GoodBoy3000

WEBVTT
Kind: captions
Language: en

00:00:00.485 --> 00:00:01.650
(soft music)

00:00:01.650 --> 00:00:03.580
- [Narrator] Every morning,
your neural chip alarm

00:00:03.580 --> 00:00:07.170
goes off at 5:00 a.m.
Metropoluxe standard time.

00:00:07.170 --> 00:00:09.000
You'd prefer to be woken up by the Sun,

00:00:09.000 --> 00:00:11.560
but nobody in your sector
of the city is allowed

00:00:11.560 --> 00:00:15.740
to venture to the upper levels
to experience real sunlight.

00:00:15.740 --> 00:00:19.300
Oh well, chip-regulated
hormones and artificial sunlight

00:00:19.300 --> 00:00:22.000
work well enough for the likes of us.

00:00:22.000 --> 00:00:23.400
Even at this early hour,

00:00:23.400 --> 00:00:26.730
you hear the hum of
Metropoluxe all around you.

00:00:26.730 --> 00:00:29.820
The artificial sun lamps
warming up their bulbs,

00:00:29.820 --> 00:00:32.010
the garbage bots circling the city floor

00:00:32.010 --> 00:00:34.010
to sweep up the trash,

00:00:34.010 --> 00:00:37.670
the workers trudging to
expresselator terminals,

00:00:37.670 --> 00:00:38.800
you roll out of your cot

00:00:38.800 --> 00:00:42.120
and switch on the tiny sunlamp
over the tiny workbench

00:00:42.120 --> 00:00:44.870
you crammed into the corner of your room.

00:00:44.870 --> 00:00:48.710
As you approach the
table, GB3 whirs awake.

00:00:48.710 --> 00:00:51.100
You greet him fondly as you pat his head.

00:00:51.100 --> 00:00:53.300
The robot nuzzles up next to you.

00:00:53.300 --> 00:00:55.890
Its metal fur is cold this morning.

00:00:55.890 --> 00:00:57.670
But you don't really mind.

00:00:57.670 --> 00:01:01.170
The GoodBoy3000 is a
canine companion robot,

00:01:01.170 --> 00:01:03.430
the most advanced technology of its kind.

00:01:03.430 --> 00:01:05.150
Well, except for the fact

00:01:05.150 --> 00:01:07.543
that they're up to the 7,000 series now,

00:01:08.610 --> 00:01:10.690
and this isn't the real deal.

00:01:10.690 --> 00:01:13.740
Those costs hundreds of
thousands of credits.

00:01:13.740 --> 00:01:17.070
You found a good as new
GoodBoy3000 neuro coil

00:01:17.070 --> 00:01:19.380
at the scrap yard a few years back,

00:01:19.380 --> 00:01:21.840
and figured you could
learn how to build one.

00:01:21.840 --> 00:01:24.560
Years of tinkering and
experimenting later,

00:01:24.560 --> 00:01:26.710
you built your best friend.

00:01:26.710 --> 00:01:29.560
Speaking of the scrapyard,
you glance at the nano clock

00:01:29.560 --> 00:01:32.060
from the corner of your heads up display.

00:01:32.060 --> 00:01:35.860
You have about 45 more
minutes left to work on GB3

00:01:35.860 --> 00:01:37.970
before you have to head
to the expresselators

00:01:37.970 --> 00:01:39.950
to get to work on time.

00:01:39.950 --> 00:01:42.160
Just thinking about your
dead circuit of a boss

00:01:42.160 --> 00:01:44.150
makes you grumble to yourself.

00:01:44.150 --> 00:01:47.080
If you're even a second
late, you're fired.

00:01:47.080 --> 00:01:48.780
You shiver at the thought.

00:01:48.780 --> 00:01:51.190
Losing your job would
mean that your parents

00:01:51.190 --> 00:01:52.770
would be furious.

00:01:52.770 --> 00:01:56.350
And what little credits you
did have would quickly dry up.

00:01:56.350 --> 00:02:00.050
You motion for GB3 to sit in
front of you on the workbench.

00:02:00.050 --> 00:02:01.830
You flip on his maintenance switch

00:02:01.830 --> 00:02:05.810
and start to examine his
auxiliary memory backup subsystem.

00:02:05.810 --> 00:02:07.630
You've been getting up
early for months now

00:02:07.630 --> 00:02:09.360
to spend time on this.

00:02:09.360 --> 00:02:11.240
It's not a feature that these robots have,

00:02:11.240 --> 00:02:13.510
but that hasn't stopped
you from taking the time

00:02:13.510 --> 00:02:16.400
and effort to figure out
the mechanics on your own.

00:02:16.400 --> 00:02:19.380
Everything from the hour
spent trolling the holodecks

00:02:19.380 --> 00:02:21.360
to study robot manuals,

00:02:21.360 --> 00:02:24.310
to showing up to your
workbench every day to tinker

00:02:24.310 --> 00:02:26.240
and experiment with parts.

00:02:26.240 --> 00:02:29.940
If you can solve it, GB3 will
be the only dog in Metropoluxe

00:02:29.940 --> 00:02:31.440
with this tech.

00:02:31.440 --> 00:02:33.230
Your hard work has paid off.

00:02:33.230 --> 00:02:36.240
Your GB3 is as good as
the newest 7,000 series

00:02:36.240 --> 00:02:38.820
they keep showing on the telebeam.

00:02:38.820 --> 00:02:39.820
You shake your head.

00:02:40.670 --> 00:02:44.650
You can't even imagine owning
something that expensive.

00:02:44.650 --> 00:02:47.070
Who even has that many credits?

00:02:47.070 --> 00:02:49.910
A solar in the upper levels, you guess.

00:02:49.910 --> 00:02:51.870
You've been trying to make a memory input

00:02:51.870 --> 00:02:54.270
of GB3 for weeks now.

00:02:54.270 --> 00:02:56.840
If something were to happen with GB3,

00:02:56.840 --> 00:02:58.950
you could initiate a restoration process

00:02:58.950 --> 00:03:00.840
that would bring him back to life.

00:03:00.840 --> 00:03:04.390
The thought of losing
him, his memory data,

00:03:04.390 --> 00:03:06.800
well, it's unthinkable.

00:03:06.800 --> 00:03:08.810
You have a promise you made to yourself

00:03:08.810 --> 00:03:10.920
that you take very seriously.

00:03:10.920 --> 00:03:13.770
Some progress, just any little thing

00:03:13.770 --> 00:03:16.880
everyday while working on GB3.

00:03:16.880 --> 00:03:19.460
But the last week or so has been rough.

00:03:19.460 --> 00:03:20.760
You check the clock again.

00:03:21.650 --> 00:03:24.030
You're running low on time for today.

00:03:24.030 --> 00:03:26.760
And while everything seems
to be connected properly,

00:03:26.760 --> 00:03:28.810
it just hasn't been working.

00:03:28.810 --> 00:03:31.300
Still, your promise.

00:03:31.300 --> 00:03:35.083
You find an empty power connection
on one of the components.

00:03:36.110 --> 00:03:38.730
That component's already
connected to the main feed.

00:03:38.730 --> 00:03:41.210
So you're not quite sure
why there's another lead

00:03:41.210 --> 00:03:44.600
going to it, but hey, it's something.

00:03:44.600 --> 00:03:48.090
You wire it up, put a dab
of sotter on the connection,

00:03:48.090 --> 00:03:50.673
and start closing the
access panels you'd opened.

00:03:52.560 --> 00:03:54.920
You even use an upgraded replacement panel

00:03:54.920 --> 00:03:58.890
over your custom enclosure
to ensure progress for today,

00:03:58.890 --> 00:04:01.270
in case the power thing was a dead end?

00:04:01.270 --> 00:04:03.160
You flip the maintenance switch back off

00:04:03.160 --> 00:04:06.740
and grab your satchel as GB3
whirs to life behind you.

00:04:06.740 --> 00:04:09.210
He greets you by rubbing
the whole length of his body

00:04:09.210 --> 00:04:11.660
along your leg, shoulder to tail,

00:04:11.660 --> 00:04:15.510
his signature move which
you taught him on your own.

00:04:15.510 --> 00:04:17.610
You give him a few quick paths on the head

00:04:17.610 --> 00:04:18.803
and hurry out the door.

00:04:19.680 --> 00:04:21.830
You check the data pad and immediately

00:04:21.830 --> 00:04:24.910
see the alerts you've been
missing while tinkering.

00:04:24.910 --> 00:04:28.047
Expresselator's out again.

00:04:28.047 --> 00:04:29.400
(upbeat music)

00:04:29.400 --> 00:04:31.130
It's not the end of the world.

00:04:31.130 --> 00:04:32.880
You only take it one stop anyhow,

00:04:32.880 --> 00:04:35.260
but the alternative is a hyper cycle path

00:04:35.260 --> 00:04:37.670
you always try to avoid.

00:04:37.670 --> 00:04:40.810
You hate showing up to
work already caked in dirt.

00:04:40.810 --> 00:04:43.030
You sigh, run to your bike,

00:04:43.030 --> 00:04:45.361
and head out to the edge of Metropoluxe.

00:04:45.361 --> 00:04:47.550
(cosmic whir)

00:04:47.550 --> 00:04:50.250
This is probably the
closest you've ever cut it.

00:04:50.250 --> 00:04:52.420
It looks like you'll
arrive just a minute or two

00:04:52.420 --> 00:04:54.340
before the start of your shift.

00:04:54.340 --> 00:04:56.650
And that's if nothing else goes wrong.

00:04:56.650 --> 00:04:58.680
Thankfully, nothing does.

00:04:58.680 --> 00:05:02.120
And a short while later you're
hopping off your hyper cycle

00:05:02.120 --> 00:05:04.750
and sprinting to the
gate of the rusty rodeo.

00:05:04.750 --> 00:05:07.830
Your satchel slapping
at the top of your leg,

00:05:07.830 --> 00:05:11.060
you skid through the door,
just as the hourly reminder

00:05:11.060 --> 00:05:14.860
blares across the telobeam
from the Metropoluxe council.

00:05:14.860 --> 00:05:17.170
- You are in sector 22

00:05:17.170 --> 00:05:20.980
of the Metropoluxe
Industrial Reclamation Zone.

00:05:20.980 --> 00:05:24.240
Travel is prohibited to solar levels.

00:05:24.240 --> 00:05:29.030
Do not attempt to interact
with the solar, thank you.

00:05:29.030 --> 00:05:32.160
- You hate being reminded how
stuck you are in this place,

00:05:32.160 --> 00:05:33.930
but since you just swiped in on time,

00:05:33.930 --> 00:05:37.620
you've still got a job
and you've got GB3 too.

00:05:37.620 --> 00:05:39.550
The scrap bot at the gate chirps.

00:05:39.550 --> 00:05:44.140
- Dangerous debris detected,
rerouting to crusher.

00:05:44.140 --> 00:05:45.270
- That's odd.

00:05:45.270 --> 00:05:48.660
You usually don't get new scrap
until later in your shift.

00:05:48.660 --> 00:05:50.580
You turn to look at
what came into the yard,

00:05:50.580 --> 00:05:54.000
but for now, all you see is
the massive magnet engaged

00:05:54.000 --> 00:05:55.850
in front of the gate.

00:05:55.850 --> 00:05:58.220
When it lifts, you drop your wrench

00:05:58.220 --> 00:06:03.220
as you see GB3 stuck to the
underside, his visor dim.

00:06:03.330 --> 00:06:07.260
A GoodBoy3000 can't survive
a magnetic field like that.

00:06:07.260 --> 00:06:09.910
They're still working on
that for the 8,000 line.

00:06:09.910 --> 00:06:11.360
And they're not even out yet.

00:06:12.340 --> 00:06:15.820
No, no, no, you must
have accidentally enabled

00:06:15.820 --> 00:06:18.470
guardian mode as you gave
him one last pat on the head

00:06:18.470 --> 00:06:19.860
out the door.

00:06:19.860 --> 00:06:21.780
Dutifully, he followed you

00:06:21.780 --> 00:06:23.840
and you took him right
to the biggest magnet

00:06:23.840 --> 00:06:25.850
you've ever seen in your life.

00:06:25.850 --> 00:06:28.350
You run over to the scrap
bot and shove it aside,

00:06:28.350 --> 00:06:31.020
overriding the controls
for the scrap magnet.

00:06:31.020 --> 00:06:33.420
Gently, you lower it
back in front of the gate

00:06:33.420 --> 00:06:36.080
and disengage the magnetic field.

00:06:36.080 --> 00:06:38.580
You hear GB3 collapse to the dirt

00:06:38.580 --> 00:06:40.870
and you start to choke up a bit.

00:06:40.870 --> 00:06:43.890
You hoist the magnet back
to its resting position

00:06:43.890 --> 00:06:45.323
and sprint over to the gate.

00:06:46.350 --> 00:06:48.110
GB3 is lifeless

00:06:48.110 --> 00:06:50.690
as you roll him over to
flip the maintenance switch

00:06:50.690 --> 00:06:52.990
and open the access panel.

00:06:52.990 --> 00:06:55.640
The upgraded panel you use
today feels like it weighs

00:06:55.640 --> 00:06:58.123
10 times as much as it
did when you put it on.

00:06:59.560 --> 00:07:04.100
Oh wow, a small amber LED glows

00:07:04.100 --> 00:07:06.660
inside the custom memory enclosure.

00:07:06.660 --> 00:07:09.610
You glance over and see a
symbol on the access panel

00:07:09.610 --> 00:07:11.640
that you didn't recognize.

00:07:11.640 --> 00:07:14.180
It feels like lightning
tearing through your brain

00:07:14.180 --> 00:07:15.890
as you make the connection.

00:07:15.890 --> 00:07:19.210
That's the symbol for magnetic shielding.

00:07:19.210 --> 00:07:21.430
You learned about it
while studying the parts

00:07:21.430 --> 00:07:24.650
you needed back when you first built GB3.

00:07:24.650 --> 00:07:27.050
Excitedly, you run for a volt pack.

00:07:27.050 --> 00:07:30.330
GB3 still out and you need to
bring in some outside power

00:07:30.330 --> 00:07:32.800
to see if this thing's going to work.

00:07:32.800 --> 00:07:34.860
You slide back next to
him on your knee pads

00:07:34.860 --> 00:07:36.430
and make the connection.

00:07:36.430 --> 00:07:38.680
The LED flashes green twice,

00:07:38.680 --> 00:07:41.060
then turns back to solid amber.

00:07:41.060 --> 00:07:44.300
You hear a click from
inside his main access.

00:07:44.300 --> 00:07:47.050
You jump up and without
even bothering to reattach

00:07:47.050 --> 00:07:49.050
the panels, you squint your eyes shut

00:07:49.050 --> 00:07:50.693
and flip the maintenance switch.

00:07:51.889 --> 00:07:52.850
(robot beeps)

00:07:52.850 --> 00:07:53.880
It worked.

00:07:53.880 --> 00:07:57.480
GB3 blinks to life, writes
himself, and displays a message

00:07:57.480 --> 00:07:59.757
in block letters across his visor,

00:07:59.757 --> 00:08:02.880
"I knew you could fix me."

00:08:02.880 --> 00:08:04.440
He wags his whole body

00:08:04.440 --> 00:08:07.960
then rubs his shoulder down
to his tail against your leg.

00:08:07.960 --> 00:08:09.383
It's really him.

00:08:10.580 --> 00:08:12.690
You knew you could fix him too.

00:08:12.690 --> 00:08:15.250
You whisper as much as you kneel back down

00:08:15.250 --> 00:08:16.970
and wrap him in your arms.

00:08:16.970 --> 00:08:19.040
You think back on everything you learned,

00:08:19.040 --> 00:08:23.360
everything you did to get to
this moment with your dog.

00:08:23.360 --> 00:08:27.010
Each one of those hours you
put in have played their part.

00:08:27.010 --> 00:08:30.030
You look lovingly into GB3's visor,

00:08:30.030 --> 00:08:32.860
and you swell with a bit
of well-earned pride.

00:08:32.860 --> 00:08:34.780
GB3 might not have all the features

00:08:34.780 --> 00:08:36.800
that the newer models come with,

00:08:36.800 --> 00:08:40.379
but he's got something even better, you.

00:08:40.379 --> 00:08:43.296
(electronic music)


The Lighthouse Keeper

WEBVTT
Kind: captions
Language: en

00:00:00.125 --> 00:00:02.500
(gentle music)

00:00:02.500 --> 00:00:06.120
- You wake to the sound of
crashing waves, swelling

00:00:06.120 --> 00:00:09.093
and breaking against the
breakwaters outside your home.

00:00:09.930 --> 00:00:11.370
They have a rhythm to them,

00:00:11.370 --> 00:00:14.393
a rhythm you've grown
accustomed to, like a heartbeat.

00:00:15.340 --> 00:00:18.770
They build, swell, and crash.

00:00:18.770 --> 00:00:22.530
Build, swell and crash again and again,

00:00:22.530 --> 00:00:24.803
splashing at the foundations of your home.

00:00:25.930 --> 00:00:29.350
This is normal, of course, after all

00:00:29.350 --> 00:00:30.700
you're a lighthouse keeper.

00:00:31.710 --> 00:00:34.290
The sun has not yet risen and it's cold

00:00:34.290 --> 00:00:37.200
in every part of the lighthouse
except your quarters.

00:00:37.200 --> 00:00:40.270
Your day begins as you throw
on your heaviest sweater,

00:00:40.270 --> 00:00:42.180
your knit cap, and the gloves

00:00:42.180 --> 00:00:44.680
your aunt knit for you this spring.

00:00:44.680 --> 00:00:47.290
This is how every day starts for you.

00:00:47.290 --> 00:00:50.240
Trudging up the stairs in
the dim pre-dawn light.

00:00:50.240 --> 00:00:52.110
You make your way to the observation room

00:00:52.110 --> 00:00:54.330
in it's grand wide windows

00:00:54.330 --> 00:00:56.670
to take weather readings
from the equipment.

00:00:56.670 --> 00:00:59.870
Your hands trace the old
pitted iron railings.

00:00:59.870 --> 00:01:02.290
As you make your way up
the tight spiral staircase

00:01:02.290 --> 00:01:04.870
in the center of the lighthouse tower.

00:01:04.870 --> 00:01:07.690
You're starting to memorize
the way the railings feel.

00:01:07.690 --> 00:01:08.920
You can tell how far up you are

00:01:08.920 --> 00:01:10.710
based on the pits in the metal,

00:01:10.710 --> 00:01:13.020
because you do this every day.

00:01:13.020 --> 00:01:14.910
Every day the same.

00:01:14.910 --> 00:01:17.300
You could do this with your eyes closed.

00:01:17.300 --> 00:01:18.840
Why do you do it?

00:01:18.840 --> 00:01:20.280
Wake up in darkness,

00:01:20.280 --> 00:01:23.480
type on an old laptop
with cold numbed fingers.

00:01:23.480 --> 00:01:25.880
Tell the weather office
that the horizons clear

00:01:25.880 --> 00:01:29.710
or that there's good visibility,
that there's heavy fog,

00:01:29.710 --> 00:01:31.860
rapidly decreasing barometric pressure,

00:01:31.860 --> 00:01:33.390
a big storm, actually.

00:01:33.390 --> 00:01:36.380
Wow, that's a huge storm,
if you're being honest.

00:01:36.380 --> 00:01:38.490
You snap out of your
reverie as the sun crests

00:01:38.490 --> 00:01:41.240
over the horizon and new readings come in.

00:01:41.240 --> 00:01:44.150
Red sky morning, sailor take warning.

00:01:44.150 --> 00:01:46.410
It's your job to keep
the bright shining lamp

00:01:46.410 --> 00:01:48.210
atop the lighthouse tower lit,

00:01:48.210 --> 00:01:50.650
to let the ships at sea
know where the ocean ends

00:01:50.650 --> 00:01:53.580
and the sharp rocks and
reefs of the coast begin.

00:01:53.580 --> 00:01:56.140
It's most important when
the visibility is bad.

00:01:56.140 --> 00:01:58.290
And right now a fog is rolling in.

00:01:58.290 --> 00:02:02.230
In an hour's time, it'll be
thick as pea soup out there.

00:02:02.230 --> 00:02:03.570
They won't be able to see a thing

00:02:03.570 --> 00:02:06.860
except for the bright beacon
of the lighthouse itself

00:02:06.860 --> 00:02:09.450
that cuts through the fog
like nothing else can.

00:02:09.450 --> 00:02:10.790
That's why you do it.

00:02:10.790 --> 00:02:14.180
People are counting on
you to keep that lamp lit.

00:02:14.180 --> 00:02:15.600
And you know what?

00:02:15.600 --> 00:02:17.280
You've got this.

00:02:17.280 --> 00:02:19.020
It's checklist time.

00:02:19.020 --> 00:02:21.610
In the supply room, you
check up on your supplies.

00:02:21.610 --> 00:02:23.760
You've got plenty of
batteries for your flashlight,

00:02:23.760 --> 00:02:28.610
spare fuses for the fuse box,
and crucially, lots of snacks.

00:02:28.610 --> 00:02:30.390
You hurry down to the basement

00:02:30.390 --> 00:02:32.200
where your diesel generator is.

00:02:32.200 --> 00:02:33.550
It's humming away.

00:02:33.550 --> 00:02:36.300
The fuel tank is full, you just checked.

00:02:36.300 --> 00:02:37.560
This is your ritual.

00:02:37.560 --> 00:02:41.700
You do this every 12
hours, every single day.

00:02:41.700 --> 00:02:44.783
It's boring, but there's
also a kind of comfort in it.

00:02:44.783 --> 00:02:48.343
A comfort in the consistency
of a simple job done well.

00:02:49.420 --> 00:02:50.810
The generator keeps the light on

00:02:50.810 --> 00:02:52.940
in the lamp at the top of the lighthouse,

00:02:52.940 --> 00:02:54.890
and it also powers the
heater in your bedroom.

00:02:54.890 --> 00:02:56.173
So there's that too.

00:02:57.110 --> 00:03:00.260
You make yourself some breakfast
and a mug of hot cocoa.

00:03:00.260 --> 00:03:02.830
The kind from a packet with
the raspberry flavoring,

00:03:02.830 --> 00:03:06.070
and you sit in your favorite
chair in the observation room,

00:03:06.070 --> 00:03:07.833
just watching the storm build.

00:03:08.690 --> 00:03:13.343
Big gray clouds, fat and heavy
with rain, swell in the sky.

00:03:14.180 --> 00:03:15.740
The wind picks up.

00:03:15.740 --> 00:03:18.670
Whitecap start blossoming
on the crests of waves.

00:03:18.670 --> 00:03:21.610
And those clouds start
moving in faster and faster,

00:03:21.610 --> 00:03:24.220
blotting out the already dim sunlight.

00:03:24.220 --> 00:03:26.650
You start to see those
sideways lightning strikes

00:03:26.650 --> 00:03:30.790
that grasp between clouds,
anvil crawlers, they call them.

00:03:30.790 --> 00:03:32.950
The sign of a really big storm.

00:03:32.950 --> 00:03:35.900
It begins to rain, and you
can just make out the rumble

00:03:35.900 --> 00:03:37.475
of the approaching storm.

00:03:37.475 --> 00:03:39.430
(thunder rumbling)

00:03:39.430 --> 00:03:41.580
You radio with some of
the bigger container ships

00:03:41.580 --> 00:03:43.610
within distance and
urge them to get to port

00:03:43.610 --> 00:03:46.410
as soon as possible, because
the radar is telling you

00:03:46.410 --> 00:03:49.283
that the really nasty
weather is coming, and fast.

00:03:50.120 --> 00:03:52.150
The wind picks up, and as the rain begins

00:03:52.150 --> 00:03:54.820
to batter the lighthouse, it
starts to get harder and harder

00:03:54.820 --> 00:03:57.507
to hear what people are
saying over the radio.

00:03:57.507 --> 00:03:59.130
(heavy rain pouring)

00:03:59.130 --> 00:04:00.560
Back on the observation deck,

00:04:00.560 --> 00:04:02.630
the sea swells are getting bigger.

00:04:02.630 --> 00:04:03.700
The winds whipping them up

00:04:03.700 --> 00:04:06.010
into these big gray blue mountains.

00:04:06.010 --> 00:04:07.260
And now they're starting to beat

00:04:07.260 --> 00:04:09.220
against the upper windows
of the lighthouse.

00:04:09.220 --> 00:04:11.280
So we're talking at least 80 feet here.

00:04:11.280 --> 00:04:13.690
Huge geysers of sea spray foaming up

00:04:13.690 --> 00:04:16.170
and splashing against the walls.

00:04:16.170 --> 00:04:18.480
When you send your weather
report into central,

00:04:18.480 --> 00:04:20.780
the satellite connection
drops out in the rain

00:04:20.780 --> 00:04:21.613
and you have to switch

00:04:21.613 --> 00:04:23.552
to the low frequency radio transmission.

00:04:23.552 --> 00:04:25.480
(dramatic music)

00:04:25.480 --> 00:04:28.170
The rain on the windows
sounds like hammer blows.

00:04:28.170 --> 00:04:31.840
The waves and the thunder are
coming so fast and so loud now

00:04:31.840 --> 00:04:34.180
that you have difficulty
telling them apart.

00:04:34.180 --> 00:04:36.640
A prodigious bolt of
lightning splits the sky

00:04:36.640 --> 00:04:39.440
and shows you the armada
of clouds on approach

00:04:39.440 --> 00:04:41.090
for just a second.

00:04:41.090 --> 00:04:42.870
It's time to check on the lamp.

00:04:42.870 --> 00:04:44.550
It's an extremely bright light

00:04:44.550 --> 00:04:48.540
refracted through this delicate,
enormous multifaceted lens

00:04:48.540 --> 00:04:50.760
that vastly increases its output,

00:04:50.760 --> 00:04:53.760
making it visible for many miles at sea.

00:04:53.760 --> 00:04:55.067
The motor's going underneath the lamp

00:04:55.067 --> 00:04:58.210
and it's rotating the
whole thing very slowly.

00:04:58.210 --> 00:05:00.843
Sending a searchlight beam
out into the gloomy dark.

00:05:02.220 --> 00:05:05.810
Modern lighthouses use LEDs,
but this definitely isn't

00:05:05.810 --> 00:05:07.380
a modern lighthouse.

00:05:07.380 --> 00:05:10.080
Plus you've grown accustomed
to the rhythmic groaning.

00:05:11.890 --> 00:05:14.170
It almost hurts to look at,
that's how bright it is,

00:05:14.170 --> 00:05:15.610
but you're glad it's there

00:05:15.610 --> 00:05:17.800
as the waves beat against the walls,

00:05:17.800 --> 00:05:20.430
as a beacon in the darkening fog.

00:05:20.430 --> 00:05:23.690
Winter brings the night on
quickly and it's growing dark.

00:05:23.690 --> 00:05:25.560
It's a good thing the light is on.

00:05:25.560 --> 00:05:29.180
At least you're not out in one
of those ships on the water.

00:05:29.180 --> 00:05:30.460
The light suddenly goes out

00:05:30.460 --> 00:05:32.610
and you are plunged into darkness.

00:05:32.610 --> 00:05:34.400
It was so bright in the lamp room, in fact

00:05:34.400 --> 00:05:36.163
that you feel completely blind.

00:05:37.090 --> 00:05:39.590
The lamp didn't burn
out, the power is gone.

00:05:39.590 --> 00:05:42.210
The motor that rotated the
lamp has stopped moving.

00:05:42.210 --> 00:05:45.713
All you can hear now is the
terrible storm and the waves.

00:05:46.550 --> 00:05:48.490
You've got to see if
the generator gave out.

00:05:48.490 --> 00:05:50.100
You grab for your sturdy flashlight,

00:05:50.100 --> 00:05:51.970
but a lightning bolt
strikes the lighthouse

00:05:51.970 --> 00:05:53.147
at the same moment.

00:05:54.480 --> 00:05:55.400
You're safe.

00:05:55.400 --> 00:05:57.060
It must've caught the lightening rod.

00:05:57.060 --> 00:05:59.100
You notice that even the roar of the storm

00:05:59.100 --> 00:06:02.470
sounds muffled and distant due
to the ringing in your ears.

00:06:02.470 --> 00:06:03.760
And it takes you a beat to realize

00:06:03.760 --> 00:06:07.240
you've dropped the
flashlight down the stairs,

00:06:07.240 --> 00:06:09.943
which you will now have to
traverse in total darkness.

00:06:10.860 --> 00:06:14.080
The wind howls, the waves
batter the lighthouse,

00:06:14.080 --> 00:06:17.680
and yet a feeling of
confidence washes over you.

00:06:17.680 --> 00:06:22.330
As soon as your hand touches
that pitted cold iron railing.

00:06:22.330 --> 00:06:24.910
You know this building
backwards and forwards.

00:06:24.910 --> 00:06:26.800
You know every inch of the stairwell

00:06:26.800 --> 00:06:29.230
based on the feel of the railing alone.

00:06:29.230 --> 00:06:31.670
You make this climb many times each day,

00:06:31.670 --> 00:06:35.100
when the sun blazes and
when it's completely dark.

00:06:35.100 --> 00:06:36.670
Every day the same.

00:06:36.670 --> 00:06:39.348
You could do this with your eyes closed.

00:06:39.348 --> 00:06:40.610
(bright music)

00:06:40.610 --> 00:06:43.210
Halfway down the stairs, you
stop off in the supply room

00:06:43.210 --> 00:06:44.730
and root around in the darkness

00:06:44.730 --> 00:06:47.030
until you find one of everything.

00:06:47.030 --> 00:06:48.580
A fuse for the fuse box,

00:06:48.580 --> 00:06:51.270
extra batteries for the
flashlight just in case,

00:06:51.270 --> 00:06:55.230
and for you a bar of chocolate,
because you'll need it.

00:06:55.230 --> 00:06:58.923
Every item was exactly in
the place you knew it'd be.

00:06:59.800 --> 00:07:01.810
On the landing just
outside the supply room,

00:07:01.810 --> 00:07:03.010
you find the flashlight.

00:07:03.010 --> 00:07:06.040
Dented, sure, but it still works.

00:07:06.040 --> 00:07:08.470
Now that you can see again,
you run down to the generator

00:07:08.470 --> 00:07:11.370
to find that it's still running,
but all the lights are out.

00:07:11.370 --> 00:07:12.870
A fuse must've broken.

00:07:12.870 --> 00:07:15.030
You hit the emergency
stop on the generator,

00:07:15.030 --> 00:07:16.450
turn off the switch that connects it

00:07:16.450 --> 00:07:17.996
to the Lighthouse's power system.

00:07:17.996 --> 00:07:20.360
Knock open the fuse box, and there it is.

00:07:20.360 --> 00:07:22.490
A blackened busted fuse.

00:07:22.490 --> 00:07:23.540
You don't know how it happened,

00:07:23.540 --> 00:07:25.040
but you've got a fresh fuse in there

00:07:25.040 --> 00:07:28.670
and replaced before you even
have time to think about it.

00:07:28.670 --> 00:07:32.180
You close the fusebox, you
turn the generator back on

00:07:32.180 --> 00:07:34.763
and you pull that switch.

00:07:34.763 --> 00:07:35.596
(flicks switch on)

00:07:35.596 --> 00:07:37.460
And the lights are on again.

00:07:37.460 --> 00:07:38.378
Up the stairs, let's go.

00:07:38.378 --> 00:07:39.250
(intense music)

00:07:39.250 --> 00:07:41.150
The lights along the
stairwell light your path,

00:07:41.150 --> 00:07:42.980
and by the time you
make it up to the lamp,

00:07:42.980 --> 00:07:45.410
you know what you'll see
before you even get there.

00:07:45.410 --> 00:07:47.030
A beacon in the darkness,

00:07:47.030 --> 00:07:49.733
sweeping out into that
rain streaked Indigo night.

00:07:50.670 --> 00:07:53.470
You did it, you were
prepared for this moment

00:07:53.470 --> 00:07:57.400
and you met it because you
were consistent and careful.

00:07:57.400 --> 00:07:59.110
That bar of chocolate tastes better

00:07:59.110 --> 00:08:01.560
than anything you've eaten in years.

00:08:01.560 --> 00:08:04.123
Because more than anything, you earned it.

00:08:05.300 --> 00:08:07.190
Your radio crackles to life,

00:08:07.190 --> 00:08:09.740
and the grateful captain of
a container ship thanks you

00:08:09.740 --> 00:08:11.260
for keeping the light on.

00:08:11.260 --> 00:08:12.590
She can see it now.

00:08:12.590 --> 00:08:14.400
And she and her navigator are using it

00:08:14.400 --> 00:08:15.863
to plot a course to shore.

00:08:17.050 --> 00:08:19.853
Remember this moment and
remember this feeling.

00:08:22.820 --> 00:08:24.370
You are a lantern (indistinct),

00:08:25.765 --> 00:08:27.690
a light for others to see by.

00:08:27.690 --> 00:08:30.893
You can do this because
you have done this.

00:08:32.160 --> 00:08:33.373
And you'll keep going.

00:08:34.232 --> 00:08:36.899
(intense music)


Theorem for limits of composite functions: when conditions aren't met

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.810
- [Tutor] In a previous
video we used this theorem

00:00:02.810 --> 00:00:05.580
to evaluate certain types
of composite functions.

00:00:05.580 --> 00:00:07.126
In this video we'll do
a few more examples,

00:00:07.126 --> 00:00:09.486
that get a little bit more involved.

00:00:09.486 --> 00:00:11.820
So let's say we wanted
to figure out the limit

00:00:11.820 --> 00:00:14.210
as x approaches zero

00:00:14.210 --> 00:00:19.210
of f of g of x,

00:00:19.270 --> 00:00:21.300
f of g of x.

00:00:21.300 --> 00:00:22.606
First of all, pause this video

00:00:22.606 --> 00:00:25.706
and think about whether
this theorem even applies.

00:00:25.706 --> 00:00:27.450
Well, the first thing to think about

00:00:27.450 --> 00:00:32.330
is what is the limit as x
approaches zero of g of x

00:00:32.330 --> 00:00:35.650
to see if we meet this first condition.

00:00:35.650 --> 00:00:37.810
So if we look at g of x, right over here

00:00:37.810 --> 00:00:40.010
as x approaches zero from the left,

00:00:40.010 --> 00:00:42.150
it looks like g is approaching two,

00:00:42.150 --> 00:00:44.310
as x approaches zero from the right,

00:00:44.310 --> 00:00:46.610
it looks like g is approaching two

00:00:46.610 --> 00:00:50.560
and so it looks like this
is going to be equal to two.

00:00:50.560 --> 00:00:51.967
So that's a check.

00:00:51.967 --> 00:00:53.820
Now let's see the second condition,

00:00:53.820 --> 00:00:57.620
is f continuous at that limit at two.

00:00:57.620 --> 00:00:59.380
So when x is equal to two,

00:00:59.380 --> 00:01:02.490
it does not look like f is continuous.

00:01:02.490 --> 00:01:05.455
So we do not meet this second
condition right over here,

00:01:05.455 --> 00:01:08.920
so we can't just directly
apply this theorem.

00:01:08.920 --> 00:01:11.490
But just because you
can't apply the theorem

00:01:11.490 --> 00:01:15.900
does not mean that the limit
doesn't necessarily exist.

00:01:15.900 --> 00:01:17.430
For example, in this situation

00:01:17.430 --> 00:01:20.410
the limit actually does exist.

00:01:20.410 --> 00:01:21.420
One way to think about it,

00:01:21.420 --> 00:01:25.380
when x approaches zero from the left,

00:01:25.380 --> 00:01:30.380
it looks like g is
approaching two from above

00:01:31.890 --> 00:01:34.530
and so that's going to be the input into f

00:01:34.530 --> 00:01:38.290
and so if we are now
approaching two from above here

00:01:38.290 --> 00:01:39.920
as the input into f,

00:01:39.920 --> 00:01:43.890
it looks like our function
is approaching zero

00:01:43.890 --> 00:01:45.830
and then we can go the other way.

00:01:45.830 --> 00:01:50.830
If we are approaching zero from
the right, right over here,

00:01:51.850 --> 00:01:54.103
it looks like the value of our function

00:01:54.103 --> 00:01:58.150
is approaching two from below.

00:01:58.150 --> 00:02:01.590
Now if we approach two from below,

00:02:01.590 --> 00:02:05.800
it looks like the value
of f is approaching zero.

00:02:05.800 --> 00:02:07.590
So in both of these scenarios,

00:02:07.590 --> 00:02:11.330
our value of our function
f is approaching zero.

00:02:11.330 --> 00:02:13.340
So I wasn't able to use this theorem,

00:02:13.340 --> 00:02:14.750
but I am able to figure out

00:02:14.750 --> 00:02:16.752
that this is going to be equal to zero.

00:02:16.752 --> 00:02:19.340
Now let me give you another example.

00:02:19.340 --> 00:02:21.660
Let's say we wanted to
figure out the limit

00:02:21.660 --> 00:02:24.180
as x approaches two

00:02:24.180 --> 00:02:28.453
of f of g of x.

00:02:29.321 --> 00:02:30.610
Pause this video,

00:02:30.610 --> 00:02:32.860
we'll first see if this
theorem even applies.

00:02:33.750 --> 00:02:36.180
Well, we first wanna see what is the limit

00:02:36.180 --> 00:02:39.619
as x approaches two of g of x.

00:02:39.619 --> 00:02:42.696
When we look at approaching
two from the left,

00:02:42.696 --> 00:02:45.363
it looks like g is
approaching negative two.

00:02:45.363 --> 00:02:48.530
When we approach x equals
two from the right,

00:02:48.530 --> 00:02:50.910
it looks like g is approaching zero.

00:02:50.910 --> 00:02:53.310
So our right and left hand
limits are not the same here,

00:02:53.310 --> 00:02:58.310
so this thing does not
exist, does not exist

00:02:59.855 --> 00:03:02.890
and so we don't meet this
condition right over here,

00:03:02.890 --> 00:03:04.622
so we can't apply the theorem.

00:03:04.622 --> 00:03:06.070
But as we've already seen,

00:03:06.070 --> 00:03:08.090
just because you can't apply the theorem

00:03:08.090 --> 00:03:11.253
does not mean that the
limit does not exist.

00:03:11.253 --> 00:03:13.240
But if you like pondering things,

00:03:13.240 --> 00:03:16.190
I encourage you to see that
this limit doesn't exist

00:03:16.190 --> 00:03:18.410
by doing very similar analysis

00:03:18.410 --> 00:03:21.453
to the one that I did
for our first example.


Theorem for limits of composite functions

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.350
- [Tutor] In this video, we're
going to try to understand

00:00:02.350 --> 00:00:04.705
limits of composite
functions, or at least a way

00:00:04.705 --> 00:00:07.230
of thinking about limits
of composite functions

00:00:07.230 --> 00:00:08.917
and in particular, we're
gonna think about the case

00:00:08.917 --> 00:00:12.360
where we're trying to find
the limit as x approaches a,

00:00:12.360 --> 00:00:16.128
of f of g of x

00:00:16.128 --> 00:00:19.680
and we're going to see
under certain circumstances,

00:00:19.680 --> 00:00:24.680
this is going to be
equal to f of the limit,

00:00:25.990 --> 00:00:30.990
the limit as x approaches a of g of x

00:00:31.610 --> 00:00:34.510
and what are those
circumstances you are asking?

00:00:34.510 --> 00:00:35.760
Well, this is going to be true

00:00:35.760 --> 00:00:38.170
if and only if two things are true,

00:00:38.170 --> 00:00:41.140
first of all, this limit needs to exist.

00:00:41.140 --> 00:00:46.140
So the limit as x approaches
a of g of x needs to exist,

00:00:48.410 --> 00:00:52.220
so that needs to exist
and then on top of that,

00:00:52.220 --> 00:00:56.000
the function f needs to be
continuous at this point

00:00:56.000 --> 00:01:00.500
and f continuous at L.

00:01:03.740 --> 00:01:05.120
So let's look at some examples

00:01:05.120 --> 00:01:07.180
and see if we can apply this idea

00:01:07.180 --> 00:01:08.603
or see if we can't apply it.

00:01:09.450 --> 00:01:12.340
So here I have two functions,

00:01:12.340 --> 00:01:15.446
that are graphically
represented right over here,

00:01:15.446 --> 00:01:18.360
let me make sure I have
enough space for them

00:01:18.360 --> 00:01:23.360
and what we see on the
left-hand side is our function f

00:01:23.780 --> 00:01:28.780
and what we see on the right-hand
side is our function g.

00:01:28.920 --> 00:01:33.060
So first let's figure
out what is the limit

00:01:33.060 --> 00:01:35.860
as x approaches negative three

00:01:35.860 --> 00:01:39.840
of f of g of x.

00:01:39.840 --> 00:01:41.630
Pause this video and see,

00:01:41.630 --> 00:01:44.440
first of all, does this theorem apply?

00:01:44.440 --> 00:01:47.517
And if it does apply, what is this limit?

00:01:47.517 --> 00:01:49.250
So the first thing we need to see

00:01:49.250 --> 00:01:51.810
is does this theorem apply?

00:01:51.810 --> 00:01:54.870
So first of all, if we
were to find the limit

00:01:54.870 --> 00:01:59.870
as x approaches negative
three of g of x, what is that?

00:02:00.150 --> 00:02:03.022
Well, when we're approaching
negative three from the right,

00:02:03.022 --> 00:02:06.522
it looks like our function
is actually at three

00:02:06.522 --> 00:02:08.770
and it looks like when we're
approaching negative three

00:02:08.770 --> 00:02:11.580
from the left, it looks like
our function is at three.

00:02:11.580 --> 00:02:13.760
So it looks like this limit is three,

00:02:13.760 --> 00:02:17.395
even though the value g of
negative three is negative two,

00:02:17.395 --> 00:02:19.940
but it's a point discontinuity.

00:02:19.940 --> 00:02:21.710
As we approach it from either side,

00:02:21.710 --> 00:02:23.790
the value of the function is at three.

00:02:23.790 --> 00:02:25.920
So this thing is going to be three,

00:02:25.920 --> 00:02:28.343
so it exists, so we meet
that first condition

00:02:28.343 --> 00:02:32.470
and then the second question
is is our function f

00:02:32.470 --> 00:02:35.750
continuous at this limit,
continuous at three?

00:02:35.750 --> 00:02:39.820
So when x equals three, yeah,
it looks like at that point,

00:02:39.820 --> 00:02:43.010
our function is definitely continuous

00:02:43.010 --> 00:02:45.400
and so we could say that this limit

00:02:45.400 --> 00:02:46.900
is going to be the same thing

00:02:46.900 --> 00:02:51.280
as this equals f of the limit

00:02:52.420 --> 00:02:56.883
as x approaches negative three of g of x,

00:02:58.020 --> 00:02:59.930
close the parentheses

00:02:59.930 --> 00:03:03.080
and we know that this is equal to three

00:03:03.080 --> 00:03:04.950
and we know that f of three

00:03:04.950 --> 00:03:07.650
is going to be equal to negative one.

00:03:07.650 --> 00:03:09.254
So this met the conditions
for this theorem

00:03:09.254 --> 00:03:10.700
and we were able to use the theorem

00:03:10.700 --> 00:03:12.150
to actually solve this limit.


Geometric series as a function

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:01.270
- [Instructor] So we have this function

00:00:01.270 --> 00:00:03.450
that's equal to 2 minus 8x squared

00:00:03.450 --> 00:00:07.000
plus 32x to the fourth
minus 128x to the sixth.

00:00:07.000 --> 00:00:08.610
And it just keeps going and going.

00:00:08.610 --> 00:00:11.170
So it's defined as an infinite series.

00:00:11.170 --> 00:00:13.130
And what I want to explore in this video,

00:00:13.130 --> 00:00:16.360
is there another way
to write this function

00:00:16.360 --> 00:00:19.800
so it's not expressed
as an infinite series?

00:00:19.800 --> 00:00:21.790
Well, some of you might be thinking,

00:00:21.790 --> 00:00:24.360
well, this looks like a geometric series

00:00:24.360 --> 00:00:26.730
on the right hand side, an
infinite geometric series,

00:00:26.730 --> 00:00:30.320
and we know what the sun of an
infinite geometric series is

00:00:30.320 --> 00:00:31.970
if it converges.

00:00:31.970 --> 00:00:34.710
So maybe that's a way
that we can express this.

00:00:34.710 --> 00:00:35.543
So let's try to do that.

00:00:35.543 --> 00:00:36.710
So first let's just confirm

00:00:36.710 --> 00:00:38.680
that this is an infinite geometric series.

00:00:38.680 --> 00:00:41.200
And in order for it to
be a geometric series,

00:00:41.200 --> 00:00:44.680
each successive term has
to be some common ratio

00:00:44.680 --> 00:00:46.220
times the previous term.

00:00:46.220 --> 00:00:48.930
So to go from 2 to negative 8x squared,

00:00:48.930 --> 00:00:50.560
what do you have to multiply by?

00:00:50.560 --> 00:00:55.560
Well you have to multiply
by negative 4x squared.

00:00:55.600 --> 00:00:57.500
Now let's see if you
multiply negative 8x squared

00:00:57.500 --> 00:00:59.700
times negative 4x
squared, what do you get?

00:00:59.700 --> 00:01:02.360
Well, negative 4 times
negative 8 is positive 32.

00:01:02.360 --> 00:01:04.220
x squared times x squared
is x to the fourth.

00:01:04.220 --> 00:01:05.130
So that works.

00:01:05.130 --> 00:01:08.200
And then you multiply that
times negative 4x squared,

00:01:08.200 --> 00:01:11.670
and you indeed would get
negative 128x to the sixth.

00:01:11.670 --> 00:01:15.540
So this indeed looks like
an infinite geometric series

00:01:15.540 --> 00:01:16.430
on the right-hand side.

00:01:16.430 --> 00:01:19.430
In fact, we can rewrite f of x

00:01:19.430 --> 00:01:21.840
as being equal to the sum

00:01:22.750 --> 00:01:27.390
from n equals 0 to infinity of,

00:01:27.390 --> 00:01:28.880
you have your first term,

00:01:28.880 --> 00:01:30.520
and then you have your common ratio,

00:01:30.520 --> 00:01:33.400
negative 4x squared

00:01:33.400 --> 00:01:35.610
to the nth power.

00:01:35.610 --> 00:01:37.890
Let's confirm that works when n equals 0

00:01:37.890 --> 00:01:39.130
this is going to be 1.

00:01:39.130 --> 00:01:43.010
So 2 times 1 is 2, and that
indeed is our first term there.

00:01:43.010 --> 00:01:44.760
And then to that, you're gonna add it

00:01:44.760 --> 00:01:46.700
to when n is equal to 1.

00:01:46.700 --> 00:01:48.690
So that's just going to be
two times negative 4X squared,

00:01:48.690 --> 00:01:52.110
which is indeed this second
term right over here.

00:01:52.110 --> 00:01:54.810
And so this looks like it works.

00:01:54.810 --> 00:01:57.350
Now what is the sum

00:01:57.350 --> 00:02:01.120
of an infinite geometric series like this?

00:02:01.120 --> 00:02:02.760
Well it's going to be a finite value,

00:02:02.760 --> 00:02:06.930
assuming the absolute
value of your common ratio

00:02:06.930 --> 00:02:09.200
is less than 1.

00:02:09.200 --> 00:02:10.680
So first of all, let's just think about

00:02:10.680 --> 00:02:12.740
under what conditions
is the absolute value

00:02:12.740 --> 00:02:14.500
of our common ratio less than 1?

00:02:14.500 --> 00:02:15.460
And then we could say, okay,

00:02:15.460 --> 00:02:19.060
that helps us to find a
radius of convergence.

00:02:19.060 --> 00:02:23.250
And then if x is in that zone,

00:02:23.250 --> 00:02:24.920
or if it's in that interval,

00:02:24.920 --> 00:02:28.860
then we can figure out a
non-infinite geometric series way

00:02:28.860 --> 00:02:31.040
of expressing this function.

00:02:31.040 --> 00:02:33.670
So if we just think about
under what circumstances

00:02:33.670 --> 00:02:36.650
will this converge, will it
come out to a finite value?

00:02:36.650 --> 00:02:39.360
That's a situation in
which the absolute value

00:02:39.360 --> 00:02:42.890
of your common ratio is less than 1.

00:02:42.890 --> 00:02:45.613
And so let's see if we can
simplify this a little bit.

00:02:46.820 --> 00:02:49.070
No matter what x is, it's
always going to be not,

00:02:49.070 --> 00:02:51.570
x squared is always
going to be non-negative.

00:02:51.570 --> 00:02:54.980
And so the only, so this entire expression

00:02:54.980 --> 00:02:56.810
is always going to be negative.

00:02:56.810 --> 00:02:59.000
And so if you take the
absolute value of it,

00:02:59.000 --> 00:03:02.840
this is going to be
evaluate as 4x squared,

00:03:02.840 --> 00:03:04.530
which is always going to be positive.

00:03:04.530 --> 00:03:07.020
So this is equivalent to 4x squared,

00:03:07.020 --> 00:03:08.980
which needs to be less than 1.

00:03:08.980 --> 00:03:13.560
Or we could say that x squared
needs to be less than 1/4.

00:03:13.560 --> 00:03:18.530
Or we could say that x needs to be

00:03:18.530 --> 00:03:20.890
less than 1/2

00:03:20.890 --> 00:03:24.180
and greater than negative 1/2.

00:03:24.180 --> 00:03:27.290
One way to think about it is
anywhere in this interval,

00:03:27.290 --> 00:03:30.350
if you square it, you're
going to be less than 1/4.

00:03:30.350 --> 00:03:33.730
At 1/2, if you square
it, it's equal to 1/4.

00:03:33.730 --> 00:03:37.150
And at negative 1/2, if you
square it, it's equal to 1/4.

00:03:37.150 --> 00:03:39.570
But for lower absolute values,

00:03:39.570 --> 00:03:41.540
it's going to be less than 1/4.

00:03:41.540 --> 00:03:43.590
And so that's what this
interval right here says.

00:03:43.590 --> 00:03:46.400
Another way to think about
it is the absolute value of x

00:03:46.400 --> 00:03:48.830
needs to be less than 1/2.

00:03:48.830 --> 00:03:51.440
And so we've just defined an interval

00:03:51.440 --> 00:03:55.680
over which this infinite
geometric series will converge.

00:03:55.680 --> 00:03:58.880
You could say this has a
radius of convergence of,

00:03:58.880 --> 00:04:02.853
let me write it this way,
radius of convergence,

00:04:04.130 --> 00:04:07.240
convergence of 1/2,

00:04:07.240 --> 00:04:11.220
you can go 1/2 above 0 and 1/2 below 0.

00:04:11.220 --> 00:04:13.460
But now that we've set the conditions

00:04:13.460 --> 00:04:16.370
under which this would
converge, let's rewrite it.

00:04:16.370 --> 00:04:19.230
So this function is going to be equal to,

00:04:19.230 --> 00:04:22.450
we know what the sum of an
infinite geometric series is.

00:04:22.450 --> 00:04:24.670
It's going to be equal to the first term

00:04:24.670 --> 00:04:29.070
over 1 minus your common ratio,

00:04:29.070 --> 00:04:32.480
1 minus negative 4x squared.

00:04:32.480 --> 00:04:35.730
And so we can rewrite
our function as f of x

00:04:35.730 --> 00:04:39.780
is equal to 2 over 1,
subtract a negative 1

00:04:39.780 --> 00:04:43.040
plus 4x squared for

00:04:43.920 --> 00:04:47.910
the absolute value of x is less than 1/2.

00:04:47.910 --> 00:04:50.900
We have the interval
over which we converge

00:04:50.900 --> 00:04:52.260
and there you have it.

00:04:52.260 --> 00:04:55.443
We are done.


Sampling distribution of the difference in sample proportions

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:02.710
- [Instructor] We're told, suppose that 8%

00:00:02.710 --> 00:00:06.840
of all cars produced at plant
A have a certain defect,

00:00:06.840 --> 00:00:11.840
and 6% of all cars produced
at plant B have this defect.

00:00:12.670 --> 00:00:15.340
Each month, a quality control manager

00:00:15.340 --> 00:00:18.640
takes separate random samples of 200

00:00:18.640 --> 00:00:22.520
of the over 3000 cars
produced from each plant.

00:00:22.520 --> 00:00:24.490
The manager looks at the difference

00:00:24.490 --> 00:00:27.640
between the proportions
of cars with the defect

00:00:27.640 --> 00:00:29.100
in each sample.

00:00:29.100 --> 00:00:30.640
So they're looking at the difference

00:00:30.640 --> 00:00:32.810
of sample proportions every month.

00:00:32.810 --> 00:00:34.300
Describe the distribution

00:00:34.300 --> 00:00:36.240
of the difference of sample proportions

00:00:36.240 --> 00:00:40.410
in terms of its mean
standard deviation and shape.

00:00:40.410 --> 00:00:42.330
So let's take these step-by-step.

00:00:42.330 --> 00:00:45.980
So first, let's think about
the mean of the difference

00:00:45.980 --> 00:00:48.160
of our sample proportions.

00:00:48.160 --> 00:00:48.993
Pause this video

00:00:48.993 --> 00:00:51.190
and try to figure out
what that's going to be.

00:00:52.150 --> 00:00:54.990
Well, we have seen this
in previous videos,

00:00:54.990 --> 00:00:57.260
that if we have the mean of the difference

00:00:57.260 --> 00:00:59.040
of two random variables,

00:00:59.040 --> 00:01:01.380
that's the same as the
difference of the means

00:01:01.380 --> 00:01:03.110
or another way to think about it is

00:01:03.110 --> 00:01:07.210
if we wanna figure out the mean of this,

00:01:07.210 --> 00:01:10.560
so sample proportion from plant A

00:01:10.560 --> 00:01:13.460
minus sample proportion from plant B,

00:01:13.460 --> 00:01:16.020
this is just going to be equal to the mean

00:01:16.020 --> 00:01:19.270
of the sample proportion from plant A,

00:01:19.270 --> 00:01:23.970
minus the mean of the sample
proportion from plant B.

00:01:23.970 --> 00:01:26.390
Now, what are these going to be equal to?

00:01:26.390 --> 00:01:30.273
Well, what's the mean of the
sample proportion of plant A.

00:01:31.220 --> 00:01:34.620
Is just going to be the
true population proportion

00:01:34.620 --> 00:01:35.453
for plant A.

00:01:35.453 --> 00:01:36.400
And they tell us that.

00:01:36.400 --> 00:01:39.750
They tell us that 8% of all
cars produced at plant A

00:01:39.750 --> 00:01:41.030
have a certain defect.

00:01:41.030 --> 00:01:44.663
So this could be 8% or we
could write it as 0.08.

00:01:46.140 --> 00:01:47.340
And then from that,

00:01:47.340 --> 00:01:52.340
we are going to subtract the
mean of the sample proportion

00:01:52.610 --> 00:01:53.950
from plant B.

00:01:53.950 --> 00:01:55.680
And we know what that mean's going to be.

00:01:55.680 --> 00:01:58.160
The mean of a sample proportion

00:01:58.160 --> 00:02:02.120
is going to be the population proportion.

00:02:02.120 --> 00:02:03.850
The parameter of the population,

00:02:03.850 --> 00:02:08.850
which we know for plant B is 6%, 0.06,

00:02:09.450 --> 00:02:13.590
and then that gets us a
mean of the difference

00:02:13.590 --> 00:02:16.990
of 0.02 or 2%

00:02:19.569 --> 00:02:24.040
or 2% difference in defect
rate would be the mean.

00:02:24.040 --> 00:02:28.280
Now let's think about
the standard deviation.

00:02:28.280 --> 00:02:30.420
So instead of thinking in
terms of standard deviation,

00:02:30.420 --> 00:02:33.210
let's think about the square
of the standard deviation,

00:02:33.210 --> 00:02:34.110
which is variance.

00:02:34.110 --> 00:02:36.760
And from there, we can go
back to standard deviation

00:02:36.760 --> 00:02:38.610
by taking a square root.

00:02:38.610 --> 00:02:41.320
So if we're looking at the variance,

00:02:41.320 --> 00:02:42.800
lemme write it this way,

00:02:42.800 --> 00:02:45.000
if we're looking at the variance

00:02:45.000 --> 00:02:48.450
of the difference of
the sample proportions,

00:02:48.450 --> 00:02:50.610
so the sample proportion from plant A

00:02:50.610 --> 00:02:53.420
minus the sample proportion from plant B,

00:02:53.420 --> 00:02:54.720
but just as a review,

00:02:54.720 --> 00:02:57.500
if you assume that we're
sampling independently

00:02:57.500 --> 00:02:58.710
from each of the plants.

00:02:58.710 --> 00:03:00.760
So what we're sampling from plant A

00:03:00.760 --> 00:03:03.040
does not affect what we're
sampling from plant B

00:03:03.040 --> 00:03:08.040
or vice versa, then we
can add the variances.

00:03:08.398 --> 00:03:11.188
So this is going to be
equal to the variance

00:03:11.188 --> 00:03:13.582
of the sample proportion from plant A

00:03:13.582 --> 00:03:18.030
plus the variance of the
sample proportion from plant B.

00:03:18.030 --> 00:03:18.863
Some of you might be saying,

00:03:18.863 --> 00:03:20.440
"Wait, aren't we taking the difference

00:03:20.440 --> 00:03:21.630
of sample proportions here?

00:03:21.630 --> 00:03:22.720
Why are we adding?"

00:03:22.720 --> 00:03:24.420
And the reminder is,

00:03:24.420 --> 00:03:27.500
remember, variance is
a measure of a spread.

00:03:27.500 --> 00:03:29.920
And whether you're now
taking the difference

00:03:29.920 --> 00:03:33.150
of random variables or you're
taking the sum of them,

00:03:33.150 --> 00:03:34.160
when you have more variables,

00:03:34.160 --> 00:03:35.710
you're going to have more spread.

00:03:35.710 --> 00:03:37.330
So regardless of whether
this is a negative

00:03:37.330 --> 00:03:40.640
or positive over here, this
is going to be a positive.

00:03:40.640 --> 00:03:42.660
So what is this going to be equal to.

00:03:42.660 --> 00:03:44.620
We can take each of these terms,

00:03:44.620 --> 00:03:45.730
what's going to be the variance

00:03:45.730 --> 00:03:48.430
of the sample proportion from plant A?

00:03:48.430 --> 00:03:52.610
Well, if every time we
looked at one of the cars,

00:03:52.610 --> 00:03:55.470
we looked at it and then we
put it back into the mix.

00:03:55.470 --> 00:03:57.600
So if we were sampling with replacement,

00:03:57.600 --> 00:03:59.730
which means that each of our observations

00:03:59.730 --> 00:04:03.130
are independent of the other
ones, we have a formula.

00:04:03.130 --> 00:04:04.360
We know that this variance

00:04:04.360 --> 00:04:07.410
would be the population
proportion of plant A

00:04:07.410 --> 00:04:10.680
times one minus the population
proportion of plant A

00:04:10.680 --> 00:04:13.850
divided by the number that
we sampled from plant A.

00:04:13.850 --> 00:04:15.930
Now, in the scenario that
we are talking about,

00:04:15.930 --> 00:04:17.810
we didn't sample with replacement,

00:04:17.810 --> 00:04:20.210
we just took 200 at a
time and looked at them.

00:04:20.210 --> 00:04:21.530
We didn't take one at a time

00:04:21.530 --> 00:04:24.080
and replace it and do that 200 times.

00:04:24.080 --> 00:04:27.210
But we also know that this is
a pretty good approximation,

00:04:27.210 --> 00:04:30.040
even when you are not
sampling with replacement.

00:04:30.040 --> 00:04:34.930
If your sample is less
than 10% of the population,

00:04:34.930 --> 00:04:38.820
and 200 is less than 10% of 3000.

00:04:38.820 --> 00:04:41.070
So this is a pretty good approximation,

00:04:41.070 --> 00:04:43.970
what you would use in a
first year statistics class.

00:04:43.970 --> 00:04:46.270
And of course, we can use the same logic.

00:04:46.270 --> 00:04:47.470
This is going to be equal

00:04:47.470 --> 00:04:49.870
to the population proportion plant B

00:04:49.870 --> 00:04:52.760
times one minus the population
proportion in plant B,

00:04:52.760 --> 00:04:55.680
all of that over your
sample size from plant B.

00:04:55.680 --> 00:04:58.270
And we know all of these things.

00:04:58.270 --> 00:05:01.490
We know that your population
proportion in plant A

00:05:01.490 --> 00:05:05.530
is 8% or 0.08.

00:05:05.530 --> 00:05:10.490
One minus that is 0.92.

00:05:10.490 --> 00:05:14.060
We're taking samples of
200 at a time from plant A.

00:05:14.060 --> 00:05:17.090
And then in plant B, we know
the population proportion,

00:05:17.090 --> 00:05:20.950
they told us is 6% or 0.06.

00:05:20.950 --> 00:05:24.253
One minus that is 0.94.

00:05:25.680 --> 00:05:28.810
And then the sample size from plant B

00:05:28.810 --> 00:05:30.670
is also going to be 200.

00:05:30.670 --> 00:05:32.260
It's going to be 200.

00:05:32.260 --> 00:05:37.260
We get 0.08 times 0.92
divided by 200 and then plus,

00:05:42.360 --> 00:05:44.510
let's open parentheses here,

00:05:44.510 --> 00:05:49.510
we get 0.06 times 0.94 divided by 200,

00:05:52.190 --> 00:05:54.950
and then actually let me
close the parentheses,

00:05:54.950 --> 00:05:57.030
and that equals this business.

00:05:57.030 --> 00:05:58.203
So 0.00065.

00:06:00.750 --> 00:06:05.220
So 0.00065.

00:06:05.220 --> 00:06:06.240
And then from this,

00:06:06.240 --> 00:06:08.440
we can figure out what the
standard is going to be.

00:06:08.440 --> 00:06:11.140
The standard deviation of the difference

00:06:11.140 --> 00:06:14.330
between our sample proportions

00:06:15.240 --> 00:06:18.090
is going to be just the
square root of this.

00:06:18.090 --> 00:06:22.463
It's going to be the
square root of 0.00065.

00:06:24.180 --> 00:06:27.410
And that is approximately equal to,

00:06:27.410 --> 00:06:31.097
let's just take the square
root, and we get this, 0.025.

00:06:34.178 --> 00:06:36.750
0.025.

00:06:36.750 --> 00:06:37.630
And there you have it,

00:06:37.630 --> 00:06:39.600
we have thought about
the standard deviation.

00:06:39.600 --> 00:06:43.430
And then last but not least,
let's think about the shape.

00:06:43.430 --> 00:06:47.200
So just as a review, we just
have to remind ourselves

00:06:47.200 --> 00:06:50.560
that the distribution of
each sample proportion

00:06:50.560 --> 00:06:52.050
is going to be normal

00:06:52.050 --> 00:06:57.050
as long as we expect at least
10 successes and 10 failures.

00:06:57.870 --> 00:06:59.010
Well, let's look at each of these.

00:06:59.010 --> 00:07:01.230
How many successes you expect

00:07:01.230 --> 00:07:03.680
where a success would
actually be a defect?

00:07:03.680 --> 00:07:04.760
But let's think about this.

00:07:04.760 --> 00:07:09.600
8% of in each case of a sample of 200,

00:07:09.600 --> 00:07:10.860
that's going to be 16.

00:07:10.860 --> 00:07:13.380
So you would expect 16 defects,

00:07:13.380 --> 00:07:15.550
and then you would expect 200 minus 16,

00:07:15.550 --> 00:07:18.490
which is a lot larger
than 10 of no defects.

00:07:18.490 --> 00:07:21.570
So both of those are
greater than or equal to 10.

00:07:21.570 --> 00:07:24.130
And then if you did the
same thing for plant B,

00:07:24.130 --> 00:07:25.050
you get the same idea.

00:07:25.050 --> 00:07:27.940
6% of 200 is 12.

00:07:27.940 --> 00:07:30.360
And then if you say the
ones that have no defects,

00:07:30.360 --> 00:07:33.660
that's 200 minus 12,
which is way more than 10,

00:07:33.660 --> 00:07:35.170
and especially in that latter case.

00:07:35.170 --> 00:07:36.760
But in every situation,

00:07:36.760 --> 00:07:40.570
we expect to have at least
10 successes and 10 failures.

00:07:40.570 --> 00:07:44.520
And so we can assume that the
distributions of each of these

00:07:44.520 --> 00:07:45.810
are going to be normal.

00:07:45.810 --> 00:07:47.250
And we also know that the difference

00:07:47.250 --> 00:07:51.820
of two normally distributed
variables is also normal,

00:07:51.820 --> 00:07:54.930
so long as they pass that
large count condition

00:07:54.930 --> 00:07:56.130
that we just talked about.

00:07:56.130 --> 00:08:00.400
And so let's draw what this
distribution might look like.

00:08:00.400 --> 00:08:02.360
It might look something like this.

00:08:02.360 --> 00:08:04.700
It's going to be a normal distribution

00:08:05.810 --> 00:08:10.293
where you have a mean right over here.

00:08:11.430 --> 00:08:12.860
I'll do that in that same color.

00:08:12.860 --> 00:08:17.740
A mean of 0.02.

00:08:17.740 --> 00:08:19.780
You can definitely take on negative values

00:08:19.780 --> 00:08:21.630
because there are some situations

00:08:21.630 --> 00:08:25.180
in which your sample
proportion from plan B

00:08:25.180 --> 00:08:27.540
actually could be larger
just by random chance

00:08:27.540 --> 00:08:28.960
than it is from plant A.

00:08:28.960 --> 00:08:31.410
So you can definitely
take on negative values.

00:08:31.410 --> 00:08:33.430
But if I wanted to show where zero is,

00:08:33.430 --> 00:08:36.300
maybe zero is right over here,

00:08:36.300 --> 00:08:39.490
so we could draw an axes right over here.

00:08:39.490 --> 00:08:42.110
And then we know what the
standard deviation is.

00:08:42.110 --> 00:08:44.730
It's 0.025 or it's approximately that.

00:08:44.730 --> 00:08:47.000
So if we were to go one
standard deviation down,

00:08:47.000 --> 00:08:48.930
we would go right about there,

00:08:48.930 --> 00:08:50.510
and if we were to go one
standard deviation up,

00:08:50.510 --> 00:08:51.650
we would go right about there.

00:08:51.650 --> 00:08:54.360
And obviously, we could go more
than one standard deviation

00:08:54.360 --> 00:08:56.573
above or below that mean.


Sampling distribution of the difference in sample proportions -Probability example

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:01.280
- [Instructor] In a previous video,

00:00:01.280 --> 00:00:04.080
we explored the sampling distribution

00:00:04.080 --> 00:00:06.320
that we got when we took the difference

00:00:06.320 --> 00:00:08.590
between sample proportions.

00:00:08.590 --> 00:00:11.010
And in that video, we
described the distribution

00:00:11.010 --> 00:00:13.960
in terms of its mean,
standard deviation, and shape.

00:00:13.960 --> 00:00:15.030
What we're going to do in this video

00:00:15.030 --> 00:00:17.210
is build on that example and try to answer

00:00:17.210 --> 00:00:18.910
a little bit more about it.

00:00:18.910 --> 00:00:20.880
So in this situation, what we wanna do

00:00:20.880 --> 00:00:22.720
is find the probability given that we,

00:00:22.720 --> 00:00:27.210
what we already know about this
sampling distribution's mean

00:00:27.210 --> 00:00:29.440
and standard deviation and shape.

00:00:29.440 --> 00:00:32.490
We wanna find the probability
that the sample proportion

00:00:32.490 --> 00:00:36.200
of defects from Plant B is greater

00:00:36.200 --> 00:00:39.480
than the sample proportion from Plant A.

00:00:39.480 --> 00:00:42.593
So pause this video and see
if you can figure this out.

00:00:43.980 --> 00:00:45.510
All right, now let's do this together.

00:00:45.510 --> 00:00:47.690
So first of all, let's just
interpret what this is.

00:00:47.690 --> 00:00:49.640
The probability that the sample proportion

00:00:49.640 --> 00:00:52.210
of defects from Plant B is greater

00:00:52.210 --> 00:00:55.090
than the sample proportion from Plant A.

00:00:55.090 --> 00:00:57.800
So the sample proportion from Plant B

00:00:57.800 --> 00:01:00.450
is greater than the
proportion from Plant A.

00:01:00.450 --> 00:01:03.390
Then the difference between
the sample proportions

00:01:03.390 --> 00:01:05.380
is going to be negative.

00:01:05.380 --> 00:01:08.100
So this is equivalent to the probability

00:01:08.100 --> 00:01:12.820
that the difference of
the sample proportions,

00:01:12.820 --> 00:01:15.210
so the sample proportion from A minus

00:01:15.210 --> 00:01:17.050
the sample proportion from B

00:01:17.050 --> 00:01:21.060
is going to be less than zero.

00:01:21.060 --> 00:01:22.570
Or another way to think about it,

00:01:22.570 --> 00:01:27.040
that's going to be this
area right over here.

00:01:27.040 --> 00:01:28.180
Now, there's a bunch of ways

00:01:28.180 --> 00:01:29.770
that we can figure out this area,

00:01:29.770 --> 00:01:31.390
but the easiest or one of the easiest,

00:01:31.390 --> 00:01:33.450
I guess there's many
different ways to do it,

00:01:33.450 --> 00:01:35.850
is to figure out, well, how many,

00:01:35.850 --> 00:01:37.330
this is up to and including

00:01:37.330 --> 00:01:40.990
how many standard
deviations below the mean,

00:01:40.990 --> 00:01:43.200
and then we could use a Z table.

00:01:43.200 --> 00:01:48.150
So we just have to do is figure
out what is the Z value here

00:01:49.410 --> 00:01:51.350
and the Z value here we just have to say,

00:01:51.350 --> 00:01:55.160
well, how many standard
deviations below the mean is this?

00:01:55.160 --> 00:01:56.690
And I'll do it up here.

00:01:56.690 --> 00:01:58.330
Let me square this off.

00:01:58.330 --> 00:02:00.500
So I don't make it too messy.

00:02:00.500 --> 00:02:02.690
Z is going to be equal to,

00:02:02.690 --> 00:02:07.690
so we are negative 0.02 from the mean

00:02:08.640 --> 00:02:10.810
or we're 0.02 to the left of the mean.

00:02:10.810 --> 00:02:13.023
So I'll just do negative 0.02,

00:02:14.060 --> 00:02:18.307
and then over the standard
deviation, which is 0.025,

00:02:20.310 --> 00:02:22.590
which is going to be equal to,

00:02:22.590 --> 00:02:24.053
I'll get a calculator here.

00:02:25.570 --> 00:02:30.570
We get 0.02 divided by
0.025 is equal to that.

00:02:33.780 --> 00:02:38.760
And we are, of course, going
to be to the left of the mean.

00:02:38.760 --> 00:02:42.910
So our Z is going to be
approximately negative 0.8 or 0.8.

00:02:42.910 --> 00:02:44.900
I'm saying approximately
'cause this was approximate

00:02:44.900 --> 00:02:47.470
over here when we figured
out the standard deviation.

00:02:47.470 --> 00:02:52.470
So it is negative 0.8 and
then we just have to use

00:02:53.810 --> 00:02:56.610
a Z lookup table.

00:02:56.610 --> 00:02:59.060
And so if we look at a Z lookup table,

00:02:59.060 --> 00:03:02.930
we see here is if we're
going to negative 0.8,

00:03:02.930 --> 00:03:07.930
negative 0.8 is right over
here, so negative 0.8.

00:03:08.300 --> 00:03:10.610
And then we have zeros after that.

00:03:10.610 --> 00:03:12.810
So we're looking at this right over here.

00:03:12.810 --> 00:03:14.780
The area under the normal curve

00:03:14.780 --> 00:03:17.020
up to and including that Z value.

00:03:17.020 --> 00:03:17.920
So we always have to make sure

00:03:17.920 --> 00:03:19.560
that we're looking at the right thing

00:03:19.560 --> 00:03:23.220
on this standard normal
probabilities table right over here.

00:03:23.220 --> 00:03:25.840
That gives us 0.21 or we could say

00:03:25.840 --> 00:03:29.130
this is approximately 21%.

00:03:29.130 --> 00:03:31.500
So let me get rid of this.

00:03:31.500 --> 00:03:33.820
And so we know that this right over here

00:03:33.820 --> 00:03:38.820
is approximately 21% or we could say 0.21.

00:03:41.010 --> 00:03:42.680
So the probability that
the sample proportion

00:03:42.680 --> 00:03:45.000
of defects from Plant B is greater

00:03:45.000 --> 00:03:47.200
than the sample proportion from Plant A.

00:03:47.200 --> 00:03:50.743
It's, give or take, it's
roughly one in five.


Path independence for line integrals

WEBVTT
Kind: captions
Language: en

00:00:00.030 --> 00:00:02.600
- [Instructor] What I wanna
do in this video is establish

00:00:02.600 --> 00:00:06.480
a reasonably powerful condition
in which we can establish

00:00:06.480 --> 00:00:09.150
that a vector field or that align integral

00:00:09.150 --> 00:00:11.660
of a vector field is path independent.

00:00:11.660 --> 00:00:13.720
And when I say that,
I mean that let's say,

00:00:13.720 --> 00:00:15.970
I were to take this line integral

00:00:15.970 --> 00:00:20.653
along the path c of f dot dr,

00:00:21.500 --> 00:00:25.360
and let's say my path looks like this.

00:00:25.360 --> 00:00:28.220
Let's say my path looks like this.

00:00:28.220 --> 00:00:33.000
that's my x and y axis, y and x,

00:00:33.000 --> 00:00:35.260
and let's say my path
looks something like this.

00:00:35.260 --> 00:00:39.840
I start there and I go over
there to point c just like

00:00:39.840 --> 00:00:43.020
or to my end point, the curve here is.

00:00:43.020 --> 00:00:45.750
And so I would evaluate
this line integral,

00:00:45.750 --> 00:00:48.740
this vector field along this path.

00:00:48.740 --> 00:00:52.310
This would be a path
independent vector field

00:00:52.310 --> 00:00:55.993
or we call that a
conservative vector field if,

00:00:57.110 --> 00:01:02.110
if this thing is equal
to the same integral,

00:01:02.890 --> 00:01:06.160
the same integral over a different path,

00:01:06.160 --> 00:01:07.360
that has the same end point.

00:01:07.360 --> 00:01:11.340
So let's call this c1 so
this is c1, and this is c2.

00:01:11.340 --> 00:01:15.140
This vector field is conservative if,

00:01:15.140 --> 00:01:17.640
if I start at the same point,
but I take a different path,

00:01:17.640 --> 00:01:20.223
let's say I go something like that.

00:01:21.340 --> 00:01:24.490
If I take a different
path and this is my c2,

00:01:24.490 --> 00:01:26.740
I still get the same value.

00:01:26.740 --> 00:01:28.430
these integrals, what's this telling me,

00:01:28.430 --> 00:01:30.900
is it all it cares about
to evaluate these integrals

00:01:30.900 --> 00:01:34.070
is my starting point and my ending point.

00:01:34.070 --> 00:01:36.040
It doesn't care what I do in between.

00:01:36.040 --> 00:01:38.490
It doesn't care how I get
from my starting point

00:01:38.490 --> 00:01:39.373
to my end point.

00:01:40.232 --> 00:01:42.120
These two integrals have
the same start point

00:01:42.120 --> 00:01:45.200
and same end point, so
irregardless of their actual path,

00:01:45.200 --> 00:01:46.430
they're going to be the same.

00:01:46.430 --> 00:01:49.810
that's what it means for f
to be a conservative field

00:01:49.810 --> 00:01:52.710
or what it means to be for this integral

00:01:52.710 --> 00:01:54.810
to be path independent.

00:01:54.810 --> 00:01:57.730
So before I prove or I
show you the conditions,

00:01:57.730 --> 00:02:00.620
let's build up our toolkit a little bit.

00:02:00.620 --> 00:02:04.680
And so you may or may
not have already seen

00:02:04.680 --> 00:02:08.950
the multivariable chain rule,
multivariable chain rule.

00:02:11.310 --> 00:02:13.563
And I'm not gonna prove it in this video,

00:02:14.410 --> 00:02:16.490
but I think it'll be
pretty intuitive for you,

00:02:16.490 --> 00:02:18.670
so maybe it doesn't need to have a proof

00:02:18.670 --> 00:02:19.530
or I'll prove it eventually,

00:02:19.530 --> 00:02:21.670
but I really just wanna
give you the intuition.

00:02:21.670 --> 00:02:24.400
And all that says is that
if I have some function,

00:02:24.400 --> 00:02:29.270
let's say I have f of x and y
but x and y are then functions

00:02:29.270 --> 00:02:34.270
of, let's say 1/3
variable t so f of x of t

00:02:34.290 --> 00:02:39.290
and y of t, that the derivative
of f with respect to t,

00:02:40.270 --> 00:02:45.270
the derivative of f with
respect to t, is multivariable.

00:02:45.820 --> 00:02:48.120
I have two variables here an x and y.

00:02:48.120 --> 00:02:49.240
this is going to be equal

00:02:49.240 --> 00:02:52.440
to the partial of f with respect to x,

00:02:52.440 --> 00:02:56.200
how fast does f change as exchanges,

00:02:56.200 --> 00:02:59.530
times the derivative
of x with respect to t.

00:02:59.530 --> 00:03:02.420
This is a single variable
function right here.

00:03:02.420 --> 00:03:04.890
So you just can take a regular derivative.

00:03:04.890 --> 00:03:09.250
So times how fast x
changes with respect to t.

00:03:09.250 --> 00:03:11.780
This is a standard derivative,
this is partial derivative,

00:03:11.780 --> 00:03:14.760
because at that level we're
dealing with two variables

00:03:14.760 --> 00:03:17.790
and we're not done plus how fast f changes

00:03:17.790 --> 00:03:19.270
with respect to y.

00:03:19.270 --> 00:03:23.470
How fast the partial of
f with respect to y times

00:03:23.470 --> 00:03:26.050
the derivative of y with respect to t.

00:03:26.050 --> 00:03:26.883
So dy, dt.

00:03:29.340 --> 00:03:30.210
I'm not gonna prove it,

00:03:30.210 --> 00:03:32.820
but I think it makes
pretty good intuition.

00:03:32.820 --> 00:03:35.810
This is saying, as I move a little bit dt,

00:03:35.810 --> 00:03:37.390
how much of a df do I get

00:03:37.390 --> 00:03:40.340
or how fast does f
change with respect to t?

00:03:40.340 --> 00:03:42.300
It says, well, there's two
ways that f can change.

00:03:42.300 --> 00:03:43.850
It can change with respect to x,

00:03:43.850 --> 00:03:45.840
and it can change with respect to y.

00:03:45.840 --> 00:03:48.310
So why don't I add those
two things together

00:03:48.310 --> 00:03:51.090
as they are both changing
with respect to t?

00:03:51.090 --> 00:03:53.840
That's all it's saying and
if you kind of imagined

00:03:53.840 --> 00:03:56.570
that you could cancel out
this partial x with this dx

00:03:56.570 --> 00:03:58.590
and this partial y with this dy,

00:03:58.590 --> 00:03:59.450
you could have kind of imagined

00:03:59.450 --> 00:04:03.360
the partial of f with respect
to t on the x side of things,

00:04:03.360 --> 00:04:06.460
and that plus the partial
of f with respect to t

00:04:06.460 --> 00:04:08.730
on the y in the y dimension,

00:04:08.730 --> 00:04:11.070
and then that'll give
you the total change of f

00:04:11.070 --> 00:04:12.250
with respect to t.

00:04:12.250 --> 00:04:15.443
Kind of a hand wavy argument
there, but at least to me,

00:04:15.443 --> 00:04:17.010
this is a pretty intuitive formula.

00:04:17.010 --> 00:04:18.400
So that's our toolkit right there,

00:04:18.400 --> 00:04:20.590
the multivariable chain rule.

00:04:20.590 --> 00:04:23.140
We're gonna put that aside for a second.

00:04:23.140 --> 00:04:27.970
Now, let's say I have some vector field f

00:04:28.920 --> 00:04:30.010
and this is a different than this f,

00:04:30.010 --> 00:04:32.290
so I'll do it in a
different color, magenta.

00:04:32.290 --> 00:04:36.800
I have some vector field f
that is a function of x and y,

00:04:36.800 --> 00:04:39.010
and let's say that it happens to be

00:04:39.010 --> 00:04:42.260
the gradient of some scalar field.

00:04:42.260 --> 00:04:47.000
Let's say it equals the
gradient of some scalar field

00:04:47.000 --> 00:04:49.410
I'll call that capital F.

00:04:49.410 --> 00:04:50.740
And this is gradient which means,

00:04:50.740 --> 00:04:54.580
and this is capital f is
also a function of x and y.

00:04:54.580 --> 00:04:58.400
So I could write f is
also, let me write it,

00:04:58.400 --> 00:05:00.190
I don't wanna write it on a new line,

00:05:00.190 --> 00:05:01.690
I could also write up here.

00:05:01.690 --> 00:05:04.890
Capital F is also a function of x and y

00:05:04.890 --> 00:05:08.630
and the gradient, all
that means is that f,

00:05:08.630 --> 00:05:12.520
the vector field f of
x, y lowercase f of x, y

00:05:12.520 --> 00:05:16.850
is equal to the partial
derivative of uppercase F

00:05:16.850 --> 00:05:21.070
with respect to x times
the eye unit vector,

00:05:21.070 --> 00:05:25.020
plus the partial of
uppercase F with respect

00:05:25.020 --> 00:05:27.810
to y times the j unit vector.

00:05:27.810 --> 00:05:30.640
This is the definition
of a gradient right here.

00:05:30.640 --> 00:05:32.300
This is the definition of a gradient,

00:05:32.300 --> 00:05:36.690
and if you imagine that uppercase
F is some type of surface,

00:05:36.690 --> 00:05:41.130
so if uppercase F is some
type of surface like that,

00:05:41.130 --> 00:05:46.130
just trying to do my, so
this is uppercase F of x, y

00:05:46.560 --> 00:05:49.900
the gradient of f of x, y is
going to be a vector field

00:05:49.900 --> 00:05:51.330
that tells you the direction

00:05:51.330 --> 00:05:53.500
of steepest ascent at any point.

00:05:53.500 --> 00:05:56.730
So it'll be defined on the x, y plane.

00:05:56.730 --> 00:05:59.890
So on the x, y plane, it'll
tell you, so let me draw.

00:05:59.890 --> 00:06:03.130
That's the vertical axis,
maybe that's the x axis,

00:06:03.130 --> 00:06:04.870
that's the y axis.

00:06:04.870 --> 00:06:05.892
So the gradient of it,

00:06:05.892 --> 00:06:08.150
if you take any point on the x, y plane

00:06:08.150 --> 00:06:10.280
it'll tell you the direction
you need to travel to go

00:06:10.280 --> 00:06:12.850
into this deepest descent
and for this gradient field

00:06:12.850 --> 00:06:15.610
it's going to be something like this.

00:06:15.610 --> 00:06:18.119
It's going to be something like this

00:06:18.119 --> 00:06:20.640
and maybe over here it starts
going in that direction

00:06:20.640 --> 00:06:22.600
'cause you would descend towards

00:06:22.600 --> 00:06:24.310
this little minimum point right here,

00:06:24.310 --> 00:06:25.587
anyway I don't wanna get
too involved in that.

00:06:25.587 --> 00:06:28.970
And the whole point of this
isn't to really you get

00:06:28.970 --> 00:06:30.340
the intuition behind ingredients

00:06:30.340 --> 00:06:32.070
there are other videos on this.

00:06:32.070 --> 00:06:34.780
The point of this is to get a a test

00:06:34.780 --> 00:06:37.990
to see whether something
is path independent

00:06:37.990 --> 00:06:39.620
whether a vector field
is path independent,

00:06:39.620 --> 00:06:42.160
whether it's conservative and it turns out

00:06:42.160 --> 00:06:45.720
that if this exists and
I'm gonna prove it now.

00:06:45.720 --> 00:06:50.720
If this exists, if f is the
gradient of some scalar field

00:06:51.850 --> 00:06:56.850
if f is equal to the
gradient of some scalar field

00:06:57.370 --> 00:07:02.370
then, then f is conservative, conservative

00:07:04.700 --> 00:07:07.780
or you could say it doesn't
matter what path we follow

00:07:07.780 --> 00:07:10.320
when we take a line integral over f,

00:07:10.320 --> 00:07:12.920
it just matters about our starting point

00:07:12.920 --> 00:07:14.980
and our ending point.

00:07:14.980 --> 00:07:18.140
Now, let me see if I
can prove that to you.

00:07:18.140 --> 00:07:19.870
So let's start with the assumption

00:07:19.870 --> 00:07:21.530
that f can be written this way

00:07:21.530 --> 00:07:25.040
as the gradient of that
lowercase f can be written

00:07:25.040 --> 00:07:28.150
as the gradient of some uppercase F.

00:07:28.150 --> 00:07:31.093
So in that case, in that case,

00:07:32.071 --> 00:07:37.071
our integral, let's see well,
let's define our path first.

00:07:37.470 --> 00:07:41.360
So our position vector
function, we always need one

00:07:41.360 --> 00:07:44.690
of those to do a line integral
or a vector line integral,

00:07:44.690 --> 00:07:49.667
r of t is going to be
equal to x of t times i

00:07:50.510 --> 00:07:55.510
plus y of t times J for
t going between a and b,

00:07:56.780 --> 00:07:58.580
we've seen this multiple times.

00:07:58.580 --> 00:08:00.200
This is the definition of

00:08:00.200 --> 00:08:01.720
this is just a very general definition

00:08:01.720 --> 00:08:04.960
of pretty much any path in two dimensions.

00:08:04.960 --> 00:08:09.960
And then we're gonna say f,
f is going to be f of x, y

00:08:12.230 --> 00:08:13.620
is going to be equal to this

00:08:13.620 --> 00:08:15.930
it's going to be the partial derivative

00:08:15.930 --> 00:08:19.810
of uppercase F with respect
to x so we're assuming

00:08:19.810 --> 00:08:22.120
that this exists, that this is true

00:08:22.120 --> 00:08:25.790
except with times i plus
the partial of uppercase F

00:08:25.790 --> 00:08:29.230
with respect to y times j.

00:08:29.230 --> 00:08:34.230
Now, given this what is f
lowercase f lowercase dot dr

00:08:42.200 --> 00:08:45.120
going to equal over this path right here

00:08:45.120 --> 00:08:48.700
this path is divined by this
position function right there.

00:08:48.700 --> 00:08:51.540
Well, it's going to be
equal to what we need to get

00:08:51.540 --> 00:08:52.890
we need to figure out what dr is

00:08:52.890 --> 00:08:54.300
and we've done that in multiple videos

00:08:54.300 --> 00:08:56.510
so I'll do that on the right over here.

00:08:56.510 --> 00:08:59.610
Dr, we've seen it multiple times I'll just

00:08:59.610 --> 00:09:00.850
actually, I'll solve it out again.

00:09:00.850 --> 00:09:04.460
DR, dt by definition was equal

00:09:04.460 --> 00:09:09.460
to dx, dt is equal to dx,
dt times i plus dy, d,

00:09:13.840 --> 00:09:15.850
I don't know why it got all fat like that.

00:09:15.850 --> 00:09:20.850
Dy, dt times j, that's what dr, dt is.

00:09:21.500 --> 00:09:24.930
So if we wanna figure out what
dr is, the differential dr,

00:09:24.930 --> 00:09:26.750
if we wanna play with
differentials in this way,

00:09:26.750 --> 00:09:29.350
multiply both sides times dt,

00:09:29.350 --> 00:09:30.970
and actually I'm gonna treat dt as,

00:09:30.970 --> 00:09:32.700
well, I'll multiply
it, I'll distribute it.

00:09:32.700 --> 00:09:37.700
It's dx, dt times dt, i
plus dy, dt times dt, j.

00:09:44.050 --> 00:09:45.460
So if we're taking the dot product

00:09:45.460 --> 00:09:49.233
of f of f with dr, what are we gonna get?

00:09:50.310 --> 00:09:51.760
What are we gonna get?

00:09:51.760 --> 00:09:53.750
We are going to get, and this is, let me,

00:09:53.750 --> 00:09:55.120
so we're gonna take,

00:09:55.120 --> 00:09:59.920
so this is going to be the
integral over the curve,

00:09:59.920 --> 00:10:03.220
over the curve from, all
right, the c right there.

00:10:03.220 --> 00:10:06.730
We can write it in terms
of the end points of t,

00:10:06.730 --> 00:10:10.350
once we feel good that we
have everything in terms of t,

00:10:10.350 --> 00:10:12.590
but it's gonna be equal to this dot that

00:10:12.590 --> 00:10:14.300
which is equal to the partial,

00:10:14.300 --> 00:10:16.410
I'll try to stay color consistent,

00:10:16.410 --> 00:10:19.240
the partial of uppercase F with respect

00:10:19.240 --> 00:10:23.187
to x times that times dx, dt.

00:10:25.080 --> 00:10:29.940
I'm gonna write this dt in a
different color, times dt plus,

00:10:32.330 --> 00:10:37.000
plus the partial of
uppercase F with respect

00:10:37.000 --> 00:10:42.000
to y times this, we're multiplying
the j components, right?

00:10:42.040 --> 00:10:44.970
We can take the dot product,
multiply the i components,

00:10:44.970 --> 00:10:48.390
and then add that to, what
you get from the product

00:10:48.390 --> 00:10:49.223
of the j component.

00:10:49.223 --> 00:10:51.360
So this j component is
partial of uppercase F

00:10:51.360 --> 00:10:52.550
with respect to y.

00:10:52.550 --> 00:10:56.620
And then we have times,
let me switch to a yellow,

00:10:56.620 --> 00:11:01.620
dy, dt times that dt right over there

00:11:01.670 --> 00:11:06.670
times that dt over there, and
then we can factor out the dt.

00:11:06.820 --> 00:11:08.930
We could factor out the dt

00:11:08.930 --> 00:11:09.950
or actually it's just so I don't have

00:11:09.950 --> 00:11:11.050
to even write it again,

00:11:11.050 --> 00:11:12.820
right now, I wrote it without,
well, let me write it again.

00:11:12.820 --> 00:11:16.180
So this is equal to the integral,

00:11:16.180 --> 00:11:18.030
and let's say we have it in terms of t.

00:11:18.030 --> 00:11:19.490
We've written everything in terms of t,

00:11:19.490 --> 00:11:22.250
so t goes from a to a.

00:11:22.250 --> 00:11:26.370
And so this is going to be
equal to, I'll write it in blue,

00:11:26.370 --> 00:11:31.370
the partial of uppercase F
with respect to x times dx, dt

00:11:33.100 --> 00:11:34.940
plus, I'm distributing this dt out,

00:11:34.940 --> 00:11:39.777
plus the partial of uppercase
F with respect to y, dy, dt,

00:11:42.240 --> 00:11:46.550
all of that times dt.

00:11:46.550 --> 00:11:49.210
This is equivalent to that.

00:11:49.210 --> 00:11:51.100
Now you might realize why

00:11:51.100 --> 00:11:53.730
I talked about the
multivariable chain rule.

00:11:53.730 --> 00:11:55.530
What is this right here?

00:11:55.530 --> 00:11:57.130
What is that right there?

00:11:57.130 --> 00:11:59.090
You can do some pattern matching.

00:11:59.090 --> 00:12:04.090
That is the same thing as
the derivative of uppercase F

00:12:04.420 --> 00:12:05.970
with respect to t.

00:12:05.970 --> 00:12:07.870
Look at this, let me, let
me copy and paste this

00:12:07.870 --> 00:12:09.170
just so you appreciate it.

00:12:10.160 --> 00:12:12.250
Let me copy and paste that.

00:12:12.250 --> 00:12:15.333
You copy it, and then let me paste it.

00:12:17.160 --> 00:12:20.910
So this is our definition or this is our,

00:12:20.910 --> 00:12:22.930
I won't say definition
one can actually prove it,

00:12:22.930 --> 00:12:24.350
you don't have to start from there,

00:12:24.350 --> 00:12:27.230
but this is our multivariable
chain rule right here, right?

00:12:27.230 --> 00:12:29.300
The derivative of any
function with respect to t

00:12:29.300 --> 00:12:30.710
is the partial of that
function with respect

00:12:30.710 --> 00:12:33.080
to x times dx, dt plus the
partial of that function

00:12:33.080 --> 00:12:34.920
with respect to y, dy, dt.

00:12:34.920 --> 00:12:38.910
I have the partial of uppercase
F with respect to x, dx, dt

00:12:38.910 --> 00:12:42.440
plus the partial uppercase
F with respect to y.

00:12:42.440 --> 00:12:45.130
And this are identical if you
just replace this lowercase f

00:12:45.130 --> 00:12:46.460
with an uppercase F.

00:12:46.460 --> 00:12:47.880
So this in blue right here,

00:12:47.880 --> 00:12:50.170
so this whole expression is equal to,

00:12:50.170 --> 00:12:54.950
the integral from a to b, t
is equal a to t is equal to b

00:12:54.950 --> 00:12:58.930
of, of, in blue here,

00:12:58.930 --> 00:13:02.660
the derivative, regular derivative of f

00:13:02.660 --> 00:13:07.220
with respect to t, dt.

00:13:07.220 --> 00:13:08.270
And how do you evaluate?

00:13:08.270 --> 00:13:10.060
Let me read the dt in green.

00:13:10.060 --> 00:13:12.000
How do you evaluate something like this?

00:13:12.000 --> 00:13:12.890
And I just wanna make a point.

00:13:12.890 --> 00:13:16.500
This is just this, from the
multivariable chain rule.

00:13:16.500 --> 00:13:19.630
And how do you evaluate a
definite integral like this?

00:13:19.630 --> 00:13:22.250
Will you take the
anti-derivative of the inside

00:13:22.250 --> 00:13:25.110
with respect to dt?

00:13:25.110 --> 00:13:26.810
So what is this going to be equal to?

00:13:26.810 --> 00:13:28.440
You take the anti-derivative
of the inside,

00:13:28.440 --> 00:13:30.740
the anti-derivative of
the inside, that's just df

00:13:30.740 --> 00:13:32.100
oh, sorry, that's just f.

00:13:32.100 --> 00:13:35.563
So this is equal to f of t.

00:13:36.540 --> 00:13:38.420
And let me be clear, let me be clear here.

00:13:38.420 --> 00:13:41.730
We wrote before that f is a function,

00:13:41.730 --> 00:13:46.060
so our uppercase F is
a function of x and y

00:13:47.196 --> 00:13:48.920
which could also be
written since each of these

00:13:48.920 --> 00:13:53.250
are functions of t of, it
could be written as f of x of t

00:13:53.250 --> 00:13:55.870
of y of t, I'm just rewriting
it in different ways.

00:13:55.870 --> 00:13:58.850
And this could be just written as f of t.

00:13:58.850 --> 00:14:00.460
This is the same thing as f of t.

00:14:00.460 --> 00:14:01.800
These are all equivalent,

00:14:01.800 --> 00:14:02.730
depending on whether you wanna include

00:14:02.730 --> 00:14:04.930
the x's and the y's only or the t's only

00:14:04.930 --> 00:14:08.130
or them both cause both of the x's and y's

00:14:08.130 --> 00:14:10.400
are functions of t.

00:14:10.400 --> 00:14:13.650
So this is the derivative
of f with respect to t.

00:14:13.650 --> 00:14:15.280
If this was just in terms of t,

00:14:15.280 --> 00:14:17.480
this is the derivative of
that with respect to t.

00:14:17.480 --> 00:14:20.460
We take it's anti-derivative,
we're left just with f,

00:14:20.460 --> 00:14:22.190
and we have to evaluate it

00:14:23.320 --> 00:14:26.550
from t is equal to a to t is equal to b.

00:14:26.550 --> 00:14:29.750
And so this is equal to,
and this is the homestretch,

00:14:29.750 --> 00:14:34.507
this is equal to f of b minus f of a.

00:14:35.840 --> 00:14:38.100
And if you wanna think
about it in these terms,

00:14:38.100 --> 00:14:39.280
this is the same thing.

00:14:39.280 --> 00:14:44.280
This is equal to f of x of b, y of b,

00:14:46.440 --> 00:14:47.440
let me re-check out all the parentheses,

00:14:47.440 --> 00:14:52.440
minus f of x of a, y of
a, these are equivalent.

00:14:53.730 --> 00:14:58.170
You give me any point on the
x, y plane, an x and a y,

00:14:58.170 --> 00:15:00.100
and it tells me where I am.

00:15:00.100 --> 00:15:04.350
This is my capital F, it gives
me a height just like that.

00:15:04.350 --> 00:15:05.830
But what this tells me this,

00:15:05.830 --> 00:15:10.040
this associates a value with
every point on the x, y plane.

00:15:10.040 --> 00:15:12.210
And what this whole exercise,

00:15:12.210 --> 00:15:15.310
remember this is the same thing as that.

00:15:15.310 --> 00:15:17.000
This is our whole thing that
we were trying to prove.

00:15:17.000 --> 00:15:21.053
That is equal to f dot dr.

00:15:21.930 --> 00:15:25.980
F dot dr, our vector field
for which is the gradient

00:15:25.980 --> 00:15:28.870
of the capital F, remember F,

00:15:28.870 --> 00:15:30.770
F was equal to the gradient of f.

00:15:30.770 --> 00:15:32.160
We assume that it's the gradient

00:15:32.160 --> 00:15:34.460
of some function, capital F.

00:15:34.460 --> 00:15:36.210
If that is the case,

00:15:36.210 --> 00:15:38.530
then we just did a little
bit of calculus or algebra

00:15:38.530 --> 00:15:39.680
or whatever you wanna call it.

00:15:39.680 --> 00:15:42.030
And we found that we can
evaluate this integral

00:15:43.014 --> 00:15:46.850
by evaluating capital
F at t is equal to b,

00:15:46.850 --> 00:15:50.600
and then subtracting from that
capital f at t is equal to a.

00:15:50.600 --> 00:15:52.210
But what that tells you
is that this integral,

00:15:52.210 --> 00:15:56.860
the value of this integral
is only dependent,

00:15:56.860 --> 00:16:01.710
is only dependent at our
starting point, t is equal to a.

00:16:01.710 --> 00:16:06.710
This is the point x of a, y
of a, and the ending point

00:16:07.960 --> 00:16:12.960
t of b, x of, or t is equal
to b which is x of b, y of b.

00:16:13.850 --> 00:16:16.060
That integral is only
dependent on these two values.

00:16:16.060 --> 00:16:17.310
How do I know that?

00:16:17.310 --> 00:16:18.830
Because to solve it,

00:16:18.830 --> 00:16:21.080
because I'm saying that this thing exists,

00:16:21.080 --> 00:16:24.230
I just had to evaluate that
thing at both those two points.

00:16:24.230 --> 00:16:26.940
I didn't care about the curve in between.

00:16:26.940 --> 00:16:31.940
So this shows that if, if
f is equal to the gradient,

00:16:34.310 --> 00:16:36.400
this is often called a potential function

00:16:36.400 --> 00:16:38.180
of capital F, although they're usually

00:16:38.180 --> 00:16:40.210
the negative of each other,
but it's the same idea.

00:16:40.210 --> 00:16:43.160
If the vector field f is the gradient

00:16:43.160 --> 00:16:47.113
of some scalar field, uppercase F, then,

00:16:48.110 --> 00:16:52.720
then we can say that f is conservative.

00:16:52.720 --> 00:16:56.740
F is conservative or that the integral,

00:16:56.740 --> 00:17:01.010
the line integral of f dot
dr is path independent.

00:17:01.010 --> 00:17:03.290
It doesn't matter what path we go on

00:17:03.290 --> 00:17:06.114
as long as our starting and
ending points are the same.

00:17:06.114 --> 00:17:07.450
Hopefully, you found that useful.

00:17:07.450 --> 00:17:08.720
I will do some examples with that.

00:17:08.720 --> 00:17:09.820
And actually in the next video,

00:17:09.820 --> 00:17:12.670
I'll prove another interesting
outcome based on this one.