:04:52.612 --> 00:04:55.710
You might wonder how do we know
anything about what happened

00:04:55.710 --> 00:04:58.207
if there's no written accounts?

00:04:58.207 --> 00:04:59.649
Well that's where the science

00:04:59.649 --> 00:05:01.560
and we get even more detective work.

00:05:01.560 --> 00:05:04.630
For example, this is a Neanderthal skull.

00:05:04.630 --> 00:05:08.992
And the type of people, the
sciences that will study this,

00:05:08.992 --> 00:05:12.242
you'll hear terms like anthropologists.

00:05:13.374 --> 00:05:16.630
Anthropologists or anthropology.

00:05:16.630 --> 00:05:19.811
This is the study of
present and past humans

00:05:19.811 --> 00:05:21.182
and human society.

00:05:21.182 --> 00:05:23.297
And then a subset of anthropology,

00:05:23.297 --> 00:05:26.808
which is really delving into pre-history

00:05:26.808 --> 00:05:30.141
and even history itself, is archaeology.

00:05:30.983 --> 00:05:34.900
Archaeology which is the
subset of anthropology

00:05:36.171 --> 00:05:40.251
that focuses on the past study
of humans and human society

00:05:40.251 --> 00:05:42.835
and they're mainly going
to do it through remains.

00:05:42.835 --> 00:05:45.781
Now there's other fields,
you might associate,

00:05:45.781 --> 00:05:48.114
the field of paleontologist.

00:05:50.982 --> 00:05:52.232
Paleontologist.

00:05:59.357 --> 00:06:00.440
Paleontology.

00:06:01.548 --> 00:06:05.751
You might associate this with
things like dinosaur bones,

00:06:05.751 --> 00:06:09.918
but their techniques are also
useful for old human remains

00:06:10.952 --> 00:06:12.946
or even pre-human remains.

00:06:12.946 --> 00:06:16.670
And so it might inform
archaeology and anthropology.

00:06:16.670 --> 00:06:18.364
So for example, an archaeologist

00:06:18.364 --> 00:06:20.626
might unearth this Neanderthal skull.

00:06:20.626 --> 00:06:23.051
They will use some science
in order to figure out

00:06:23.051 --> 00:06:26.590
when did this skull enter into the ground?

00:06:26.590 --> 00:06:30.486
They might use a technique
like radiocarbon dating.

00:06:30.486 --> 00:06:32.069
Radiocarbon dating,

00:06:34.754 --> 00:06:36.214
which could be used for things

00:06:36.214 --> 00:06:39.376
up to around 50,000 years old,

00:06:39.376 --> 00:06:43.197
so around that time span on our timeline.

00:06:43.197 --> 00:06:44.295
And it's based on this idea

00:06:44.295 --> 00:06:48.098
that you have this atmospheric carbon 14

00:06:48.098 --> 00:06:51.824
that actually comes from nitrogen 14

00:06:51.824 --> 00:06:55.662
that gets between interactions
with the solar wind

00:06:55.662 --> 00:06:56.942
and these cosmic particles

00:06:56.942 --> 00:06:59.610
becomes this radioactive carbon 14.

00:06:59.610 --> 00:07:02.532
And the carbon 14 which can
become part of carbon dioxide

00:07:02.532 --> 00:07:04.609
incorporate into plants
through photosynthesis

00:07:04.609 --> 00:07:06.342
which then get eaten into animals.

00:07:06.342 --> 00:07:07.983
So while something is living,

00:07:07.983 --> 00:07:12.089
they'll have a certain amount
of carbon 14 in their tissue.

00:07:12.089 --> 00:07:13.576
But then once they die,

00:07:13.576 --> 00:07:16.323
they're now no longer
adding more carbon 14.

00:07:16.323 --> 00:07:21.080
And the carbon 14 decays into
the more stable carbon 12.

00:07:21.080 --> 00:07:22.598
And so based on the ratio

00:07:22.598 --> 00:07:25.470
between the carbon 14 to the carbon 12,

00:07:25.470 --> 00:07:29.637
and it takes roughly 5,000
years, 5,730 years to be exact,

00:07:30.883 --> 00:07:34.241
for roughly half of the carbon
14 to decay into carbon 12.

00:07:34.241 --> 00:07:35.595
So based on this ratio,

00:07:35.595 --> 00:07:37.713
and I go into much more
detail in other videos,

00:07:37.713 --> 00:07:40.327
you can figure out how
old these things are

00:07:40.327 --> 00:07:42.054
and you get reasonably precise,

00:07:42.054 --> 00:07:45.611
within a couple of a
hundred, hundreds of years.

00:07:45.611 --> 00:07:48.085
If you want to go further into the past,

00:07:48.085 --> 00:07:51.668
there's things like
potassium-argon dating,

00:07:53.194 --> 00:07:57.112
which is once again taking a
radioactive form of potassium

00:07:57.112 --> 00:08:00.862
and using the idea that
it decays into argon.

00:08:01.952 --> 00:08:06.119
And that when a volcano
releases the argon in that rock,

00:08:07.210 --> 00:08:10.081
is able to go into the atmosphere,
but then once it hardens,

00:08:10.081 --> 00:08:11.816
you have the decay and so you can see

00:08:11.816 --> 00:08:15.904
how long since that volcanic
eruption are we looking at.

00:08:15.904 --> 00:08:17.265
And so for example, you can dig,

00:08:17.265 --> 00:08:22.262
you can do stratographic
techniques right over here.

00:08:22.262 --> 00:08:25.278
Stratigraphy, I have
trouble saying these words.

00:08:25.278 --> 00:08:28.122
That's looking at the
various layers of the earth.

00:08:28.122 --> 00:08:29.868
And you might use some dating techniques,

00:08:29.868 --> 00:08:31.717
for example potassium-argon.

00:08:31.717 --> 00:08:33.985
Say, "Okay, this is that,
a certain amount of age,

00:08:33.985 --> 00:08:36.141
"that is a certain amount of age,

00:08:36.141 --> 00:08:39.938
"this was volcanic rock
from a volcanic eruption."

00:08:39.938 --> 00:08:41.554
And then you can look at the fossils.

00:08:41.554 --> 00:08:43.202
You can say, "Okay, a
fossil that I found here

00:08:43.202 --> 00:08:45.605
"is going to be newer than the stuff here

00:08:45.605 --> 00:08:47.279
"and it's gonna be older
than the stuff here

00:08:47.279 --> 00:08:49.665
"and this might be the newest of all."

00:08:49.665 --> 00:08:50.993
So you can look at relative dating

00:08:50.993 --> 00:08:52.751
and if you're lucky enough
to have some volcanic rock,

00:08:52.751 --> 00:08:55.790
you could do some of this
potassium-argon dating

00:08:55.790 --> 00:08:58.744
and there's many many
many other techniques.

00:08:58.744 --> 00:09:01.330
And it isn't just about
saying, "Oh, this skull

00:09:01.330 --> 00:09:04.418
"was in this place in
the world at this time."

00:09:04.418 --> 00:09:06.388
You can start to infer other things.

00:09:06.388 --> 00:09:08.900
You can look for fossils
of the type of animals,

00:09:08.900 --> 00:09:11.379
the type of plants near
these burial sites.

00:09:11.379 --> 00:09:13.441
You could see how dense
these burial sites were,

00:09:13.441 --> 00:09:15.049
what type of cultures were there.

00:09:15.049 --> 00:09:16.779
You could start to make inferences.

00:09:16.779 --> 00:09:19.695
You can try to infer
how these people died.

00:09:19.695 --> 00:09:22.033
You might have some trauma fractures here

00:09:22.033 --> 00:09:23.689
and you might say, "Okay,
that was a violent death."

00:09:23.689 --> 00:09:25.612
You might look at their
teeth to think about the type

00:09:25.612 --> 00:09:28.869
of things they might have
eaten or their general health.

00:09:28.869 --> 00:09:32.107
You might look at the tools
that are buried near them.

00:09:32.107 --> 00:09:36.095
This right over here, these
are Paleolithic arrow,

00:09:36.095 --> 00:09:38.663
I guess, spearheads or
tools right over here.

00:09:38.663 --> 00:09:41.688
Paleolithic is defined by
more of these harder edges.

00:09:41.688 --> 00:09:43.270
You have Neolithic tools

00:09:43.270 --> 00:09:45.239
which have more of these smoother edges.

00:09:45.239 --> 00:09:48.561
Then Old Stone Age, New
Stone Age right over here.

00:09:48.561 --> 00:09:50.153
And by looking at all of that,

00:09:50.153 --> 00:09:52.822
you have all of these
scientists, you have these

00:09:52.822 --> 00:09:55.164
anthropologists,
archaeologists, paleontologists,

00:09:55.164 --> 00:09:58.417
who are starting to piece
together pre-history.

00:09:58.417 --> 00:10:00.217
And sometimes these
techniques are done together.

00:10:00.217 --> 00:10:02.882
Sometimes we have writing
and we have these technique

00:10:02.882 --> 00:10:05.843
to try to get a more complete picture.

00:10:05.843 --> 00:10:09.332
Now I want to end with
just a note of caution.

00:10:09.332 --> 00:10:11.529
Even though we have all
of these techniques,

00:10:11.529 --> 00:10:14.743
we're learning more
everyday, our understanding

00:10:14.743 --> 00:10:18.687
of all that has happened
is very very incomplete.

00:10:18.687 --> 00:10:21.332
And even more, it
constantly gets challenged

00:10:21.332 --> 00:10:22.509
the more we learn.

00:10:22.509 --> 00:10:25.022
There are things that very
serious people believed

00:10:25.022 --> 00:10:28.872
50 or 100 years ago that we
have now proven to be false.

00:10:28.872 --> 00:10:31.102
And things that we now
take very seriously,

00:10:31.102 --> 00:10:32.879
it's likely that in 50 or 100 years

00:10:32.879 --> 00:10:35.239
people might prove some of that wrong.

00:10:35.239 --> 00:10:37.330
So history, even though
it's about the past,

00:10:37.330 --> 00:10:39.035
it is constantly evolving.

00:10:39.035 --> 00:10:40.555
We're constantly learning more

00:10:40.555 --> 00:10:43.346
and we should have a very solid humility

00:10:43.346 --> 00:10:46.846
about what we know and what we don't know.


Origins of agriculture

WEBVTT
Kind: captions
Language: en

00:00:00.209 --> 00:00:03.726
- [Sal] This timeline
here covers 200,000 years,

00:00:03.726 --> 00:00:08.190
from 200,000 years into
the past, to the present.

00:00:08.190 --> 00:00:10.616
And just to get a sense
of the scale of this,

00:00:10.616 --> 00:00:12.486
if we were to go 2,000 years ago

00:00:12.486 --> 00:00:15.258
to the time of the Roman Empire,

00:00:15.258 --> 00:00:18.554
that would be roughly
here on the timeline.

00:00:18.554 --> 00:00:20.432
If I were to say, when
were the pyramids built?

00:00:20.432 --> 00:00:23.653
That would be roughly
there on the timeline.

00:00:23.653 --> 00:00:27.447
So by human standards this is
a very long period of time.

00:00:27.447 --> 00:00:30.567
And I didn't choose this
time span arbitrarily.

00:00:30.567 --> 00:00:33.538
200,000 years is about how long we believe

00:00:33.538 --> 00:00:37.584
anatomically modern humans
have been on our planet,

00:00:37.584 --> 00:00:40.917
our sub-species of Homo sapiens sapiens.

00:00:41.913 --> 00:00:44.429
Now the reason why I
show these stone tools

00:00:44.429 --> 00:00:48.596
is because the ages, the
periods, of modern humanity,

00:00:50.137 --> 00:00:51.552
or even pre-modern humanity,

00:00:51.552 --> 00:00:53.552
are named after the types of tools

00:00:53.552 --> 00:00:55.854
that have been found
in archaeological digs.

00:00:55.854 --> 00:00:59.867
So most of even pre-human,
or near pre-human,

00:00:59.867 --> 00:01:01.967
and human history,

00:01:01.967 --> 00:01:06.134
has been the Paleolithic
period, or old stone age.

00:01:08.591 --> 00:01:10.341
Paleo, paleo for old,

00:01:11.432 --> 00:01:14.432
lithic coming from lithos for stone.

00:01:15.805 --> 00:01:17.055
So Paleolithic.

00:01:17.891 --> 00:01:21.930
The old stone age is the
great bulk of human history.

00:01:21.930 --> 00:01:23.430
And there's also a Mesolithic

00:01:23.430 --> 00:01:26.847
that comes about 15,000, 20,000 years ago.

00:01:26.847 --> 00:01:29.628
But then around 10,000 years ago,

00:01:29.628 --> 00:01:33.209
the stones have a much more
polished appearance to them.

00:01:33.209 --> 00:01:34.542
Things like this.

00:01:34.542 --> 00:01:37.640
And so that period from
about 10,000 years,

00:01:37.640 --> 00:01:39.966
starting with about 10,000, 15,000 years,

00:01:39.966 --> 00:01:41.754
depending on what part
of the world you look at,

00:01:41.754 --> 00:01:44.792
is referred to as the Neolithic period.

00:01:44.792 --> 00:01:45.625
Neolithic,

00:01:46.509 --> 00:01:48.426
referring to new stone.

00:01:49.716 --> 00:01:52.475
And on top of this timeline,

00:01:52.475 --> 00:01:53.933
I have also shown what's happening

00:01:53.933 --> 00:01:55.933
at a very large scale, climactically,

00:01:55.933 --> 00:01:57.731
on the Earth.

00:01:57.731 --> 00:02:00.525
So these blue periods are ice ages,

00:02:00.525 --> 00:02:02.107
and these reddish orange periods

00:02:02.107 --> 00:02:04.628
are the periods in between ice ages.

00:02:04.628 --> 00:02:06.876
And so you can see the last ice age ended

00:02:06.876 --> 00:02:09.293
roughly 15,000 years ago,

00:02:09.293 --> 00:02:12.543
and it began roughly 110,000 years ago.

00:02:13.844 --> 00:02:15.562
Now I'm giving you all this context

00:02:15.562 --> 00:02:18.370
about these Paleolithic,
Neolithic, and the ice ages,

00:02:18.370 --> 00:02:20.946
because we're gonna talk about probably

00:02:20.946 --> 00:02:23.948
the most important series of events,

00:02:23.948 --> 00:02:27.283
or innovation, in all of human history.

00:02:27.283 --> 00:02:30.080
And that is agriculture.

00:02:30.080 --> 00:02:31.080
Agriculture.

00:02:32.879 --> 00:02:34.763
For most of human history,

00:02:34.763 --> 00:02:36.911
over this Paleolithic period,

00:02:36.911 --> 00:02:38.580
over most of this timeline

00:02:38.580 --> 00:02:42.880
going up until about
10,000 or 15,000 years ago,

00:02:42.880 --> 00:02:45.298
our ancestors were hunter gatherers.

00:02:45.298 --> 00:02:47.900
They would have to chase the
game wherever it might be,

00:02:47.900 --> 00:02:49.729
they couldn't settle down in one place.

00:02:49.729 --> 00:02:51.071
Maybe there were a few that were near

00:02:51.071 --> 00:02:54.574
some sources of fish where
they might be able to do

00:02:54.574 --> 00:02:56.197
some basic fishing.

00:02:56.197 --> 00:02:58.026
But they would have
done hunting like this.

00:02:58.026 --> 00:02:59.195
They would have done gathering,

00:02:59.195 --> 00:03:00.945
which means getting berries or mushrooms

00:03:00.945 --> 00:03:02.111
where they could find it.

00:03:02.111 --> 00:03:04.611
It probably was eating a
lot of things like insects,

00:03:04.611 --> 00:03:07.952
and that is most of human history.

00:03:07.952 --> 00:03:11.535
But then around 10,000
or 15,000 years ago,

00:03:12.742 --> 00:03:15.575
we have the advent of human beings

00:03:16.576 --> 00:03:19.274
taking nature into their own hands.

00:03:19.274 --> 00:03:21.026
Instead of saying, let's
just follow the game

00:03:21.026 --> 00:03:23.516
wherever the game might migrate to,

00:03:23.516 --> 00:03:26.193
let's actually domesticate these animals.

00:03:26.193 --> 00:03:27.991
Let's take some of them,
start breeding them

00:03:27.991 --> 00:03:30.364
so they're more suitable
for human consumption,

00:03:30.364 --> 00:03:33.249
so that they are easier to
raise, maybe more robust,

00:03:33.249 --> 00:03:35.830
and we will breed them,

00:03:35.830 --> 00:03:39.365
and we will raise them for milk, for meat.

00:03:39.365 --> 00:03:42.080
We also started to domesticate plants.

00:03:42.080 --> 00:03:43.663
Instead of saying, okay let's just gather

00:03:43.663 --> 00:03:45.914
those berries there where
it happens to emerge,

00:03:45.914 --> 00:03:48.662
oh let's actually start to plant things.

00:03:48.662 --> 00:03:52.164
And on a very predictable
way, be able to harvest them,

00:03:52.164 --> 00:03:54.253
and so be able to have a
more predictable food supply.

00:03:54.253 --> 00:03:56.836
Both of these things allowed human beings

00:03:56.836 --> 00:03:58.602
to have a higher population density,

00:03:58.602 --> 00:04:02.477
to start to settle down, to
have a more sedentary life.

00:04:02.477 --> 00:04:05.226
And this is a huge, huge, huge deal.

00:04:05.226 --> 00:04:06.649
Just to get a sense.

00:04:06.649 --> 00:04:08.935
We believe that the carrying capacity

00:04:08.935 --> 00:04:10.185
for the planet,

00:04:11.067 --> 00:04:13.443
for human beings as hunter gatherers,

00:04:13.443 --> 00:04:15.193
is 10 million people.

00:04:16.399 --> 00:04:19.733
And that is what we estimate
the world population was

00:04:19.733 --> 00:04:23.321
around the time of the last
ice age, or shortly afterwards.

00:04:23.321 --> 00:04:27.417
And that's because a tribe
of 100 hunter gatherers

00:04:27.417 --> 00:04:30.500
is going to need 50 square kilometers

00:04:31.419 --> 00:04:34.044
to 100 square kilometers

00:04:34.044 --> 00:04:36.080
to hunt and gather from,

00:04:36.080 --> 00:04:40.252
or actually 500, from some
of the estimates I've seen.

00:04:40.252 --> 00:04:42.250
And it might seem like a lot of area

00:04:42.250 --> 00:04:43.952
that you would need to
hunt and gather from,

00:04:43.952 --> 00:04:45.200
but imagine that you and your family

00:04:45.200 --> 00:04:46.948
had to go live in the woods now.

00:04:46.948 --> 00:04:49.108
How much food could you actually find?

00:04:49.108 --> 00:04:51.925
You'd have to walk
miles and miles per day,

00:04:51.925 --> 00:04:53.549
if you're trying to hunt,

00:04:53.549 --> 00:04:56.095
you'd have to walk miles and miles per day

00:04:56.095 --> 00:05:00.873
to get whatever wild grains
or berries, or whatever,

00:05:00.873 --> 00:05:03.581
or mushrooms, or whatever, or insects,

00:05:03.581 --> 00:05:05.330
that you might consume.

00:05:05.330 --> 00:05:07.537
But then with the advent of agriculture

00:05:07.537 --> 00:05:11.838
it allowed for a much, much
higher density population.

00:05:11.838 --> 00:05:15.505
In fact, going from the
birth of agriculture,

00:05:15.505 --> 00:05:17.638
which happens in the Neolithic period,

00:05:17.638 --> 00:05:21.563
the dawn of agriculture coincides
with the Neolithic period,

00:05:21.563 --> 00:05:23.936
they're often used
somewhat interchangeably.

00:05:23.936 --> 00:05:27.531
Going from that period
10,000, 15,000 years ago,

00:05:27.531 --> 00:05:29.369
to the time of ancient Rome,

00:05:29.369 --> 00:05:32.165
so we're on the order of 10,000 years,

00:05:32.165 --> 00:05:35.367
the world population
with agriculture exploded

00:05:35.367 --> 00:05:39.450
from 10 million to roughly 250 million.

00:05:39.450 --> 00:05:40.700
And that's 25x.

00:05:41.705 --> 00:05:44.568
And we know that from the
time of Rome 'til now,

00:05:44.568 --> 00:05:46.866
another 2,000 years with agriculture,

00:05:46.866 --> 00:05:49.789
our population has gone up 28x,

00:05:49.789 --> 00:05:53.082
it's growing exponentially,
to seven billion.

00:05:53.082 --> 00:05:54.840
And there's no way that we could have had

00:05:54.840 --> 00:05:56.391
this level of density

00:05:56.391 --> 00:05:58.058
without agriculture.

00:05:58.980 --> 00:06:00.479
Now a key question is,

00:06:00.479 --> 00:06:03.730
is why did agriculture emerge right then?

00:06:03.730 --> 00:06:07.325
Well one theory is is,
well it seems to be only,

00:06:07.325 --> 00:06:09.399
and I'll say only in quotes,

00:06:09.399 --> 00:06:12.901
only a few thousand years
after the end of the ice age.

00:06:12.901 --> 00:06:15.445
Maybe you had positive climate change,

00:06:15.445 --> 00:06:17.211
at least from a human point of view,

00:06:17.211 --> 00:06:20.628
that allowed land to support agriculture.

00:06:23.385 --> 00:06:25.392
That seems like a reasonable theory.

00:06:25.392 --> 00:06:27.112
Although you might say,
well we've had other periods

00:06:27.112 --> 00:06:28.610
where we didn't have an ice age,

00:06:28.610 --> 00:06:31.366
how come we see no evidence
of agriculture then?

00:06:31.366 --> 00:06:35.405
And one counterargument or
explanation I've heard for that

00:06:35.405 --> 00:06:37.901
is, anatomically modern humans

00:06:37.901 --> 00:06:40.899
started to emerge around
200,000 years ago,

00:06:40.899 --> 00:06:43.874
but that doesn't mean
that the way they thought,

00:06:43.874 --> 00:06:46.702
or that their brains were
the same as modern humans.

00:06:46.702 --> 00:06:49.536
And so maybe in this
period the human brain

00:06:49.536 --> 00:06:52.369
just wasn't capable of performing,

00:06:53.579 --> 00:06:55.751
or thinking of agriculture.

00:06:55.751 --> 00:06:57.793
Other theories are is
that the human population

00:06:57.793 --> 00:07:01.550
through hunter gathering
kept increasing over time,

00:07:01.550 --> 00:07:03.425
and it was right after the last ice age

00:07:03.425 --> 00:07:05.466
that you really got close to this

00:07:05.466 --> 00:07:08.423
critical mass of population,

00:07:08.423 --> 00:07:12.100
at which point, for every
extra human to be born,

00:07:12.100 --> 00:07:14.888
another human would not be able to live,

00:07:14.888 --> 00:07:16.096
or not be able to be born,

00:07:16.096 --> 00:07:17.556
because there was a fixed supply of food

00:07:17.556 --> 00:07:19.888
as hunter gatherers.

00:07:19.888 --> 00:07:21.996
And so you could say out of necessity,

00:07:21.996 --> 00:07:24.202
maybe a tribe here or
a little group there,

00:07:24.202 --> 00:07:26.993
started to say hey, well
what if we started to

00:07:26.993 --> 00:07:28.998
domesticate some of this cattle?

00:07:28.998 --> 00:07:30.698
Or if we started to plant crops?

00:07:30.698 --> 00:07:33.199
All of a sudden they would
start to have a higher density

00:07:33.199 --> 00:07:34.654
and their population could increase.

00:07:34.654 --> 00:07:36.408
And not only would they
be able to survive,

00:07:36.408 --> 00:07:39.382
but they could also be very dominant

00:07:39.382 --> 00:07:41.181
in things like conflict.

00:07:41.181 --> 00:07:42.803
Once you start having agriculture,

00:07:42.803 --> 00:07:44.263
and as agriculture advances,

00:07:44.263 --> 00:07:46.894
as I mentioned, these people
could be more sedentary.

00:07:46.894 --> 00:07:48.765
They wouldn't have to move around.

00:07:48.765 --> 00:07:51.927
They wouldn't have to move
around all of the time.

00:07:51.927 --> 00:07:54.011
That allows them to

00:07:54.011 --> 00:07:56.970
create even defensive structures.

00:07:56.970 --> 00:07:58.634
You could have specialization,

00:07:58.634 --> 00:08:02.483
where not everyone is
having to worry about,

00:08:02.483 --> 00:08:04.333
specialization,

00:08:04.333 --> 00:08:07.918
not everyone is having to worry
about food all of the time.

00:08:07.918 --> 00:08:09.960
And so you could develop warriors,

00:08:09.960 --> 00:08:13.594
you could have technology
developing, especially weapons,

00:08:13.594 --> 00:08:16.174
and so not only would
that tribe or that group

00:08:16.174 --> 00:08:17.882
be able to get higher density,

00:08:17.882 --> 00:08:19.917
but they would be able
to defend their property.

00:08:19.917 --> 00:08:21.790
In fact they would
probably care a lot more

00:08:21.790 --> 00:08:23.038
about property,

00:08:23.038 --> 00:08:24.786
because they use it to farm,

00:08:24.786 --> 00:08:26.079
versus hunter gatherers

00:08:26.079 --> 00:08:28.039
might just view that as their territory

00:08:28.039 --> 00:08:29.334
that they wander over

00:08:29.334 --> 00:08:31.747
in order to find food.

00:08:31.747 --> 00:08:36.715
Now whatever the actual cause
of the birth of agriculture,

00:08:36.715 --> 00:08:40.715
it has had profound
implications on our society.

00:08:41.961 --> 00:08:46.010
You would not have had
the birth of city-states,

00:08:46.010 --> 00:08:47.635
without agriculture.

00:08:47.635 --> 00:08:50.592
City-states had high density populations.

00:08:50.592 --> 00:08:53.134
They could not have been
supported with hunter gathering.

00:08:53.134 --> 00:08:56.142
You could not have had the
development of technology,

00:08:56.142 --> 00:08:57.851
which, people needed more time,

00:08:57.851 --> 00:09:01.175
you needed specialization,
in order to have that.

00:09:01.175 --> 00:09:03.811
It's, I would say, fair to say,

00:09:03.811 --> 00:09:06.937
we wouldn't even have our modern,

00:09:06.937 --> 00:09:09.473
our modern civilization
would not have been possible

00:09:09.473 --> 00:09:11.057
without agriculture.

00:09:11.057 --> 00:09:13.350
I would not be making this
video to you right now.

00:09:13.350 --> 00:09:14.599
In fact most of us wouldn't even have been

00:09:14.599 --> 00:09:15.941
around on the planet

00:09:15.941 --> 00:09:17.897
because the planet wouldn't
have been able to support

00:09:17.897 --> 00:09:20.558
our really immensely large population

00:09:20.558 --> 00:09:24.725
without not only agriculture,
but an ever advancing,

00:09:26.221 --> 00:09:29.070
technologically advancing agriculture.

00:09:29.070 --> 00:09:30.820
So it might seem like a small thing.

00:09:30.820 --> 00:09:32.487
You go from hunter
gatherer to agriculture,

00:09:32.487 --> 00:09:34.942
but it's been one of the
most, maybe the most profound,

00:09:34.942 --> 00:09:37.326
well it's probably up there
with language and writing,

00:09:37.326 --> 00:09:39.702
things that have defined

00:09:39.702 --> 00:09:41.995
what makes humans humans.

00:09:41.995 --> 00:09:46.162
And to get a sense of where
this agriculture was born,

00:09:47.667 --> 00:09:50.709
here's the various regions that we believe

00:09:50.709 --> 00:09:52.675
agriculture emerged from.

00:09:52.675 --> 00:09:56.128
The Levant right over here
in the eastern Mediterranean,

00:09:56.128 --> 00:09:57.506
modern day Middle East,

00:09:57.506 --> 00:10:00.644
Syria, Israel, Palestine, Iraq,

00:10:00.644 --> 00:10:02.288
southeastern Turkey,

00:10:02.288 --> 00:10:04.005
this is one of the areas where we think

00:10:04.005 --> 00:10:05.602
agriculture first emerged

00:10:05.602 --> 00:10:09.185
on the order of 10,000
to 15,000 years ago.

00:10:10.028 --> 00:10:12.578
We believe rice came from China.

00:10:12.578 --> 00:10:16.961
I've seen estimates anywhere
from 6,000 to 13,000 years ago.

00:10:16.961 --> 00:10:19.677
You have the potato and other crops

00:10:19.677 --> 00:10:22.266
coming from the Andes
thousands of years ago,

00:10:22.266 --> 00:10:24.315
and this is just a sample of the areas

00:10:24.315 --> 00:10:25.987
where we see agriculture,

00:10:25.987 --> 00:10:29.976
both domestication of crops
and animals first emerging.

00:10:29.976 --> 00:10:31.312
And what's interesting is,

00:10:31.312 --> 00:10:33.268
over time we're likely
to discover other places

00:10:33.268 --> 00:10:35.265
as we have more archaeological digs

00:10:35.265 --> 00:10:36.760
and we find more fossil evidence of

00:10:36.760 --> 00:10:40.148
ancient agriculture and ancient peoples.

00:10:40.148 --> 00:10:43.108
And even the time frame where
we believe this happened,

00:10:43.108 --> 00:10:46.026
the 10,000, 12,000, 15,000 years ago,

00:10:46.026 --> 00:10:47.983
that's likely to maybe
move back a little bit

00:10:47.983 --> 00:10:51.773
as we discover older and older evidence.

00:10:51.773 --> 00:10:52.990
So I'll leave you there,

00:10:52.990 --> 00:10:54.829
but the big takeaway is

00:10:54.829 --> 00:10:56.998
most of human history

00:10:56.998 --> 00:10:59.372
we were not only making rough stones

00:10:59.372 --> 00:11:01.083
for our weapons and our tools,

00:11:01.083 --> 00:11:03.215
but we were chasing animals
wherever we could find 'em.

00:11:03.215 --> 00:11:05.297
We didn't have predictable food sources.

00:11:05.297 --> 00:11:08.297
We were gathering fruits and grains,

00:11:09.935 --> 00:11:11.685
and insects wherever we could

00:11:11.685 --> 00:11:12.985
to support our families,

00:11:12.985 --> 00:11:15.907
and there was a limit on how
many humans could be around.

00:11:15.907 --> 00:11:19.283
But shortly after the end of
the ice age, it all changed.

00:11:19.283 --> 00:11:21.004
You had a population explosion,

00:11:21.004 --> 00:11:24.337
and we think that is due to agriculture.


Roman Empire and Christianity

WEBVTT
Kind: captions
Language: en

00:00:00.208 --> 00:00:02.391
- [Narrator] As we've talked
about in multiple videos,

00:00:02.391 --> 00:00:03.809
Christianity is a religion

00:00:03.809 --> 00:00:06.724
that grew out of the
fringes of the Roman Empire.

00:00:06.724 --> 00:00:11.465
It starts as a Jewish
sect in Judea and Galilee

00:00:11.465 --> 00:00:15.126
with the teachings of Jesus
and his early ministry.

00:00:15.126 --> 00:00:16.363
But it's important to keep in mind

00:00:16.363 --> 00:00:18.909
that at the time of Jesus's crucifixion,

00:00:18.909 --> 00:00:21.992
around 30 C.E., he only had, at most,

00:00:23.584 --> 00:00:27.439
we're talking about in
the hundreds of followers.

00:00:27.439 --> 00:00:29.863
But in other videos, we
talk about his apostles

00:00:29.863 --> 00:00:31.340
who start spreading the faith,

00:00:31.340 --> 00:00:33.378
not only to other Jews,

00:00:33.378 --> 00:00:36.211
but also to non-Jews, to gentiles.

00:00:37.330 --> 00:00:39.441
We talked about Emperor Nero

00:00:39.441 --> 00:00:42.212
and his famous purges of Christians,

00:00:42.212 --> 00:00:44.711
in particular, you have
the famous Roman Fire,

00:00:44.711 --> 00:00:47.372
or The Fire of Rome in 64 C.E.

00:00:47.372 --> 00:00:49.123
that Nero blames on Christians,

00:00:49.123 --> 00:00:52.102
and he uses this to do
significant purge in Rome.

00:00:52.102 --> 00:00:55.760
Many historians believe that
the apostles Peter and Paul

00:00:55.760 --> 00:00:57.919
were killed during those purges.

00:00:57.919 --> 00:01:00.778
This is a painting of
Nero using Christians

00:01:00.778 --> 00:01:05.361
lighting them alive to
punish them for that fire.

00:01:05.361 --> 00:01:08.012
Now, other imperial acts that continue

00:01:08.012 --> 00:01:09.929
to have a significant effect

00:01:09.929 --> 00:01:12.246
on this very early Christian church.

00:01:12.246 --> 00:01:16.090
We talked about the destroying
of the second temple

00:01:16.090 --> 00:01:18.090
in Jerusalem in 70 C.E.,

00:01:19.761 --> 00:01:22.064
and that's under the reign of Vespasian.

00:01:22.064 --> 00:01:24.826
And this of course is a
very significant event

00:01:24.826 --> 00:01:26.540
for the Jewish people,

00:01:26.540 --> 00:01:29.166
but it also has a significant
effect on Christianity,

00:01:29.166 --> 00:01:31.053
because remember Christianity was really

00:01:31.053 --> 00:01:32.373
centered at Jerusalem.

00:01:32.373 --> 00:01:34.836
It was a sect of Christianity,

00:01:34.836 --> 00:01:37.233
or it was, it got its start

00:01:37.233 --> 00:01:38.772
as a sect of Christianity,

00:01:38.772 --> 00:01:41.156
but now, with the
destruction of the temple,

00:01:41.156 --> 00:01:43.233
the Jews were dispersed from Jerusalem,

00:01:43.233 --> 00:01:46.737
and so were many of
these early Christians.

00:01:46.737 --> 00:01:48.430
And so the centers of Christianity

00:01:48.430 --> 00:01:50.544
became throughout the empire,

00:01:50.544 --> 00:01:52.895
places like Antioch.

00:01:52.895 --> 00:01:56.674
And you continue to have a
discomfort with Christians

00:01:56.674 --> 00:01:59.338
especially as they became
more and more in number,

00:01:59.338 --> 00:02:01.213
and they started to have
more and more power.

00:02:01.213 --> 00:02:02.650
And here you have these people

00:02:02.650 --> 00:02:06.131
who refused to obey the Roman gods,

00:02:06.131 --> 00:02:08.878
to do their rituals, to
worship the Roman gods,

00:02:08.878 --> 00:02:11.808
and remember, even, there
was emperor worship here.

00:02:11.808 --> 00:02:15.984
And the issue was more so
than even the Jewish people,

00:02:15.984 --> 00:02:18.179
who, to a certain degree,
kept to themselves,

00:02:18.179 --> 00:02:19.631
these Christians were missionaries.

00:02:19.631 --> 00:02:23.567
They were spreading their
faith to Roman citizens.

00:02:23.567 --> 00:02:25.451
Families were being split apart,

00:02:25.451 --> 00:02:27.081
where some decided to become Christians

00:02:27.081 --> 00:02:29.686
and no longer follow
these Roman practices.

00:02:29.686 --> 00:02:31.940
And so you have these
various persecutions,

00:02:31.940 --> 00:02:33.706
and these various purges,

00:02:33.706 --> 00:02:36.071
and they get more and more significant

00:02:36.071 --> 00:02:37.175
as you go into the second

00:02:37.175 --> 00:02:39.561
and even to some degree
in the third century

00:02:39.561 --> 00:02:41.626
as Christians become
more and more prominent

00:02:41.626 --> 00:02:42.971
and more and more in number.

00:02:42.971 --> 00:02:45.454
Remember, they have this missionary zeal,

00:02:45.454 --> 00:02:48.085
and it really becomes a significant purge

00:02:48.085 --> 00:02:50.680
under the reign of Diocletian.

00:02:50.680 --> 00:02:52.948
In previous videos, we
talk about Diocletian.

00:02:52.948 --> 00:02:55.635
He's the one that
famously split the empire

00:02:55.635 --> 00:02:57.027
between east and west,

00:02:57.027 --> 00:03:00.461
but he's also famous for
his purge of the Christians.

00:03:00.461 --> 00:03:04.912
The most extreme state-sponsored
purges of Christians

00:03:04.912 --> 00:03:06.626
in the Roman Empire.

00:03:06.626 --> 00:03:09.231
But what's interesting is right after,

00:03:09.231 --> 00:03:11.091
or shortly after those purges,

00:03:11.091 --> 00:03:12.725
we're talking a few decades here,

00:03:12.725 --> 00:03:14.501
the next empire that would,

00:03:14.501 --> 00:03:16.803
the next emperor who would have reign

00:03:16.803 --> 00:03:18.387
over the entire Roman Empire,

00:03:18.387 --> 00:03:19.781
would be Constantine,

00:03:19.781 --> 00:03:21.941
and at least relative to Christians,

00:03:21.941 --> 00:03:24.950
he would be the opposite of Diocletian,

00:03:24.950 --> 00:03:28.011
because as he's trying
to consolidate his power

00:03:28.011 --> 00:03:29.011
in 312 C.E.,

00:03:30.077 --> 00:03:33.532
you have the famous battle
at the Milvian Bridge

00:03:33.532 --> 00:03:36.365
outside of, or entering into Rome.

00:03:37.611 --> 00:03:40.084
And in this battle, Constantine,

00:03:40.084 --> 00:03:42.278
he sees a vision of the Christian cross

00:03:42.278 --> 00:03:43.504
centered on the sun,

00:03:43.504 --> 00:03:45.437
and he thinks that that is a sign

00:03:45.437 --> 00:03:47.522
that he should fight under the cross.

00:03:47.522 --> 00:03:51.688
And he gets the symbols painted
on his soldier's shield,

00:03:51.688 --> 00:03:53.009
and he's victorious.

00:03:53.009 --> 00:03:55.592
This date, 312, Milvian Bridge,

00:03:56.872 --> 00:03:58.567
The Battle at Milvian Bridge,

00:03:58.567 --> 00:03:59.900
is viewed as the beginning

00:03:59.900 --> 00:04:02.252
of Constantine's
conversion to Christianity,

00:04:02.252 --> 00:04:04.835
viewing it as his chosen faith.

00:04:05.793 --> 00:04:09.429
In 313, the Roman emperor, or the empire,

00:04:09.429 --> 00:04:11.303
issues the edict of Milan,

00:04:11.303 --> 00:04:13.926
which officially makes Christianity OK.

00:04:13.926 --> 00:04:16.402
It legalizes Christianity,
which is a big deal,

00:04:16.402 --> 00:04:18.645
because these folks have been
persecuted all this time.

00:04:18.645 --> 00:04:21.549
They haven't been
worshiping the Roman gods.

00:04:21.549 --> 00:04:25.078
Then in 325, Constantine
goes even further,

00:04:25.078 --> 00:04:27.599
and he convenes the council of Nicea,

00:04:27.599 --> 00:04:28.920
where he brings the bishops

00:04:28.920 --> 00:04:31.617
from all of Christendom together,

00:04:31.617 --> 00:04:34.562
to start having a more
unified belief system.

00:04:34.562 --> 00:04:36.565
They come up with a Nicean creed.

00:04:36.565 --> 00:04:38.808
A lot of the debate is whether Jesus,

00:04:38.808 --> 00:04:41.929
as son of God, should be
considered equal to God

00:04:41.929 --> 00:04:44.747
or as God or a little less,

00:04:44.747 --> 00:04:45.757
as being the son,

00:04:45.757 --> 00:04:46.969
maybe if he is the son,

00:04:46.969 --> 00:04:48.416
maybe there was a time
where there wasn't a son,

00:04:48.416 --> 00:04:49.290
and there was just God,

00:04:49.290 --> 00:04:52.254
but the Nicean Creed says no,
they are one and the same,

00:04:52.254 --> 00:04:54.529
these two things are equivalent.

00:04:54.529 --> 00:04:56.517
And then Constantine, on his death bed,

00:04:56.517 --> 00:04:58.589
famously gets baptized,

00:04:58.589 --> 00:05:00.229
and becomes a Christian,

00:05:00.229 --> 00:05:02.616
the first Christian emperor

00:05:02.616 --> 00:05:05.116
of this gigantic Roman Empire.

00:05:07.341 --> 00:05:08.174
And so you can imagine,

00:05:08.174 --> 00:05:09.979
you have more and more
followers of Christianity,

00:05:09.979 --> 00:05:12.830
but now, this is, so to speak, prime time.

00:05:12.830 --> 00:05:14.776
The emperor of the Roman,

00:05:14.776 --> 00:05:17.456
the emperor of the Roman Empire

00:05:17.456 --> 00:05:19.266
has converted to Christianity,

00:05:19.266 --> 00:05:22.394
and it's a legalized religion.

00:05:22.394 --> 00:05:24.654
And then that gets even more momentum

00:05:24.654 --> 00:05:27.256
when we get to the end
of the fourth century.

00:05:27.256 --> 00:05:29.454
Emperor Theodosius, who is Christian,

00:05:29.454 --> 00:05:32.056
under him, it becomes
the official religion

00:05:32.056 --> 00:05:32.978
of the Roman Empire,

00:05:32.978 --> 00:05:34.893
the same empire that at one point,

00:05:34.893 --> 00:05:36.629
Christianity was illegal,

00:05:36.629 --> 00:05:38.031
and it was being persecuted,

00:05:38.031 --> 00:05:40.731
now it was the official religion

00:05:40.731 --> 00:05:41.866
of the Roman Empire.

00:05:41.866 --> 00:05:43.829
Theodosius, I mention him in other videos,

00:05:43.829 --> 00:05:47.036
he was incidentally also the last emperor

00:05:47.036 --> 00:05:51.203
to rule over both Eastern
and Western Roman Empires.

00:05:52.061 --> 00:05:55.650
So you have this empire that
starts off very anti-Christian,

00:05:55.650 --> 00:05:56.749
actually in the beginning,

00:05:56.749 --> 00:05:58.911
they don't really take much
note of what's going on

00:05:58.911 --> 00:06:00.413
in Judea and Galilee,

00:06:00.413 --> 00:06:02.460
but more and more, they
start to persecute them.

00:06:02.460 --> 00:06:04.662
But we go a few hundred years into this,

00:06:04.662 --> 00:06:07.495
and it becomes a Christian empire.


Paul and the apostles Christianity

WEBVTT
Kind: captions
Language: en

00:00:00.348 --> 00:00:01.335
- [Instructor] The central figure

00:00:01.335 --> 00:00:04.136
in Christianity is, clearly, Jesus,

00:00:04.136 --> 00:00:05.837
but it's important to note that

00:00:05.837 --> 00:00:09.335
he does not establish the
religion all by himself.

00:00:09.335 --> 00:00:12.448
In fact, at the time if his crucifixion,

00:00:12.448 --> 00:00:14.487
and, according to Christian beliefs,

00:00:14.487 --> 00:00:16.406
resurrection and ascension,

00:00:16.406 --> 00:00:20.573
the number of followers he has
numbers in the low hundreds.

00:00:21.687 --> 00:00:25.498
The actual work of spreading and creating

00:00:25.498 --> 00:00:28.571
the church falls on his disciples,

00:00:28.571 --> 00:00:33.157
and, particular, the ones that
are referred to as apostles.

00:00:33.157 --> 00:00:35.395
When we looked at the Gospels,

00:00:35.395 --> 00:00:37.768
which are the life and
teachings of Jesus Christ,

00:00:37.768 --> 00:00:39.608
the first four books of the New Testament,

00:00:39.608 --> 00:00:41.320
Matthew, Mark, Luke, and John,

00:00:41.320 --> 00:00:44.230
we talk about the 12 disciples,

00:00:44.230 --> 00:00:46.979
those primary followers of Jesus.

00:00:46.979 --> 00:00:50.880
They're often referred to
as the 12 Apostles, as well,

00:00:50.880 --> 00:00:53.214
because according to Christian beliefs,

00:00:53.214 --> 00:00:57.193
between the resurrection
and the ascension of Jesus,

00:00:57.193 --> 00:01:00.294
he tells the apostles,
look, go spread the word.

00:01:00.294 --> 00:01:03.742
Apostle comes from Greek for
the one who spreads the word.

00:01:03.742 --> 00:01:06.715
These were the first
Christian missionaries.

00:01:06.715 --> 00:01:11.332
Now, first amongst these apostles
is often considered Peter.

00:01:11.332 --> 00:01:13.919
This is Peter, right over here.

00:01:13.919 --> 00:01:15.668
He is mentioned in the Gospels.

00:01:15.668 --> 00:01:19.807
He is beside Jesus at the
time of the Transfiguration.

00:01:19.807 --> 00:01:23.255
Amongst Roman Catholics, he
is viewed as the first Pope,

00:01:23.255 --> 00:01:25.057
and he is really the head of

00:01:25.057 --> 00:01:27.568
the church of the Christian followers,

00:01:27.568 --> 00:01:29.590
and they're not even called
Christians at this point,

00:01:29.590 --> 00:01:32.090
in Jerusalem, right over here.

00:01:33.717 --> 00:01:35.067
But we're gonna talk about another

00:01:35.067 --> 00:01:37.493
significant apostle, as well,

00:01:37.493 --> 00:01:40.160
in this video, and that is Paul.

00:01:41.043 --> 00:01:43.713
As we'll see, Paul is not mentioned in

00:01:43.713 --> 00:01:45.705
the first four books of the Bible,

00:01:45.705 --> 00:01:48.593
he wasn't a contemporary of Jesus,

00:01:48.593 --> 00:01:52.318
or, at least, he didn't
know Jesus during his life.

00:01:52.318 --> 00:01:54.101
He did live at the same time,

00:01:54.101 --> 00:01:55.780
and you can see it on this timeline

00:01:55.780 --> 00:01:59.947
where Jesus is born slightly
before the Common Era,

00:02:01.266 --> 00:02:03.718
I start at year one, because
there isn't a year zero,

00:02:03.718 --> 00:02:05.554
and you can see that Peter is born,

00:02:05.554 --> 00:02:07.689
historians believe, around 1 AD,

00:02:07.689 --> 00:02:10.568
and Paul would have been
born shortly after that.

00:02:10.568 --> 00:02:12.357
What's interesting about Paul is,

00:02:12.357 --> 00:02:17.066
Paul starts his life as a
very conservative Pharisee.

00:02:17.066 --> 00:02:21.155
He starts his life
persecuting Christian people,

00:02:21.155 --> 00:02:24.457
and then he has a fairly dramatic switch.

00:02:24.457 --> 00:02:26.374
You'll often hear Paul,

00:02:27.729 --> 00:02:32.018
or you might also hear the
name Saul, or Paul of Tarsus,

00:02:32.018 --> 00:02:34.441
these are all referring
to the same person.

00:02:34.441 --> 00:02:38.254
Tarsus is right over here, in
modern day Southern Turkey.

00:02:38.254 --> 00:02:40.119
Paul, as I mentioned, he grows up

00:02:40.119 --> 00:02:41.857
in a conservative Jewish family,

00:02:41.857 --> 00:02:43.918
but he also has Roman citizenship.

00:02:43.918 --> 00:02:45.932
That's why he has this Roman name, Paul,

00:02:45.932 --> 00:02:47.767
and he has his Hebrew name, Saul.

00:02:47.767 --> 00:02:50.082
He's educated in Jerusalem.

00:02:50.082 --> 00:02:53.354
He doesn't enter into
the story until we get

00:02:53.354 --> 00:02:55.455
to the fifth book of the New Testament,

00:02:55.455 --> 00:02:57.781
the Acts of the Apostles.

00:02:57.781 --> 00:03:00.056
The Acts of the Apostles
talks a lot about Peter,

00:03:00.056 --> 00:03:02.995
how he spreads the church,
how he is persecuted,

00:03:02.995 --> 00:03:05.745
how he performs various miracles,

00:03:07.327 --> 00:03:11.204
but it also is where we
get introduced to Paul.

00:03:11.204 --> 00:03:12.753
When we first see him, as I mentioned,

00:03:12.753 --> 00:03:14.891
he is persecuting Christians.

00:03:14.891 --> 00:03:19.629
As this Church of Jerusalem
starts to get established,

00:03:19.629 --> 00:03:23.129
Peter puts a fellow by the name of Stephen

00:03:24.180 --> 00:03:27.557
in charge of distributing alms to widows

00:03:27.557 --> 00:03:29.857
that are coming to them.

00:03:29.857 --> 00:03:31.642
Stephen, and this is a narrative

00:03:31.642 --> 00:03:33.594
that you'll hear often
in the New Testament,

00:03:33.594 --> 00:03:36.243
he starts to get on the nerves

00:03:36.243 --> 00:03:38.282
of the religious establishment,

00:03:38.282 --> 00:03:39.820
the Jewish religious establishment,

00:03:39.820 --> 00:03:42.044
who think that he's a blasphemer,

00:03:42.044 --> 00:03:43.691
that his beliefs are threatening them,

00:03:43.691 --> 00:03:46.557
and so there's this
famous trial, once again,

00:03:46.557 --> 00:03:49.881
with the Sanhedrin, and
the trial of Stephen.

00:03:49.881 --> 00:03:53.567
Here I have an account
for him, of that trial,

00:03:53.567 --> 00:03:58.278
or near the end of that
trial, from the book of Acts,

00:03:58.278 --> 00:04:01.291
and during that trial Stephen goes into

00:04:01.291 --> 00:04:04.139
this long account of what
the teachings of Jesus are,

00:04:04.139 --> 00:04:05.917
and how they're justified by

00:04:05.917 --> 00:04:08.001
what is mentioned in the Old Testament.

00:04:08.001 --> 00:04:11.203
He actually does a fairly
long account of it.

00:04:11.203 --> 00:04:13.703
He also is fairly rebellious.

00:04:13.703 --> 00:04:16.602
He says, look, you, the establishment,

00:04:16.602 --> 00:04:20.952
what prophet have you
not persecuted before?

00:04:20.952 --> 00:04:22.766
This really angers them, and so

00:04:22.766 --> 00:04:24.203
we read from the book of Acts,

00:04:24.203 --> 00:04:26.829
they all rushed at him,
dragged him out of the city,

00:04:26.829 --> 00:04:28.778
and began to stone him.

00:04:28.778 --> 00:04:30.577
Meanwhile, the witnesses lay their coats

00:04:30.577 --> 00:04:33.863
at the feet of a young man named Saul,

00:04:33.863 --> 00:04:38.253
and we see him right over here,
depicted in this painting.

00:04:38.253 --> 00:04:41.338
Saul approved of their killing him.

00:04:41.338 --> 00:04:43.983
On that day, a great persecution

00:04:43.983 --> 00:04:47.045
broke out against the church in Jerusalem.

00:04:47.045 --> 00:04:49.171
All except the apostles were scattered

00:04:49.171 --> 00:04:51.930
throughout Judea and Samaria.

00:04:51.930 --> 00:04:55.596
Godly men buried Stephen
and mourned deeply for him.

00:04:55.596 --> 00:04:58.346
But Saul began to destroy the church.

00:04:58.346 --> 00:05:00.355
Going from house to house,

00:05:00.355 --> 00:05:04.706
he dragged off both men and
women and put them in prison.

00:05:04.706 --> 00:05:08.548
He is a pretty fervent
persecutor of Christians.

00:05:08.548 --> 00:05:10.298
It goes even further.

00:05:11.861 --> 00:05:15.381
Then, we read, we go on to
read in the book of Acts,

00:05:15.381 --> 00:05:18.218
Meanwhile, Saul was still breathing out

00:05:18.218 --> 00:05:20.880
murderous threats against
the Lord's disciples.

00:05:20.880 --> 00:05:23.409
He went to the high priest and asked him

00:05:23.409 --> 00:05:26.184
for letters to the synagogues in Damascus,

00:05:26.184 --> 00:05:29.756
so that if he found any there
who belonged to the Way,

00:05:29.756 --> 00:05:31.757
whether men or women, he might take them

00:05:31.757 --> 00:05:33.234
as prisoners to Jerusalem.

00:05:33.234 --> 00:05:35.632
As he neared Damascus on his journey,

00:05:35.632 --> 00:05:36.909
so this is going to be interesting thing.

00:05:36.909 --> 00:05:39.133
He's so fervent, he
goes to the high priest,

00:05:39.133 --> 00:05:41.785
and says, I wanna persecute

00:05:41.785 --> 00:05:43.610
these Christians wherever they are.

00:05:43.610 --> 00:05:45.006
I wanna go to Damascus,

00:05:45.006 --> 00:05:46.647
and see if I can persecute them there.

00:05:46.647 --> 00:05:48.607
As he neared Damascus on his journey,

00:05:48.607 --> 00:05:51.882
suddenly a light from
heaven flashed around him.

00:05:51.882 --> 00:05:55.157
He fell to the ground and
heard a voice say to him,

00:05:55.157 --> 00:05:58.660
"Saul, Saul, why do you persecute me?"

00:05:58.660 --> 00:06:03.583
This is an image of that,
this narrative in the Bible.

00:06:03.583 --> 00:06:05.419
"Who are you, Lord?", Saul asked.

00:06:05.419 --> 00:06:09.247
"I am Jesus, whom you are
persecuting," he replied.

00:06:09.247 --> 00:06:11.333
"Now get up and go into the city,"

00:06:11.333 --> 00:06:12.421
the city being Damascus,

00:06:12.421 --> 00:06:14.920
"and you will be told what you must do."

00:06:14.920 --> 00:06:17.420
Paul is blinded by this event.

00:06:18.270 --> 00:06:19.598
He's blind for three days.

00:06:19.598 --> 00:06:20.534
He goes to Damascus,

00:06:20.534 --> 00:06:23.951
and he is healed by a follower of Christ.

00:06:25.797 --> 00:06:28.820
At that moment, he completely does a 180.

00:06:28.820 --> 00:06:29.807
He goes from being one of

00:06:29.807 --> 00:06:31.581
the chief persecutors of the Christians,

00:06:31.581 --> 00:06:34.306
to being one of the chief believers,

00:06:34.306 --> 00:06:38.594
and he starts to aggressively
spread the faith.

00:06:38.594 --> 00:06:42.177
So that brings us, so
let's go to this map,

00:06:43.183 --> 00:06:46.516
because what's significant about Paul is

00:06:47.378 --> 00:06:49.446
Peter was, you can really view him

00:06:49.446 --> 00:06:53.584
as the first head of the
church, especially after Jesus,

00:06:53.584 --> 00:06:56.481
but he primarily viewed it as something

00:06:56.481 --> 00:06:58.354
to spread amongst the Jewish people.

00:06:58.354 --> 00:06:59.569
There are events, for example,

00:06:59.569 --> 00:07:00.755
in Acts where he does realize,

00:07:00.755 --> 00:07:02.729
hey, maybe I should be,

00:07:02.729 --> 00:07:05.791
the word of Jesus should go
beyond the Jewish people.

00:07:05.791 --> 00:07:07.075
It should go to the non-Jews,

00:07:07.075 --> 00:07:09.317
who are referred to as Gentiles.

00:07:09.317 --> 00:07:11.505
But Paul, and you can see on the map here,

00:07:11.505 --> 00:07:14.525
he has this revelation,
I guess you could say,

00:07:14.525 --> 00:07:16.980
as he's approaching Damascus.

00:07:16.980 --> 00:07:19.147
He goes back to Jerusalem,

00:07:20.267 --> 00:07:24.017
and he tries to convince
the other followers,

00:07:24.929 --> 00:07:26.355
and you can imagine
they're quite skeptical

00:07:26.355 --> 00:07:29.892
of this person who is shortly
before persecuting them.

00:07:29.892 --> 00:07:31.667
But he eventually convinces them,

00:07:31.667 --> 00:07:34.402
and he goes on multiple
missionary journeys,

00:07:34.402 --> 00:07:35.879
spreading the faith.

00:07:35.879 --> 00:07:38.767
His center of operations is at Antioch,

00:07:38.767 --> 00:07:39.902
and he's a significant figure

00:07:39.902 --> 00:07:42.451
who helps build the church at Antioch.

00:07:42.451 --> 00:07:44.863
What's interesting about
the Church of Antioch,

00:07:44.863 --> 00:07:47.062
it ends up being a very
significant church,

00:07:47.062 --> 00:07:51.229
is that at Antioch, the
Christians, or the Christian faith,

00:07:52.162 --> 00:07:54.277
is not just spread to the Jewish people,

00:07:54.277 --> 00:07:57.277
they're spread to the
non-Jews, to the Gentiles.

00:07:57.277 --> 00:07:59.124
It's actually at the Church of Antioch

00:07:59.124 --> 00:08:00.612
that the followers of Jesus Christ,

00:08:00.612 --> 00:08:02.447
according to Biblical accounts,

00:08:02.447 --> 00:08:06.101
were first referred to as Christians.

00:08:06.101 --> 00:08:09.122
What you see on this map, here,

00:08:09.122 --> 00:08:13.038
are the various missionary
journeys that Paul,

00:08:13.038 --> 00:08:15.186
who is a Roman citizen, was able,

00:08:15.186 --> 00:08:17.276
or Saul, depending on
which name you want to use,

00:08:17.276 --> 00:08:21.443
was able to do through, especially
the Eastern Roman Empire.

00:08:22.638 --> 00:08:25.055
In that mid 40s here, let me,

00:08:26.110 --> 00:08:28.686
in the mid 40s, right over here,

00:08:28.686 --> 00:08:31.610
he does a missionary journey.

00:08:31.610 --> 00:08:33.674
Once again, Antioch is his base,

00:08:33.674 --> 00:08:36.848
he goes to Cyprus, and southern, mid,

00:08:36.848 --> 00:08:39.711
what we would call Turkey today.

00:08:39.711 --> 00:08:43.561
In the early 50s, he goes
on this fairly long journey

00:08:43.561 --> 00:08:46.224
through modern day Turkey,

00:08:46.224 --> 00:08:49.638
through Macedonia, through Greece.

00:08:49.638 --> 00:08:53.524
As he's doing this, he's spreading
the word of Jesus Christ.

00:08:53.524 --> 00:08:55.451
He's starting to help establish churches

00:08:55.451 --> 00:08:57.199
in these significant places,

00:08:57.199 --> 00:08:59.438
Philippi, Thessalonica, Corinth,

00:08:59.438 --> 00:09:01.100
all of these various places.

00:09:01.100 --> 00:09:04.448
You see his third journey, here,

00:09:04.448 --> 00:09:08.615
very similar in path in
yellow to the second journey.

00:09:09.472 --> 00:09:12.198
He does that in the late 50s CE.

00:09:12.198 --> 00:09:16.389
Once again, he's constantly
establishing these churches.

00:09:16.389 --> 00:09:18.761
The important thing to realize about Paul,

00:09:18.761 --> 00:09:20.514
it's a controversy.

00:09:20.514 --> 00:09:23.561
To what degree should
the teachings of Jesus

00:09:23.561 --> 00:09:27.186
be spread to the Gentiles,
to the non-Jewish people?

00:09:27.186 --> 00:09:29.198
There are these famous debates,

00:09:29.198 --> 00:09:32.900
and Paul's view is, no, we
should not force the Gentiles

00:09:32.900 --> 00:09:35.686
to do everything that the Jews have to do,

00:09:35.686 --> 00:09:39.699
the circumcision, the Jewish dietary laws.

00:09:39.699 --> 00:09:44.546
You could imagine, this doubly
angers the church leaders.

00:09:44.546 --> 00:09:48.287
Not only is he spreading
the word of Jesus,

00:09:48.287 --> 00:09:50.714
that has annoyed them to no end,

00:09:50.714 --> 00:09:53.326
but he also wants to
spread it to the Gentiles,

00:09:53.326 --> 00:09:55.064
but he doesn't think that the Gentiles

00:09:55.064 --> 00:09:56.853
need to perform all of the things

00:09:56.853 --> 00:09:59.516
that the Jewish leaders
believe are dictated

00:09:59.516 --> 00:10:03.016
by the laws of Moses, by the Jewish faith.

00:10:05.238 --> 00:10:06.989
And circumcision is
actually a major debate.

00:10:06.989 --> 00:10:10.466
Circumcision, the dietary
habits, et cetera, et cetera.

00:10:10.466 --> 00:10:11.928
Eventually, he goes back

00:10:11.928 --> 00:10:15.491
to Jerusalem to face this leadership,

00:10:15.491 --> 00:10:18.241
and now this is in the early 60s,

00:10:19.202 --> 00:10:23.335
and they, once again,
they put him on trial.

00:10:23.335 --> 00:10:25.276
As a Jewish citizen, he says, hey,

00:10:25.276 --> 00:10:26.926
I have a right to see the Emperor.

00:10:26.926 --> 00:10:29.589
He is taken to Rome, and along the way

00:10:29.589 --> 00:10:32.275
they get caught in a storm,

00:10:32.275 --> 00:10:34.752
but eventually he makes his way to Rome,

00:10:34.752 --> 00:10:37.038
where he is held prisoner.

00:10:37.038 --> 00:10:41.205
It is believed that both Peter
and Paul were killed by Nero.

00:10:42.637 --> 00:10:44.389
We talk in previous videos,

00:10:44.389 --> 00:10:47.238
the fire in 64 CE in Rome that

00:10:47.238 --> 00:10:48.902
destroys a large chunk of the city.

00:10:48.902 --> 00:10:51.500
Some people think Nero
actually set the fire

00:10:51.500 --> 00:10:54.576
to clear some land so that
he could build his palace.

00:10:54.576 --> 00:10:57.751
He famously blames the fire on Christians.

00:10:57.751 --> 00:11:01.013
There's a large purge of
Christians that occur after that.

00:11:01.013 --> 00:11:04.013
A lot of accounts say that
Peter was crucified by Nero.

00:11:04.013 --> 00:11:06.012
He requests, according to these accounts,

00:11:06.012 --> 00:11:07.775
to be crucified upside down,

00:11:07.775 --> 00:11:09.023
because he doesn't deserve to get

00:11:09.023 --> 00:11:11.625
the same crucifixion as Jesus,

00:11:11.625 --> 00:11:14.825
and Paul, some accounts
say, that he was beheaded.

00:11:14.825 --> 00:11:16.323
Once again, they're dying in these

00:11:16.323 --> 00:11:20.198
Christian purges that Nero is performing.

00:11:20.198 --> 00:11:24.378
You can see it right over here,
if those accounts are true.

00:11:24.378 --> 00:11:25.526
Now it's really interesting,

00:11:25.526 --> 00:11:26.817
I already talked about Paul

00:11:26.817 --> 00:11:29.367
being this missionary to the Gentiles.

00:11:29.367 --> 00:11:31.493
He is the one, especially
at the Church of Antioch,

00:11:31.493 --> 00:11:35.541
where Christianity starts to
really separate from Judaism,

00:11:35.541 --> 00:11:37.929
starts to become a
faith in it's own right.

00:11:37.929 --> 00:11:39.417
That was even further accelerated

00:11:39.417 --> 00:11:41.865
in a few years after their death,

00:11:41.865 --> 00:11:44.380
or it might have been roughly
coincident with their death,

00:11:44.380 --> 00:11:45.802
depending on when their deaths were.

00:11:45.802 --> 00:11:49.552
From 66 to 70, you have
the Jewish rebellion,

00:11:52.616 --> 00:11:55.227
sometimes called the
First Roman-Jewish War,

00:11:55.227 --> 00:11:58.627
and it ends with the
Roman Empire destroying

00:11:58.627 --> 00:12:01.478
the temple at Jerusalem, the
center of the Jewish faith.

00:12:01.478 --> 00:12:02.967
The second temple of Jerusalem,

00:12:02.967 --> 00:12:07.404
the one that was rebuilt,
partially, by Cyrus the Great.

00:12:07.404 --> 00:12:10.517
So this causes the Jewish
people to have to leave,

00:12:10.517 --> 00:12:14.684
and they no longer have
this center of their people.

00:12:15.591 --> 00:12:17.541
You have the Jewish diaspora that

00:12:17.541 --> 00:12:19.217
gets spread throughout the Roman Empire,

00:12:19.217 --> 00:12:23.129
and other empires, like
the Parthians in Persia.

00:12:23.129 --> 00:12:26.931
They really, there's a famous
battle in 73 of Masada,

00:12:26.931 --> 00:12:29.914
this mountain fortress, where the Romans,

00:12:29.914 --> 00:12:31.978
essentially, well, it's a mass suicide,

00:12:31.978 --> 00:12:34.405
but it's this bloody extermination of

00:12:34.405 --> 00:12:37.866
the Jewish people in and around Jerusalem,

00:12:37.866 --> 00:12:39.143
so they are spread.

00:12:39.143 --> 00:12:41.204
That's, obviously, a significant series

00:12:41.204 --> 00:12:43.026
of events for the Jewish people,

00:12:43.026 --> 00:12:45.416
but it's also significant
for Christianity,

00:12:45.416 --> 00:12:49.043
because Jerusalem was a
center of the Christian faith.

00:12:49.043 --> 00:12:51.753
Now the Christian followers
are going to be spread around,

00:12:51.753 --> 00:12:55.212
and many of these other places
that Paul spread the Gospel,

00:12:55.212 --> 00:12:58.229
becomes centers of Christianity.

00:12:58.229 --> 00:13:01.844
Now to emphasize the importance
of Paul to Christianity,

00:13:01.844 --> 00:13:05.580
this is a listing of
the books of the Bible.

00:13:05.580 --> 00:13:07.002
Jesus, his life and teachings,

00:13:07.002 --> 00:13:10.182
is primarily coming from
the Gospels, right here.

00:13:10.182 --> 00:13:11.655
Most of what I told you in this video,

00:13:11.655 --> 00:13:15.105
this is excerpts from the fifth
book of the New Testament,

00:13:15.105 --> 00:13:17.229
Acts of the Apostles.

00:13:17.229 --> 00:13:21.431
Then, most modern orderings of the Bible,

00:13:21.431 --> 00:13:23.381
the next several, or many books,

00:13:23.381 --> 00:13:24.815
are called epistles, or letters.

00:13:24.815 --> 00:13:27.259
They're primarily the epistles of Paul,

00:13:27.259 --> 00:13:29.672
letters that he wrote to his
followers in different places.

00:13:29.672 --> 00:13:31.382
You can see most of these names,

00:13:31.382 --> 00:13:35.034
these are cities that you see
where Paul was a missionary.

00:13:35.034 --> 00:13:36.957
Corinthians, Corinth is right here,

00:13:36.957 --> 00:13:39.384
these are letters that Paul is
sending to the church there.

00:13:39.384 --> 00:13:40.534
Galatians, you see Galatia

00:13:40.534 --> 00:13:42.947
right over here in
central modern day Turkey.

00:13:42.947 --> 00:13:44.997
Ephesians in Ephesus,

00:13:44.997 --> 00:13:47.860
Philippians in Philippi, right over here,

00:13:47.860 --> 00:13:49.507
Colossians in Colossae,

00:13:49.507 --> 00:13:53.424
Thessalonians in
Thessalonica, right over here.

00:13:54.332 --> 00:13:56.447
This is a major chunk of the Bible

00:13:56.447 --> 00:13:58.933
that is either written by Paul,

00:13:58.933 --> 00:14:01.472
or some people think
it was written by Paul,

00:14:01.472 --> 00:14:03.558
or is ascribed to Paul.

00:14:03.558 --> 00:14:05.783
What I have here in this deeper orange,

00:14:05.783 --> 00:14:07.520
these are the books that historians

00:14:07.520 --> 00:14:09.922
think were actually written by Paul.

00:14:09.922 --> 00:14:12.059
This lighter orange,
there are some debate,

00:14:12.059 --> 00:14:13.547
and the yellow ones are the ones

00:14:13.547 --> 00:14:16.721
where people think it probably
wasn't written by Paul.

00:14:16.721 --> 00:14:17.632
As you can imagine, this is

00:14:17.632 --> 00:14:20.259
a significant chunk of the New Testament

00:14:20.259 --> 00:14:24.645
that was actually written
or ascribed to Paul.

00:14:24.645 --> 00:14:27.072
He was really the significant missionary

00:14:27.072 --> 00:14:31.239
that created the Christian
faith beyond the Jewish people.


The Jacobian Determinant

WEBVTT
Kind: captions
Language: en

00:00:00.351 --> 00:00:01.563
- [Voiceover] In this
video, I want to talk about

00:00:01.563 --> 00:00:04.192
something called the Jacobian determinant.

00:00:04.192 --> 00:00:05.861
And it's, more or less,
just what it sounds like.

00:00:05.861 --> 00:00:08.210
It's the determinant
of the Jacobian matrix

00:00:08.210 --> 00:00:09.484
that I've been talking to you

00:00:09.484 --> 00:00:11.428
the last couple videos about.

00:00:11.428 --> 00:00:12.467
And before we jump into it,

00:00:12.467 --> 00:00:13.587
I just want to give a quick review

00:00:13.587 --> 00:00:15.710
of how you think about
the determinant itself,

00:00:15.710 --> 00:00:18.568
just in an ordinary
linear algebra context.

00:00:18.568 --> 00:00:23.532
So if I'm taking the determinant
of some kind of matrix,

00:00:23.532 --> 00:00:26.282
let's say, three, zero, one, two,

00:00:28.605 --> 00:00:30.312
something like this,

00:00:30.312 --> 00:00:31.775
to compute the determinant,

00:00:31.775 --> 00:00:33.916
you take these diagonal terms here,

00:00:33.916 --> 00:00:37.995
so you take three multiplied by that two,

00:00:37.995 --> 00:00:40.704
and then you subtract
off the other diagonal,

00:00:40.704 --> 00:00:43.699
subtract off one multiplied by zero.

00:00:43.699 --> 00:00:47.097
And in this case, that evaluates to six.

00:00:47.097 --> 00:00:47.930
But there is, of course,

00:00:47.930 --> 00:00:50.435
much more than just a
computation going on here.

00:00:50.435 --> 00:00:52.674
There's a really nice geometric intuition.

00:00:52.674 --> 00:00:55.721
Namely, if we think of this
matrix, three, zero, one, two,

00:00:55.721 --> 00:00:57.488
as a linear transformation,

00:00:57.488 --> 00:01:00.041
as something that's gonna
take this first basis vector

00:01:00.041 --> 00:01:02.003
over to the coordinates three, zero,

00:01:02.003 --> 00:01:03.321
and that second basis vector

00:01:03.321 --> 00:01:05.270
over to the coordinates one, two,

00:01:05.270 --> 00:01:06.908
you know, thinking about the columns,

00:01:06.908 --> 00:01:09.203
you can think of the
determinant as measuring

00:01:09.203 --> 00:01:13.655
how much this transformation
stretches or squishes space.

00:01:13.655 --> 00:01:14.913
And in particular,

00:01:14.913 --> 00:01:17.800
you'll notice how I have this
yellow region highlighted,

00:01:17.800 --> 00:01:20.689
and this region starts
off as the unit square,

00:01:20.689 --> 00:01:23.610
a square with side lengths
one so its area is one.

00:01:23.610 --> 00:01:25.694
And there's nothing special
about this particular region.

00:01:25.694 --> 00:01:28.138
It's just nice as a canonical shape,

00:01:28.138 --> 00:01:29.698
with an area of one,

00:01:29.698 --> 00:01:31.131
so that we can compare it to what happens

00:01:31.131 --> 00:01:32.441
after the transformation.

00:01:32.441 --> 00:01:35.368
Ask, how much does that
area get stretched out?

00:01:35.368 --> 00:01:36.378
And the answer is,

00:01:36.378 --> 00:01:39.759
it gets stretched out by a
factor of the determinant.

00:01:39.759 --> 00:01:41.767
That's kind of what the determinant means,

00:01:41.767 --> 00:01:44.409
is that all areas, if you were
to draw up any kind of shape,

00:01:44.409 --> 00:01:45.796
not just that one square,

00:01:45.796 --> 00:01:48.297
are gonna get stretched
out by a factor of six.

00:01:48.297 --> 00:01:49.400
And we can actually verify,

00:01:49.400 --> 00:01:52.424
looking at this parallelogram
that the square turned into.

00:01:52.424 --> 00:01:54.381
It has a base of three

00:01:54.381 --> 00:01:56.099
and then the height is two.

00:01:56.099 --> 00:01:57.833
And three times two is six.

00:01:57.833 --> 00:01:59.832
And that has everything
to do with the fact that

00:01:59.832 --> 00:02:03.443
this three showed up here
and this two showed up there.

00:02:03.443 --> 00:02:05.003
So now, let's think about
what this might mean

00:02:05.003 --> 00:02:07.152
in the context of what
I've been describing

00:02:07.152 --> 00:02:08.789
in the last couple videos.

00:02:08.789 --> 00:02:11.500
And if you'll remember, we
had a multivariable function,

00:02:11.500 --> 00:02:15.623
something that you can write out as f one

00:02:15.623 --> 00:02:17.013
with two inputs

00:02:17.013 --> 00:02:18.232
and then the second component,

00:02:18.232 --> 00:02:20.565
f two, also with two inputs.

00:02:22.551 --> 00:02:24.118
And the function that I was looking at,

00:02:24.118 --> 00:02:27.373
that we were kind of analyzing
to learn about the Jacobian,

00:02:27.373 --> 00:02:30.853
had the first component, x plus sine of y,

00:02:30.853 --> 00:02:31.979
x plus sine y,

00:02:31.979 --> 00:02:36.146
and the second component
was y plus the sine of x.

00:02:37.528 --> 00:02:40.330
And the idea was that this
function is not at all linear.

00:02:40.330 --> 00:02:43.078
It's gonna make everything
very curvy and complicated.

00:02:43.078 --> 00:02:45.684
However, if we zoom in
around a particular region,

00:02:45.684 --> 00:02:49.069
which is what this outer yellow
box represents, zooming in,

00:02:49.069 --> 00:02:51.133
it will look like a linear transformation.

00:02:51.133 --> 00:02:53.112
In fact, I can kind of play this forward,

00:02:53.112 --> 00:02:55.405
and we see that even
though everything is crazy,

00:02:55.405 --> 00:02:56.850
inside that zoomed in version,

00:02:56.850 --> 00:02:59.821
things loosely look
like a linear function.

00:02:59.821 --> 00:03:03.248
And you'll notice I have this
inner yellow box highlighted,

00:03:03.248 --> 00:03:05.684
and this yellow box inside
corresponds to the unit square

00:03:05.684 --> 00:03:07.546
that I was showing in the last animation.

00:03:07.546 --> 00:03:10.415
And again, it's just a
placeholder as something to watch

00:03:10.415 --> 00:03:14.136
to see how much the area of
any kind of blob in that region

00:03:14.136 --> 00:03:15.920
gets stretched.

00:03:15.920 --> 00:03:16.940
So, in this particular case,

00:03:16.940 --> 00:03:18.368
when you play out the animation,

00:03:18.368 --> 00:03:20.134
areas don't really change that much.

00:03:20.134 --> 00:03:21.705
They get stretched out a little bit,

00:03:21.705 --> 00:03:24.233
but it's not that dramatic.

00:03:24.233 --> 00:03:27.162
So, if we know the matrix that
describes the transformation

00:03:27.162 --> 00:03:29.083
that this looks like zoomed in,

00:03:29.083 --> 00:03:31.100
the determinant of that
matrix will tell us

00:03:31.100 --> 00:03:34.274
the factor by which areas
tend to get stretched out.

00:03:34.274 --> 00:03:36.990
And in particular, you can
think of this little yellow box

00:03:36.990 --> 00:03:39.495
and the factor by which it gets stretched.

00:03:39.495 --> 00:03:40.328
And as a reminder,

00:03:40.328 --> 00:03:43.177
the matrix describing that
zoomed in transformation

00:03:43.177 --> 00:03:44.412
is the Jacobian.

00:03:44.412 --> 00:03:46.306
It is this thing that kind of holds

00:03:46.306 --> 00:03:48.860
all of the partial
differential information.

00:03:48.860 --> 00:03:51.854
You take the partial derivative of f,

00:03:51.854 --> 00:03:53.173
with respect to x,

00:03:53.173 --> 00:03:55.819
sorry, partial of f one
of that first component,

00:03:55.819 --> 00:03:59.279
and then the partial derivative
of the second component,

00:03:59.279 --> 00:04:01.189
with respect to x,

00:04:01.189 --> 00:04:03.040
and then on the other column,

00:04:03.040 --> 00:04:06.020
we have the partial derivative
of that first component,

00:04:06.020 --> 00:04:08.046
with respect to y,

00:04:08.046 --> 00:04:11.121
and the partial derivative
of that second component,

00:04:11.121 --> 00:04:12.621
with respect to y.

00:04:13.496 --> 00:04:14.622
And if you...

00:04:14.622 --> 00:04:16.962
Let's see, I'm gonna close this off.

00:04:16.962 --> 00:04:18.529
Close off this matrix.

00:04:18.529 --> 00:04:21.136
And if you evaluate each one
of these partial derivatives

00:04:21.136 --> 00:04:22.410
at a particular point,

00:04:22.410 --> 00:04:25.083
at whatever point we happen to zoom in on,

00:04:25.083 --> 00:04:27.330
in this case, it was negative two, one,

00:04:27.330 --> 00:04:28.726
once you plug that into all of these,

00:04:28.726 --> 00:04:32.123
you get some matrix that's
just full of numbers.

00:04:32.123 --> 00:04:34.096
And what turns out to
be a very useful thing

00:04:34.096 --> 00:04:36.396
later on in multivariable calc concepts,

00:04:36.396 --> 00:04:39.130
is to take the determinant of that matrix,

00:04:39.130 --> 00:04:41.696
to kind of analyze how much
space is getting stretched

00:04:41.696 --> 00:04:44.285
or squished in that region.

00:04:44.285 --> 00:04:45.704
So in the last video, we worked this out

00:04:45.704 --> 00:04:47.900
for this specific example here,

00:04:47.900 --> 00:04:51.214
where that top left function
turned out just to be

00:04:51.214 --> 00:04:52.795
the constant function, one, right,

00:04:52.795 --> 00:04:54.923
because we were taking the
partial derivative of this guy

00:04:54.923 --> 00:04:56.986
with respect to x and that was one.

00:04:56.986 --> 00:04:59.859
And likewise, in the bottom right,

00:04:59.859 --> 00:05:02.168
that was also a constant function of one.

00:05:02.168 --> 00:05:04.515
And then the others were cosine functions.

00:05:04.515 --> 00:05:06.778
This one was cosine x

00:05:06.778 --> 00:05:08.361
because we were taking
the partial derivative

00:05:08.361 --> 00:05:11.678
of this second component
here with respect to x.

00:05:11.678 --> 00:05:13.818
And then the top right of our matrix

00:05:13.818 --> 00:05:15.151
was cosine of y.

00:05:17.058 --> 00:05:18.869
And these are, in general,
functions of x and y

00:05:18.869 --> 00:05:20.351
because you know, you're gonna plug in

00:05:20.351 --> 00:05:22.765
whatever the input point
you're zooming in on.

00:05:22.765 --> 00:05:24.304
And when we're thinking
about the determinant here,

00:05:24.304 --> 00:05:26.137
let's just go ahead and
take the determinant

00:05:26.137 --> 00:05:28.808
in this form, in the form as a function.

00:05:28.808 --> 00:05:32.488
So I'm going to ask about the
determinant of this matrix,

00:05:32.488 --> 00:05:36.065
or maybe you think of it as
a matrix-valued function.

00:05:36.065 --> 00:05:37.678
And in this case, we do the same thing.

00:05:37.678 --> 00:05:39.919
I mean, procedurally, you know
how to take a determinant.

00:05:39.919 --> 00:05:41.765
We take these diagonals,

00:05:41.765 --> 00:05:45.020
so that's just gonna be one times one,

00:05:45.020 --> 00:05:48.522
and then we subtract off the
product of the other diagonal,

00:05:48.522 --> 00:05:50.522
subtract off cosine of x

00:05:52.504 --> 00:05:54.671
multiplied by cosine of y.

00:05:57.531 --> 00:05:59.239
And as an example, let's
plug in this point here

00:05:59.239 --> 00:06:01.426
that we're zooming in
on, negative two, one.

00:06:01.426 --> 00:06:04.580
So I'm going to plug in x
is equal to negative two,

00:06:04.580 --> 00:06:07.140
and y is equal to one.

00:06:07.140 --> 00:06:09.518
And when you plug in
cosine of negative two,

00:06:09.518 --> 00:06:13.685
that's gonna come out to be
approximately negative 0.42.

00:06:16.387 --> 00:06:17.961
And when you plug in cosine of y,

00:06:17.961 --> 00:06:20.445
cosine of one in this case,

00:06:20.445 --> 00:06:23.695
that's gonna come out to be about 0.54.

00:06:27.289 --> 00:06:28.613
And when we multiply those,

00:06:28.613 --> 00:06:31.805
when we take one minus
the product of those,

00:06:31.805 --> 00:06:34.722
it's gonna be about negative 0.227.

00:06:37.871 --> 00:06:39.499
And that's all stuff that you
can plug into your calculator

00:06:39.499 --> 00:06:40.572
if you want.

00:06:40.572 --> 00:06:42.767
And what that means is
that the total determinant,

00:06:42.767 --> 00:06:45.423
evaluated at that point,
the Jacobian determinant

00:06:45.423 --> 00:06:48.006
at the point negative two, one,

00:06:49.623 --> 00:06:50.873
is about 1.227.

00:06:53.460 --> 00:06:56.334
So that's telling you that
areas tend to get stretched out

00:06:56.334 --> 00:06:58.932
by this factor around that point.

00:06:58.932 --> 00:07:00.345
And that kind of lines
up with what we see.

00:07:00.345 --> 00:07:03.750
We see that areas get stretched
out maybe a little bit,

00:07:03.750 --> 00:07:05.045
but not that much, right?

00:07:05.045 --> 00:07:07.596
It's only by a factor of about 1.2.

00:07:07.596 --> 00:07:09.442
And now, let's contrast this.

00:07:09.442 --> 00:07:12.007
If instead we zoom in at the
point where x is equal to zero

00:07:12.007 --> 00:07:13.418
and y is equal to one,

00:07:13.418 --> 00:07:15.838
so I'm gonna go over here
and all I'm gonna change,

00:07:15.838 --> 00:07:18.195
all I'm gonna change is
that x is equal to zero

00:07:18.195 --> 00:07:19.936
and y will still equal one,

00:07:19.936 --> 00:07:22.375
and what that means is that cosine of x,

00:07:22.375 --> 00:07:25.626
instead of being negative 0.42,

00:07:25.626 --> 00:07:27.658
well what's cosine of zero,

00:07:27.658 --> 00:07:31.586
that's actually precisely
equal to one, right?

00:07:31.586 --> 00:07:33.445
We don't have to approximate on this one,

00:07:33.445 --> 00:07:37.253
which means when we multiply
them, one times 0.54,

00:07:37.253 --> 00:07:41.724
well that, that's gonna
now be about 0.54, right?

00:07:41.724 --> 00:07:44.759
So this one, once we actually
perform the subtraction,

00:07:44.759 --> 00:07:48.074
instead when you take one minus 0.54,

00:07:48.074 --> 00:07:50.241
that's gonna give us 0.46.

00:07:52.880 --> 00:07:55.777
So even before watching,
because this determinant

00:07:55.777 --> 00:08:00.462
of the Jacobian around the point
zero, one is less than one,

00:08:00.462 --> 00:08:02.435
this is telling us we should expect areas

00:08:02.435 --> 00:08:03.940
to get squished down.

00:08:03.940 --> 00:08:07.401
Precisely, they should be
squished by a factor of 0.46.

00:08:07.401 --> 00:08:08.759
And let's see if this looks right, right?

00:08:08.759 --> 00:08:11.108
We're looking at the zoomed
in version around that point,

00:08:11.108 --> 00:08:14.814
and areas should tend
to contract around that.

00:08:14.814 --> 00:08:16.126
And indeed, they do.

00:08:16.126 --> 00:08:17.484
You see it got squished down,

00:08:17.484 --> 00:08:18.940
it looks like by a fair bit,

00:08:18.940 --> 00:08:19.881
and from our calculation,

00:08:19.881 --> 00:08:22.289
we can conclude that they
got scaled down precisely

00:08:22.289 --> 00:08:24.503
by a factor of 0.46.

00:08:24.503 --> 00:08:27.032
That's what the determinant means.

00:08:27.032 --> 00:08:29.022
So like I said, this is
actually a very nice notion

00:08:29.022 --> 00:08:30.722
throughout multivariable calculus,

00:08:30.722 --> 00:08:33.033
is that you look at a tiny
little local neighborhood

00:08:33.033 --> 00:08:34.396
around a point,

00:08:34.396 --> 00:08:35.932
and if you just want to
get a general feel for,

00:08:35.932 --> 00:08:37.730
does this function, as a transformation,

00:08:37.730 --> 00:08:40.830
tend to stretch out that region
or to squish it together,

00:08:40.830 --> 00:08:43.972
how much do areas change in
that little neighborhood,

00:08:43.972 --> 00:08:47.048
that's exactly what this
Jacobian determinant is,

00:08:47.048 --> 00:08:49.131
you know, built to solve.

00:08:50.334 --> 00:08:52.430
So with that, I'll see
you guys next video.


Computing a Jacobian matrix

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.289
- [Teacher] So, just as a
reminder of where we are,

00:00:02.289 --> 00:00:04.571
we've got this very
non-linear transformation

00:00:04.571 --> 00:00:07.558
and we showed that if you zoom
in on a specific point while

00:00:07.558 --> 00:00:09.046
that transformation is happening,

00:00:09.046 --> 00:00:10.968
it looks a lot like something linear

00:00:10.968 --> 00:00:12.772
and we reason that you can figure out

00:00:12.772 --> 00:00:14.854
what linear transformation that looks like

00:00:14.854 --> 00:00:17.922
by taking the partial derivatives
of your given function,

00:00:17.922 --> 00:00:19.897
the one that I defined up here,

00:00:19.897 --> 00:00:22.514
and then turning that into a matrix.

00:00:22.514 --> 00:00:24.599
And what I want to do here
is basically just finish up

00:00:24.599 --> 00:00:26.315
what I was talking about by computing all

00:00:26.315 --> 00:00:27.802
of those partial derivatives.

00:00:27.802 --> 00:00:30.298
So, first of all, let me
just rewrite the function

00:00:30.298 --> 00:00:31.553
back on the screen so we have it

00:00:31.553 --> 00:00:33.598
in a convenient place to look at.

00:00:33.598 --> 00:00:37.884
The first component is x plus sin of y.

00:00:37.884 --> 00:00:40.217
Sin of y and then y plus sin

00:00:41.823 --> 00:00:45.097
of x was the second component.

00:00:45.097 --> 00:00:47.390
So, what I want to do
here is just compute all

00:00:47.390 --> 00:00:48.819
of those partial derivatives to show

00:00:48.819 --> 00:00:50.627
what kind of thing this looks like.

00:00:50.627 --> 00:00:54.270
So, let's go ahead and
get rid of this word

00:00:54.270 --> 00:00:58.401
and I'll go ahead and kind
of redraw the matrix here.

00:00:58.401 --> 00:00:59.868
So, for that upper left component,

00:00:59.868 --> 00:01:01.723
we're taking the partial derivative

00:01:01.723 --> 00:01:04.231
with respect to x of the first component.

00:01:04.231 --> 00:01:05.799
So, we look up at this first component

00:01:05.799 --> 00:01:09.335
and the partial derivative
with respect to x is just one.

00:01:09.335 --> 00:01:12.049
Since there's one times x plus
something that has nothing

00:01:12.049 --> 00:01:14.005
to do with x and then below that,

00:01:14.005 --> 00:01:15.064
we take the partial derivative

00:01:15.064 --> 00:01:17.829
of the second component
with respect to x down here.

00:01:17.829 --> 00:01:20.593
And that guy, the y, well
that looks like a constant

00:01:20.593 --> 00:01:22.995
so nothing happens, and
the derivative of sin

00:01:22.995 --> 00:01:25.328
of x becomes cosine of x.

00:01:28.144 --> 00:01:30.563
And then up here, we're
taking the partial derivative

00:01:30.563 --> 00:01:32.826
with respect to y of the first component;

00:01:32.826 --> 00:01:34.739
that upper one here, and for that,

00:01:34.739 --> 00:01:37.779
partial derivative of x,
with respect to y, is zero

00:01:37.779 --> 00:01:39.626
and partial derivative of sin of y,

00:01:39.626 --> 00:01:42.459
with respect to y, is cosine of y.

00:01:43.379 --> 00:01:46.343
And then, finally, the partial derivative

00:01:46.343 --> 00:01:50.506
of the second component with
respect to y looks like one

00:01:50.506 --> 00:01:54.151
because it's just one
times y plus some constant.

00:01:54.151 --> 00:01:57.461
And this is the general Jacobian
as a function of x and y,

00:01:57.461 --> 00:01:58.986
but if we want to understand what happens

00:01:58.986 --> 00:02:01.383
around this specific
point that started off

00:02:01.383 --> 00:02:04.110
at, well, I think I recorded
it here at negative two, one,

00:02:04.110 --> 00:02:06.779
we plug that in to each
one of these values.

00:02:06.779 --> 00:02:10.367
So, when we plug in negative two, one.

00:02:10.367 --> 00:02:12.409
So go ahead and just
kind of again, rewrite it

00:02:12.409 --> 00:02:15.979
to remember we're plugging
in negative two, one

00:02:15.979 --> 00:02:20.133
as our specific point,
that matrix as a function,

00:02:20.133 --> 00:02:22.257
kind of a matrix valued function,

00:02:22.257 --> 00:02:25.307
becomes one, and then next we have cosine,

00:02:25.307 --> 00:02:27.705
but we're plugging in negative two for x,

00:02:27.705 --> 00:02:30.036
cosine of negative two,
and if you're curious,

00:02:30.036 --> 00:02:32.580
that is approximately equal to,

00:02:32.580 --> 00:02:34.249
I calculated this earlier.

00:02:34.249 --> 00:02:38.158
Negative zero point four
two, if you just want

00:02:38.158 --> 00:02:40.758
to think in terms of a number there.

00:02:40.758 --> 00:02:42.993
Then for the upper right,
we have cosine again,

00:02:42.993 --> 00:02:45.118
but now we're plugging in the value for y,

00:02:45.118 --> 00:02:48.550
which is one and cosine of
one is approximately equal

00:02:48.550 --> 00:02:52.467
to zero point five four;
and then bottom right,

00:02:53.459 --> 00:02:56.212
that's just another constant: one.

00:02:56.212 --> 00:03:00.250
So, that is the matrix, just
as a matrix full of numbers,

00:03:00.250 --> 00:03:02.361
and just as kind of a gut
check we can take a look

00:03:02.361 --> 00:03:05.148
at the linear transformation
this was supposed to look like,

00:03:05.148 --> 00:03:07.008
and notice how the first basis factor,

00:03:07.008 --> 00:03:10.615
the thing it got turned into,
which is this factor here,

00:03:10.615 --> 00:03:13.182
does look like it has coordinates one

00:03:13.182 --> 00:03:15.210
and negative zero point four two, right?

00:03:15.210 --> 00:03:16.669
It's got this rightward component

00:03:16.669 --> 00:03:18.792
that's about as long as
the vector itself started

00:03:18.792 --> 00:03:20.188
and then this downward component,

00:03:20.188 --> 00:03:21.920
which I think that's pretty believable

00:03:21.920 --> 00:03:24.523
that that's negative zero point four two.

00:03:24.523 --> 00:03:26.821
And then, likewise, this
second column is telling us

00:03:26.821 --> 00:03:29.120
what happened to that second basis factor,

00:03:29.120 --> 00:03:32.115
which is the one that looks like this.

00:03:32.115 --> 00:03:35.529
And again, its y component
is about as long as

00:03:35.529 --> 00:03:37.982
how it started, right, the length of one.

00:03:37.982 --> 00:03:40.833
And then the rightward component
is around half of that,

00:03:40.833 --> 00:03:42.663
and we actually see that in the diagram,

00:03:42.663 --> 00:03:43.847
but this is something you compute.

00:03:43.847 --> 00:03:44.801
Again, it's pretty straightforward.

00:03:44.801 --> 00:03:47.696
You just take all of the
possible partial derivatives,

00:03:47.696 --> 00:03:50.172
and you organize them
into a grid like this.

00:03:50.172 --> 00:03:52.405
So, with that, I'll see
you guys next video.


Addressing treating differentials algebraically

WEBVTT
Kind: captions
Language: en

00:00:00.419 --> 00:00:01.732
- [Instructor] So when
you first learn calculus,

00:00:01.732 --> 00:00:04.354
you learn that the derivative
of some function f,

00:00:04.354 --> 00:00:07.058
could be written as f prime of x is equal

00:00:07.058 --> 00:00:09.530
to the limit as, then
there's multiple ways

00:00:09.530 --> 00:00:12.347
of doing this, the change
in x approaches zero

00:00:12.347 --> 00:00:16.347
of f of x plus our change
in x, minus f of x,

00:00:18.569 --> 00:00:20.319
over our change in x.

00:00:21.841 --> 00:00:24.175
And you learn multiple notations for this.

00:00:24.175 --> 00:00:28.397
For example, if you
that y is equal f of x,

00:00:28.397 --> 00:00:31.502
you might write this as y prime.

00:00:31.502 --> 00:00:34.252
You might write this as d-y, d-x,

00:00:35.388 --> 00:00:37.886
which you'll often hear me
say is the derivative of y,

00:00:37.886 --> 00:00:40.149
with respect to x, and that you can do

00:00:40.149 --> 00:00:41.955
the derivative of f with respect to x

00:00:41.955 --> 00:00:44.906
because y is equal to our function.

00:00:44.906 --> 00:00:46.512
But then later on when
you, especially when you

00:00:46.512 --> 00:00:49.051
start getting into differential equations,

00:00:49.051 --> 00:00:51.987
you see people start
to treat this notation

00:00:51.987 --> 00:00:54.911
as an actual algebraic expression.

00:00:54.911 --> 00:00:59.357
For example, you will learn or
you might have already seen.

00:00:59.357 --> 00:01:02.285
If you're trying to solve
the differential equation,

00:01:02.285 --> 00:01:06.691
the derivative of y with
respect to x, is equal to y.

00:01:06.691 --> 00:01:08.551
So the rate of change
of y with respect to x

00:01:08.551 --> 00:01:11.533
is equal to the value of y itself.

00:01:11.533 --> 00:01:14.733
This is one of, the most
basic differential equations

00:01:14.733 --> 00:01:15.903
you might see.

00:01:15.903 --> 00:01:17.771
You'll see this technique,
where people just say,

00:01:17.771 --> 00:01:20.351
"Well, let's just multiply
both sides by d-x."

00:01:20.351 --> 00:01:24.228
Just treating d-x like as if
it's some algebraic expression.

00:01:24.228 --> 00:01:28.318
So you multiply both sides
by d-x and then you have,

00:01:28.318 --> 00:01:30.517
so that would cancel out algebraically,

00:01:30.517 --> 00:01:32.232
and so you see people treat it like that.

00:01:32.232 --> 00:01:35.538
So you have d-y is equal to y times d-x,

00:01:35.538 --> 00:01:38.240
and then they'll say, "Okay
let's divide both sides by y."

00:01:38.240 --> 00:01:39.530
Which is a reasonable thing to do.

00:01:39.530 --> 00:01:41.148
Y is an algebraic expression.

00:01:41.148 --> 00:01:46.085
So if you divide both sides
by y, you get one over y,

00:01:46.085 --> 00:01:47.752
d-y is equal to d-x.

00:01:49.425 --> 00:01:51.529
And then folks will integrate both sides

00:01:51.529 --> 00:01:54.416
to find a general solution to
this differential equation.

00:01:54.416 --> 00:01:56.477
But my point on this video isn't to think

00:01:56.477 --> 00:01:59.237
about how do you solve a
differential equation here,

00:01:59.237 --> 00:02:01.671
but to think about this notion of using,

00:02:01.671 --> 00:02:03.780
what we call differentials.

00:02:03.780 --> 00:02:06.123
So a d-x, or a d-y,

00:02:06.123 --> 00:02:08.333
and treating them algebraically like this.

00:02:08.333 --> 00:02:10.026
Treating them as algebraic expressions,

00:02:10.026 --> 00:02:13.114
where I can just multiply
both sides by just d-x or d-y,

00:02:13.114 --> 00:02:16.327
or divide both sides by d-x or d-y.

00:02:16.327 --> 00:02:21.204
And I don't normally say
this, but the rigor you need

00:02:21.204 --> 00:02:24.153
to show that this is
okay in this situation,

00:02:24.153 --> 00:02:27.282
is not an easy thing to say.

00:02:27.282 --> 00:02:31.069
And so to just feel reasonably
okay about doing this,

00:02:31.069 --> 00:02:32.846
this is a little bit hand wavy,

00:02:32.846 --> 00:02:35.008
it's not super mathematically rigorous.

00:02:35.008 --> 00:02:37.822
But it has proven to be
a useful tool for us,

00:02:37.822 --> 00:02:40.126
to find these solutions.

00:02:40.126 --> 00:02:43.944
And conceptually the way
that I think about a d-y, or

00:02:43.944 --> 00:02:47.745
a d-x, is this is the
super small change in y,

00:02:47.745 --> 00:02:50.590
in response to a super small change in x.

00:02:50.590 --> 00:02:52.983
And that's essentially
what this definition

00:02:52.983 --> 00:02:54.216
of the limit is telling us.

00:02:54.216 --> 00:02:56.978
Especially as Delta x approaches zero,

00:02:56.978 --> 00:02:59.679
we're going to have a
super small change in x,

00:02:59.679 --> 00:03:01.603
as Delta x approaches zero.

00:03:01.603 --> 00:03:02.984
And then we're going to have a resulting

00:03:02.984 --> 00:03:05.327
super small change in y.

00:03:05.327 --> 00:03:07.169
So that's one way that you can feel

00:03:07.169 --> 00:03:09.157
a little bit better, of...

00:03:09.157 --> 00:03:11.136
And this is actually one
of the justifications

00:03:11.136 --> 00:03:12.941
for this type of notation.

00:03:12.941 --> 00:03:15.533
As you can view this, what's
the resulting super small,

00:03:15.533 --> 00:03:17.349
or what's the super small change in y,

00:03:17.349 --> 00:03:19.718
for a given super small change in x?

00:03:19.718 --> 00:03:22.886
Which is giving us the sense
of what's the limiting value

00:03:22.886 --> 00:03:24.507
of the slope?

00:03:24.507 --> 00:03:25.695
As we go from the slope

00:03:25.695 --> 00:03:27.561
of the secant line to a tangent line?

00:03:27.561 --> 00:03:29.188
And if you view it that way you might feel

00:03:29.188 --> 00:03:33.104
a little bit better about
using the differentials,

00:03:33.104 --> 00:03:34.559
or creating them algebraically.

00:03:34.559 --> 00:03:36.292
Let me just multiply both sides by

00:03:36.292 --> 00:03:39.372
that super small change in x.

00:03:39.372 --> 00:03:41.800
So the big picture is, this is a technique

00:03:41.800 --> 00:03:43.469
that you'll often see,

00:03:43.469 --> 00:03:47.087
in introductory differential
equations classes.

00:03:47.087 --> 00:03:49.303
Introductory multi-variable classes and

00:03:49.303 --> 00:03:51.127
introductory calculus classes.

00:03:51.127 --> 00:03:53.573
But it's not very mathematically rigorous,

00:03:53.573 --> 00:03:57.206
to just treat differentials
like algebraic expressions.

00:03:57.206 --> 00:03:59.076
But even thought it's
not very mathematically

00:03:59.076 --> 00:04:01.778
rigorous to do it willy nilly like that.

00:04:01.778 --> 00:04:04.266
It has proven to be very useful.

00:04:04.266 --> 00:04:06.370
Now as you get more sophisticated

00:04:06.370 --> 00:04:08.686
in your mathematics there
are rigorous definitions

00:04:08.686 --> 00:04:10.267
of a differential.

00:04:10.267 --> 00:04:11.889
Where you can get a better sense of where

00:04:11.889 --> 00:04:13.778
it is mathematically rigorous to use it

00:04:13.778 --> 00:04:14.801
and where it isn't.

00:04:14.801 --> 00:04:16.357
But the whole point here is, if you felt

00:04:16.357 --> 00:04:18.609
a little weird feeling about multiplying

00:04:18.609 --> 00:04:22.384
both sides by d-x or dividing
both sides by d-x or d-y.

00:04:22.384 --> 00:04:25.426
Your feeling was mathematically justified,

00:04:25.426 --> 00:04:27.833
because it's not a very
rigorous thing to do.

00:04:27.833 --> 00:04:31.908
At least until you have
more rigor behind it,

00:04:31.908 --> 00:04:33.755
but I will tell you that if you're

00:04:33.755 --> 00:04:36.509
an introductory student
it is a reasonable thing

00:04:36.509 --> 00:04:39.470
to do as you explore and manipulate

00:04:39.470 --> 00:04:42.862
some of these basic
differential equations.