Prelude to the Peloponnesian War

WEBVTT
Kind: captions
Language: en

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- [Narrator] In the last few videos

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we talked about the Greco Persian wars

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or we could say the
Persian invasion of Greece.

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And in the first wave, the
first Persian invasion,

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the Athenians were able
to stop them at Marathon.

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And then in the second Persian invasion

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led by Xerxes they were able to route

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the Persian navy at Salamis

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and eventually get rid of
the Persians all together

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at Plataea in 479 BCE.

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So at that point the Greeks,
especially the city-states,

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the powerful city-states
of Sparta and Athens,

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they're fighting on the same side.

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But as we'll see in this video

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this all starts to break down

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and they will eventually
be at war with each other

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and a very ugly war.

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You know a lot of times when
we think about these things

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thousands of years later it
might seem somewhat romantic,

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the stuff of stories and adventures,

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but these things are ugly,
people are getting killed,

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people are getting tortured,
civilians are dying,

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whole cities are being destroyed.

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Now whenever we talk about history

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it's very important to think about

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where your story is coming from

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and you should always
have a nice dose of doubt

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on what were the motivations
of the person who wrote it

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or how much can they
really know about things.

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And when we talked about
the Persian invasions

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of Greece, or the Greco-Persian wars,

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that was coming to us from
someone who is often known

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as the father of history,

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I guess that's a very
Greco-centric perspective,

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but that is Herodutus.

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And now we're going to
start getting our accounts

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from someone who wrote the famous history

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of the Peloponnesian War
and that is Thucydides.

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And they're so closely linked,

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especially because they're giving history

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of essentially this fifth century BCE

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that this is actually a
double bust of the two,

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Herodutus and Thucydides.

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So once again a lot of the story

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that I'm about to tell,
is coming via Thucydides.

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So as soon as the Persians are,

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they've been defeated, they leave.

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You might remember that the Athenians

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had to escape to Salamis,
that's where you also had

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the naval route of the Persian navy,

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and so the Athenians decide hey let's move

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back to Athens.

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And in deciding to move back to Athens,

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right over there, they
say okay well we want

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to build a wall around
Athens in case anyone else

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wants to invade us.

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And you might say okay maybe
they are somewhat concerned

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about the Persians, but
you could also argue

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that they are somewhat
concerned about the Spartans

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who are famous for having
this very powerful army.

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And so they start the construction

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on what's often called the long wall

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and it's a way for them to have access

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to the sea and to protect their city

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in the event of a siege.

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Now if you take the other side,

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if you think about what's going on

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in the minds of the Spartans,

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they are thinking okay
you know for a long time

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we were really the dominant military power

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but what we've just seen
with the Greco-Persian wars

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is this Athenian navy
is actually quite strong

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and in some ways, in fact in a lot of ways

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you can argue that they are the ones

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that were able to beat off
the Persians at Salamis.

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That the navy was essential there.

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So Sparta's feeling a
little bit threatened.

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And in particular they don't like the idea

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of this wall because here you
have the city-state Athens

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with this very strong navy,
now they're building a wall.

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Sparta says hey if they have a wall

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it's going to be very hard
for us if they don't do

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what, you know, if they start
exerting too much influence,

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it's going to be hard for
us to take down Athens.

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They instead suggest hey Athens
why don't you build a wall

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at the Isthmus of
Corinth, right over here.

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You might notice the geography of Greece.

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You have this little, I
guess you could call it

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this little strip of land that connects

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the mainland I guess you could say,

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to this peninsula, this
very large peninsula

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which is called Peloponnese,

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which is where Sparta is.

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And you could imagine that this is

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a very strategic location.

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Anyone who wants to
conquer the Peloponnese

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by land is going to have
to go through there.

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The Athenians are of
course no, we care about

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protecting ourselves and so they start

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to build this wall to protect themselves

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which the Spartans already
are not so happy about.

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Now the other thing that the Athenians do,

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they continue to build
up their naval power.

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They see hey this is a very useful thing.

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They start connecting
with a bunch of allies

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around the Aegean, some on islands,

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some on the coast of Anatolia over here.

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And they meet on this island of Delos

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which you can't see here.

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It's a very small island.

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And they all agree to
contribute their shared navies

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and to contribute money
in order to continue

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to fight off the Persians
wherever they are.

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You know these Persians
did this injustice to us.

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Now let's go on the offensive.

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And they form what is
called the Delian League.

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Now once again Sparta is
not so happy about this

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because more and more it becomes clear

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that this Delian League, Athens isn't just

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a participant in the Delian League,

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Athens is starting to
run the Delian League.

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And when people try to drop
out of the Delian League

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over time Athens starts to exert itself.

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So the Delian League is really starting

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more and more and more
into an Athenian empire.

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So once again Athens proved itself,

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its navy proved itself to be
very powerful in this war.

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Sparta's getting insecure.

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Athens doesn't listen to
Sparta about this wall

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and decides to start building it.

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Sparta's feeling even more insecure.

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Athens is starting to
create this Delian League

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which is really an Athenian empire,

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even more insecure.

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And it kind of gets a little
bit more intense in 464.

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So there are some accounts that Sparta

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around this time was already planning

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to invade Athens.

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To say hey these folks
are getting too powerful,

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let's just put them in their place.

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But at around that time, in 464,

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you have a major earthquake in Sparta.

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And so it demolishes
large parts of the city.

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And when that earthquake happens,

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the Spartan Helots who I guess

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the best way you can view
them, they are slaves

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and I encourage you to
read more about them.

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They were treated horribly.

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And in Sparta the accounts
are that there were seven

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of these Helots, of these
slaves, for every Spartan.

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And so the Helots start a revolt

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and Sparta goes to the other city-states

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of Greece and says help
us put down this revolt

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that happened right after our earthquake.

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And so Athens agrees.

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They send 4,000 citizen
soldiers to Sparta.

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The citizen soldiers are famous Hoplites

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to help put down the revolt.

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But once again Sparta is feeling insecure.

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They say hey why did
they send these soldiers.

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Maybe they say they want
to put down this rebellion,

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but we Sparta we're at kind
of a fairly vulnerable state

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right here, what if they
Athenians took the side

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of the Helots.

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Well then they might be
able to take over Sparta.

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So they send the Athenian Hoplites back.

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They said you know we
don't need your help.

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And so you can imagine the Athenians now

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are starting to take significant offense

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to how the Spartans are treating them.

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And then you fast
forward a little bit more

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to 459 BCE and you have
two allies of Sparta,

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Megara and Corinth and they're right

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in this very strategic location,

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that little land, that land connection

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that land bridge between the mainland

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and the Peloponnese and there is a dispute

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between these two allies of Sparta,

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between Corinth and Megara.

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And Athens decides to intervene

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and form an alliance with Megara.

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Well you can imagine
you know this insecurity

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has been building up in Sparta

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for at this point over a decade

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and so it all comes to head

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and they start having skirmishes

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and they start essentially going

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to war with each other.

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They have these sporadic skirmishes

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over the next 15 years.

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And this period right over here

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from roughly 460 or 459,
for the next 15 years,

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this is often referred to as
the First Peloponnesian War

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because once again you have Sparta

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and its allies starting to get into

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all of these skirmishes
with Athens and its allies.

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But this isn't the Peloponnesian War.

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The Peloponnesian War is going to come

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a little bit about 15 years later.

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The first Peloponnesian
War that we were just

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talking about, that eventually ends

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what's called a 30 years
peace, peace treaty

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which only lasts for 15 years.

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But the whole time the Athenians,

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their navy is growing more powerful,

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they keep getting in each others way

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and annoying each other and
so the Peloponnesian War

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which we'll talk about in more detail

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will occur at the end
of the fifth century BCE

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and it will conclude with
Sparta being victorious.

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But in the process most of Greece

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is significantly hurt.

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And all of the city-states or
the significant city-states

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of Greece are significantly weakened.

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And so that sets things up for

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the fourth century BCE where it leaves

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the city-states of Greece vulnerable

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to attack from the north,
in particular Macedonia.

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And we'll see that actually
several videos in the future.


Second Persian Invasion

WEBVTT
Kind: captions
Language: en

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- [Instructor] In the last videos,

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we saw a dominant Persia
have to put down a rebellion

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by the Ionians in the Anatolian Peninsula,

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and they were really, really mad

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that these Ionians were
helped by the Athenians

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and the Eretrians,

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and so Darius, the King of Kings goes off

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to try to conquer and put down

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the Athenians and the Eretrians.

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The first time he sends
a fleet in 492 BCE,

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it's destroyed by a storm.

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And we'll see this is not the last time

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that at least part of a Persian fleet

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is destroyed by a storm.

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Then in 490, he sends a fleet again,

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and then he is defeated by the Athenians

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at Marathon,

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and we saw that right over there.

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Let me do that in color
you can actually see.

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But as I mentioned in the last video,

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the Persians were not done.

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Darius would not live to see

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another round with the Greeks,

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but his successor, Xerxes, would not only

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try another attempt, but he
would amass a huge, huge force

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against the Greeks to, in his
mind, finally take them over,

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and he wants it done so badly

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that he leads the forces himself.

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And so we are now 10 years after,

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10 years after the first Persian invasion.

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We are now at 480 BCE, where Xerxes

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is going to try to invade
Greece by land and by sea,

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but as we will see, he is also
not going to be successful.

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But this second invasion
is the stuff of legend,

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and once again, the
historical accounts come to us

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primarily from Herodotus, who was Greek,

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and who was not a direct observer of this.

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So you might want to take all
of this with a grain of salt,

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because it does make the
Greeks look awfully good.

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But we do think that
most of this happened,

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but obviously Herodotus probably

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added a little bit of bias there.

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We are likely to never know.

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So let's think about, or let's see

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what Xerxes attempts to do.

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So we see this magenta line.

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This is the line of
attack of Xerxes in 480,

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and you can see, there's one magenta line

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that is going by sea,

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and one magenta line that
is going by sea here,

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and another magenta line
that is going by land.

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And let's zoom in a little bit more.

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I have another map here.

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So let's zoom in to
this map right over here

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to give us a clearer picture
of what's about to happen

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in this next Persian invasion.

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So just to reorient ourselves,

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here are the land forces,

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and according to Herodotus
and historians of that time,

00:03:01.215 --> 00:03:03.189
they numbered this force

00:03:03.189 --> 00:03:05.464
in potentially millions of soldiers,

00:03:05.464 --> 00:03:08.088
modern historians think it
was closer to 50 thousand

00:03:08.088 --> 00:03:10.317
to 300 thousand, we really don't know,

00:03:10.317 --> 00:03:13.057
but we think it's in the
approximately hundred thousand,

00:03:13.057 --> 00:03:14.997
or a few hundred thousands, not millions,

00:03:14.997 --> 00:03:17.110
but by any measure, that is a huge,

00:03:17.110 --> 00:03:18.943
a huge military force.

00:03:19.815 --> 00:03:23.078
So this is hundreds of
thousands right over here.

00:03:23.078 --> 00:03:25.161
So hundreds of thousands.

00:03:26.108 --> 00:03:27.722
Or let me write this.

00:03:27.722 --> 00:03:31.820
Hundreds of thousands, maybe
between 50 and 300 thousand.

00:03:31.820 --> 00:03:35.036
They're coming this
way, the Persian forces.

00:03:35.036 --> 00:03:39.203
And they also have a fleet
of approximately 1,200 ships.

00:03:40.551 --> 00:03:43.418
Now the Persians really
are not having good luck

00:03:43.418 --> 00:03:45.891
with weather whenever
they try to attack Greece.

00:03:45.891 --> 00:03:47.865
They face a storm, and about a third

00:03:47.865 --> 00:03:49.699
of their fleet is destroyed.

00:03:49.699 --> 00:03:51.522
And so up here they are left with about,

00:03:51.522 --> 00:03:53.229
and these are all approximate,

00:03:53.229 --> 00:03:56.201
remember this happened
over 2,000 years ago,

00:03:56.201 --> 00:03:58.024
roughly 2,500 years ago,

00:03:58.024 --> 00:04:01.193
so it's amazing that we know
anything about it at all,

00:04:01.193 --> 00:04:03.829
and obviously we have to rely on Herodotus

00:04:03.829 --> 00:04:07.393
and whatever historical
accounts we can find.

00:04:07.393 --> 00:04:11.143
So the Persians are
invading by land and sea,

00:04:12.292 --> 00:04:15.253
and the Greeks have a strategy of

00:04:15.253 --> 00:04:19.142
let's try to stop them
by land at Thermopylae,

00:04:19.142 --> 00:04:23.949
and stop them by sea at
the Straight of Artemisium

00:04:23.949 --> 00:04:25.282
right over here.

00:04:27.107 --> 00:04:29.510
And there is some historical debate of

00:04:29.510 --> 00:04:32.737
was this a big, grand
strategy to eventually

00:04:32.737 --> 00:04:36.720
try to defeat the Persians at Salamis,

00:04:36.720 --> 00:04:37.974
which eventually happens,

00:04:37.974 --> 00:04:41.387
or were they generally
trying to stop them there,

00:04:41.387 --> 00:04:43.186
and because they couldn't,

00:04:43.186 --> 00:04:45.764
they had to retreat back to Salamis.

00:04:45.764 --> 00:04:47.412
As we'll see, that second narrative

00:04:47.412 --> 00:04:48.573
is what actually happens.

00:04:48.573 --> 00:04:51.000
Well I guess in some level,
both of those narratives happen.

00:04:51.000 --> 00:04:54.123
That they are able to at
least slow down the Persians

00:04:54.123 --> 00:04:56.375
at both of these places, and the Persians

00:04:56.375 --> 00:05:00.323
are eventually defeated at
the Straight of Salamis.

00:05:00.323 --> 00:05:04.409
Now Thermopylae is the stuff of legends.

00:05:04.409 --> 00:05:07.254
It is, if you've ever seen the movie 300,

00:05:07.254 --> 00:05:10.254
it is about the 300 Spartan soldiers

00:05:11.190 --> 00:05:14.290
led by King Leonitus, along with roughly

00:05:14.290 --> 00:05:16.925
7,000 other Greeks that
they're able to collect

00:05:16.925 --> 00:05:19.967
to stop the Persians at Thermopylae.

00:05:19.967 --> 00:05:22.846
And Thermopylae, you're
in this coastal area

00:05:22.846 --> 00:05:25.830
where there's a very limited area for

00:05:25.830 --> 00:05:28.976
this massive Persian army
to be funneled through,

00:05:28.976 --> 00:05:32.540
and the Greeks are trying to stop them

00:05:32.540 --> 00:05:34.108
at the Pass of Thermopylae.

00:05:34.108 --> 00:05:37.103
And you can see that right over there.

00:05:37.103 --> 00:05:39.402
And, by Herodotus's accounts,

00:05:39.402 --> 00:05:41.573
they are actually quite successful

00:05:41.573 --> 00:05:44.243
because they are funneling
that Persian army

00:05:44.243 --> 00:05:45.787
into a very narrow space,

00:05:45.787 --> 00:05:47.808
the Spartans along with the other Greeks

00:05:47.808 --> 00:05:48.980
are able to push back.

00:05:48.980 --> 00:05:51.070
And this is a massive outnumbering.

00:05:51.070 --> 00:05:54.553
Roughly 7,000 versus many
tens or hundreds of thousands,

00:05:54.553 --> 00:05:57.136
but as Herodotus's accounts go,

00:05:59.882 --> 00:06:01.716
there was a traitor amongst the Greeks

00:06:01.716 --> 00:06:03.411
who would go to the Persians

00:06:03.411 --> 00:06:06.639
and show them another way around,

00:06:06.639 --> 00:06:09.309
and show them another way around.

00:06:09.309 --> 00:06:12.084
And so the Persians are
essentially able to,

00:06:12.084 --> 00:06:13.628
not only get around the Greeks,

00:06:13.628 --> 00:06:15.277
but by surrounding them, are
able to defeat the Greeks

00:06:15.277 --> 00:06:18.829
at Thermopylae and continue their march.

00:06:18.829 --> 00:06:21.929
And remember, they were able
to get to Eretria before,

00:06:21.929 --> 00:06:23.926
10 years before, but
they really want to seek

00:06:23.926 --> 00:06:26.353
their revenge on Athens.

00:06:26.353 --> 00:06:28.895
And they are able to go to go Athens,

00:06:28.895 --> 00:06:30.834
but by the time they go there,

00:06:30.834 --> 00:06:33.087
they see that the town has been,

00:06:33.087 --> 00:06:34.700
for the most part, evacuated.

00:06:34.700 --> 00:06:36.802
That the Athenians, when they saw

00:06:36.802 --> 00:06:38.741
that the Persians were coming,

00:06:38.741 --> 00:06:42.537
they went to Salamis right over here.

00:06:42.537 --> 00:06:44.871
So even though Athens
was sacked and destroyed,

00:06:44.871 --> 00:06:47.494
the Athenian people were not destroyed.

00:06:47.494 --> 00:06:50.327
Now simultaneous with Thermopylae,

00:06:51.918 --> 00:06:56.085
you had a naval battle happen
in the Straight of Artemisium.

00:06:57.119 --> 00:06:58.977
And once again, even
though there was about

00:06:58.977 --> 00:07:02.425
600 Persian ships that
were in this battle,

00:07:02.425 --> 00:07:05.316
there was on the order of
about 200 or 300 Greek ships,

00:07:05.316 --> 00:07:07.417
so the Greeks were once again outnumbered,

00:07:07.417 --> 00:07:09.089
and they were able to
slow down the Persians,

00:07:09.089 --> 00:07:10.482
but not stop them.

00:07:10.482 --> 00:07:12.224
And the Persians kept having bad luck,

00:07:12.224 --> 00:07:13.849
especially with these storms,

00:07:13.849 --> 00:07:16.322
because even we had these
first ships get destroyed,

00:07:16.322 --> 00:07:19.747
they sent some 200 ships around Euboea,

00:07:19.747 --> 00:07:22.359
I'm probably not pronouncing
it perfectly right here,

00:07:22.359 --> 00:07:25.656
but then they get destroyed by a storm.

00:07:25.656 --> 00:07:26.922
So now you have the ships

00:07:26.922 --> 00:07:31.055
that engaged the Greeks in
the Straight of Artemisium,

00:07:31.055 --> 00:07:33.900
the Greeks pull back because
they know they're outnumbered,

00:07:33.900 --> 00:07:36.988
and they essentially go in retreat,

00:07:36.988 --> 00:07:38.927
or what the Persians think are retreat,

00:07:38.927 --> 00:07:40.587
so the Persians follow the Greek fleet

00:07:40.587 --> 00:07:44.140
all the way back to the
Straight of Salamis.

00:07:44.140 --> 00:07:48.307
And it's over there that
the Greek fleet is able to

00:07:50.816 --> 00:07:53.916
plan a defeat of the Persian fleet.

00:07:53.916 --> 00:07:56.273
So you have the Persian fleet gets

00:07:56.273 --> 00:07:58.084
destroyed decisively at Salamis,

00:07:58.084 --> 00:08:00.952
you have the Persian army still in Athens

00:08:00.952 --> 00:08:02.647
is able to destroy Athens,

00:08:02.647 --> 00:08:04.981
but the Athenian people
have not been destroyed,

00:08:04.981 --> 00:08:06.827
and so there's a question for Xerxes,

00:08:06.827 --> 00:08:08.092
what to do at this point.

00:08:08.092 --> 00:08:10.009
This is all in 480 BCE,

00:08:11.575 --> 00:08:13.421
and what Xerxes decides to do,

00:08:13.421 --> 00:08:16.451
he says hey I don't want
to get stranded in Europe

00:08:16.451 --> 00:08:18.924
at the edge of my empire,

00:08:18.924 --> 00:08:21.571
and so Xerxes heads back,

00:08:21.571 --> 00:08:25.404
but he leaves some of
his ground forces there,

00:08:26.494 --> 00:08:30.661
and they eventually face a
decisive defeat at Plataea,

00:08:31.556 --> 00:08:32.889
right over here.

00:08:34.435 --> 00:08:37.935
And so that is the last significant threat

00:08:38.777 --> 00:08:40.774
of the Persians against the Greeks.

00:08:40.774 --> 00:08:43.712
From then there the
Greco-Persian wars continue

00:08:43.712 --> 00:08:45.070
for the next several decades,

00:08:45.070 --> 00:08:46.602
for the next 20 or 30 years,

00:08:46.602 --> 00:08:49.667
but at that point it's more of
the Greeks on the offensive,

00:08:49.667 --> 00:08:51.072
and this really is the beginning

00:08:51.072 --> 00:08:54.739
of the Golden Age of
the Greek civilization.


Beginning of the Greco Persian Wars

WEBVTT
Kind: captions
Language: en

00:00:00.492 --> 00:00:02.023
- [Instructor] This right here is a map

00:00:02.023 --> 00:00:05.457
of the Persian empire in 490 BCE,

00:00:05.457 --> 00:00:07.046
before the Common Era.

00:00:07.046 --> 00:00:10.004
And you see that it is
an extensive empire.

00:00:10.004 --> 00:00:12.775
It was established by Cyrus the Great

00:00:12.775 --> 00:00:14.014
and then this successors.

00:00:14.014 --> 00:00:15.566
We talked about it in previous videos

00:00:15.566 --> 00:00:18.268
how they were able to conquer
the Neo-Babylonian Empire

00:00:18.268 --> 00:00:21.268
in 539 BCE and then go on to conquer

00:00:22.138 --> 00:00:23.755
much of the middle east.

00:00:23.755 --> 00:00:28.251
They were able to conquer the
long-lived Egyptian empire.

00:00:28.251 --> 00:00:32.418
They seemed somewhat unstoppable
at this point in 490 BCE.

00:00:33.325 --> 00:00:36.570
At this point, we have Darius is ruler

00:00:36.570 --> 00:00:38.325
of the Persian Empire,

00:00:38.325 --> 00:00:42.385
the Achaemenic Empire, as
it's often referred to.

00:00:42.385 --> 00:00:44.517
And they're able to keep expanding

00:00:44.517 --> 00:00:47.767
until they come across some unexpected,

00:00:49.315 --> 00:00:51.991
unexpectedly resistant people,

00:00:51.991 --> 00:00:53.599
and that's the Greeks.

00:00:53.599 --> 00:00:56.034
This is probably one of the most famous

00:00:56.034 --> 00:00:59.978
and legendary stories
in history, the rivalry,

00:00:59.978 --> 00:01:04.007
the conflicts between the
Persians and the Greeks.

00:01:04.007 --> 00:01:06.509
You should maybe take it with
a little bit of grain of salt,

00:01:06.509 --> 00:01:07.808
because what we know of it,

00:01:07.808 --> 00:01:09.651
the history that I'm about to give you,

00:01:09.651 --> 00:01:12.238
comes to us from Herodotus.

00:01:12.238 --> 00:01:15.483
And Herodotus who's often
known as the father of history,

00:01:15.483 --> 00:01:17.275
he was Greek.

00:01:17.275 --> 00:01:21.020
And he also lived after a lot
of these conflicts happened

00:01:21.020 --> 00:01:22.282
and he wrote about these conflicts,

00:01:22.282 --> 00:01:24.515
his adult life was well
after these happened,

00:01:24.515 --> 00:01:27.100
so he had to talk to people
who were around that time

00:01:27.100 --> 00:01:29.775
and obviously, he is getting
the Greek side of the story.

00:01:29.775 --> 00:01:32.748
There are no surviving
Persian accounts for it,

00:01:32.748 --> 00:01:34.245
so you might want to take
it with a grain of salt

00:01:34.245 --> 00:01:36.764
because the accounts of Herodotus do tend

00:01:36.764 --> 00:01:39.281
to make the Greeks look pretty good.

00:01:39.281 --> 00:01:40.826
But let's just talk about what happened,

00:01:40.826 --> 00:01:43.071
because taking it with a grain of salt,

00:01:43.071 --> 00:01:47.023
it is a fascinating series of stories.

00:01:47.023 --> 00:01:48.547
So you have Darius as king,

00:01:48.547 --> 00:01:53.004
you can see the extent of
the Persian Empire in 490 BC

00:01:53.004 --> 00:01:54.470
or really at the turn of the century

00:01:54.470 --> 00:01:57.970
if we were to go back 10 years to 500 BCE,

00:01:59.001 --> 00:02:02.250
but they are in control of some folks

00:02:02.250 --> 00:02:05.002
who are are a little bit more rebellious

00:02:05.002 --> 00:02:08.469
than most of the people that
are within the Persian Empire.

00:02:08.469 --> 00:02:10.243
And they are in this
region right over here,

00:02:10.243 --> 00:02:13.990
known as Ionia, where
you had Greek settlements

00:02:13.990 --> 00:02:16.771
that are now under Persian control.

00:02:16.771 --> 00:02:17.771
From 500 BCE

00:02:19.517 --> 00:02:20.934
to about 494 BCE,

00:02:22.029 --> 00:02:26.196
so kind of the first 10 years
of the fifth century BCE,

00:02:27.845 --> 00:02:31.012
you have a series of revolts in Ionia.

00:02:32.738 --> 00:02:34.610
Let me zoom in on that a little bit,

00:02:34.610 --> 00:02:36.743
so we can see that,

00:02:36.743 --> 00:02:38.973
you can see our timeline,

00:02:38.973 --> 00:02:40.306
right over here.

00:02:41.547 --> 00:02:44.266
So this area, we're now zoomed in,

00:02:44.266 --> 00:02:47.308
this area is referred to as Ionia.

00:02:47.308 --> 00:02:49.002
This would be modern-day,

00:02:49.002 --> 00:02:52.126
the coast of modern-day Turkey.

00:02:52.126 --> 00:02:56.297
We're out here on the Anatolian Peninsula.

00:02:56.297 --> 00:02:59.730
And these rebellions in 498 BCE,

00:02:59.730 --> 00:03:03.530
the rebels are able to take
over and burn down Sardis

00:03:03.530 --> 00:03:05.770
which is under Persian control.

00:03:05.770 --> 00:03:08.778
The Persians under Darius
are not happy about this.

00:03:08.778 --> 00:03:10.778
So they send a force to,

00:03:12.986 --> 00:03:14.520
to take back control

00:03:14.520 --> 00:03:16.494
and also to get a little bit of revenge,

00:03:16.494 --> 00:03:19.628
because when the rebels were
able to take over Sardis,

00:03:19.628 --> 00:03:22.317
they had help from some
of their Greek brethren.

00:03:22.317 --> 00:03:24.743
It's important to note
that the the Greeks weren't

00:03:24.743 --> 00:03:26.806
some type of a unified state.

00:03:26.806 --> 00:03:30.194
They weren't a unified Empire
like the Persians were.

00:03:30.194 --> 00:03:33.363
They were a fragmented
group of city-states.

00:03:33.363 --> 00:03:36.758
You have the famous
Athens, Corinth, Sparta,

00:03:36.758 --> 00:03:37.776
all of these city-states,

00:03:37.776 --> 00:03:39.527
they shared a common language.

00:03:39.527 --> 00:03:41.778
They shared similar religion,

00:03:41.778 --> 00:03:45.030
but they were not one unified Empire.

00:03:45.030 --> 00:03:48.802
But when their brethren, the Greek rebels,

00:03:48.802 --> 00:03:51.649
these Ionic rebels, I guess you could say,

00:03:51.649 --> 00:03:55.033
rose up against the their Persian rulers,

00:03:55.033 --> 00:03:57.866
they had assistance from Athenians

00:03:58.714 --> 00:04:00.881
and from folks in Eritrea.

00:04:01.799 --> 00:04:04.519
So when Darius finds out about this,

00:04:04.519 --> 00:04:07.024
not only does he want to take
back what the rebels had,

00:04:07.024 --> 00:04:08.257
he wants to suppress them,

00:04:08.257 --> 00:04:11.130
he wants to put down that rebellion.

00:04:11.130 --> 00:04:15.464
He also wants to have
revenge on the Athenians

00:04:15.464 --> 00:04:17.371
and the Eritreans.

00:04:17.371 --> 00:04:18.988
So the first thing he does

00:04:18.988 --> 00:04:21.905
is he sends out a fleet in 492 BCE.

00:04:24.268 --> 00:04:27.761
You see his fleet here in green.

00:04:27.761 --> 00:04:30.446
This fleet that I am tracing.

00:04:30.446 --> 00:04:33.748
He sends it out in order to in order to,

00:04:33.748 --> 00:04:37.761
in order to seek revenge on the Athenians

00:04:37.761 --> 00:04:39.105
and the Eritreans.

00:04:39.105 --> 00:04:41.207
But he is unlucky and once again,

00:04:41.207 --> 00:04:42.863
these are the accounts of Herodotus.

00:04:42.863 --> 00:04:46.158
His fleet, his large
fleet, runs into a storm

00:04:46.158 --> 00:04:48.175
and is mostly destroyed,

00:04:48.175 --> 00:04:49.864
and is mostly destroyed.

00:04:49.864 --> 00:04:54.766
And so in 492 BCE, the
Persians are unsuccessful.

00:04:54.766 --> 00:04:56.245
But they are not satisfied.

00:04:56.245 --> 00:04:59.016
Darius is not happy with that idea

00:04:59.016 --> 00:05:02.099
that we are the largest empire known,

00:05:02.982 --> 00:05:06.510
we should be able to take
on these fragmented Greeks,

00:05:06.510 --> 00:05:08.501
with a much smaller population,

00:05:08.501 --> 00:05:11.552
much less wealth than
the mighty Persians do.

00:05:11.552 --> 00:05:12.802
So in 490 BCE,

00:05:14.277 --> 00:05:17.777
he sends out another attempt to take over,

00:05:18.752 --> 00:05:20.015
to seek revenge

00:05:20.015 --> 00:05:23.578
and to subjugate the
Eritreans and the Athenians.

00:05:23.578 --> 00:05:25.992
This time he is a little
bit more successful.

00:05:25.992 --> 00:05:28.244
His fleet is not destroyed.

00:05:28.244 --> 00:05:30.727
So this is in this brown color,

00:05:30.727 --> 00:05:32.087
I'll try to color it in.

00:05:32.087 --> 00:05:33.490
This brown color right over here,

00:05:33.490 --> 00:05:37.157
you can see the path
of the fleet this time.

00:05:38.071 --> 00:05:40.207
And he is eventually,

00:05:40.207 --> 00:05:44.248
he is able to subjugate
and take over Eritrea,

00:05:44.248 --> 00:05:46.800
but as he's going to Athens,

00:05:46.800 --> 00:05:49.267
which you can see right here,

00:05:49.267 --> 00:05:51.934
they decide to stop at Marathon.

00:05:53.023 --> 00:05:55.052
That word might be familiar to you

00:05:55.052 --> 00:05:58.035
and we'll talk about
in a second why it is.

00:05:58.035 --> 00:06:01.760
That is where they're
engaged by the Athenians.

00:06:01.760 --> 00:06:05.593
The Athenians are able
to defeat the Persians.

00:06:07.219 --> 00:06:09.322
This is a very big deal in history.

00:06:09.322 --> 00:06:11.258
Once again, this is Herodotus' account,

00:06:11.258 --> 00:06:14.261
but up to that point, the
Persians seemed invincible.

00:06:14.261 --> 00:06:15.834
But now the Athenians were able

00:06:15.834 --> 00:06:18.917
to actually destroy them in a battle.

00:06:19.945 --> 00:06:22.750
And the Persian fleet, they
go back to their boats,

00:06:22.750 --> 00:06:25.060
they run back to their
boats according to Herodotus

00:06:25.060 --> 00:06:27.850
and many of them are killed
as they as they retreat.

00:06:27.850 --> 00:06:29.758
They're going to go back
around the peninsula,

00:06:29.758 --> 00:06:32.243
potentially to go attack Athens.

00:06:32.243 --> 00:06:36.110
but they decide not to and they retreat.

00:06:36.110 --> 00:06:37.782
Now the word Marathon,

00:06:37.782 --> 00:06:39.802
this is a very big deal
in history first of all,

00:06:39.802 --> 00:06:44.495
is that this mighty empire
is put back in this battle.

00:06:44.495 --> 00:06:47.675
It oftentimes, historians will mark it

00:06:47.675 --> 00:06:51.985
as the beginning of a
kind of Greek ascendancy,

00:06:51.985 --> 00:06:54.385
the ascendancy of Greek civilization,

00:06:54.385 --> 00:06:56.059
at the end of these Greco-Persian Wars.

00:06:56.059 --> 00:06:58.769
We're really just in the
beginning periods of right now

00:06:58.769 --> 00:07:02.136
that's really the beginning
of the Greek golden age.

00:07:02.136 --> 00:07:05.996
But as we'll see, the Greco-Persian wars

00:07:05.996 --> 00:07:09.110
are not over with the battle at Marathon.

00:07:09.110 --> 00:07:13.004
That was just the first major interaction

00:07:13.004 --> 00:07:15.017
over the next few decades.

00:07:15.017 --> 00:07:17.015
Now one thing that you might be saying is,

00:07:17.015 --> 00:07:20.265
hey, that word marathon,
that seems familiar.

00:07:20.265 --> 00:07:22.021
Maybe some of you have run a marathon

00:07:22.021 --> 00:07:25.799
or you want to run a
marathon going 26.2 miles.

00:07:25.799 --> 00:07:29.992
You might say, is there any
relation to this battle,

00:07:29.992 --> 00:07:34.484
to this location in
Greece called, Marathon.

00:07:34.484 --> 00:07:36.747
The simple answer is, yes.

00:07:36.747 --> 00:07:38.508
So there's an apocryphal story,

00:07:38.508 --> 00:07:39.758
and when people say
apocryphal, they're like,

00:07:39.758 --> 00:07:41.035
"Well, we're not sure
if it really happened,

00:07:41.035 --> 00:07:42.526
"or maybe it didn't happen."

00:07:42.526 --> 00:07:44.739
And this one, people are
pretty sure it didn't happen.

00:07:44.739 --> 00:07:46.501
The story goes something like,

00:07:46.501 --> 00:07:48.039
when they had the battle,

00:07:48.039 --> 00:07:51.467
a messenger was sent
from Marathon to Athens

00:07:51.467 --> 00:07:54.009
and the distance from Marathon to Athens

00:07:54.009 --> 00:07:56.286
is roughly the distance
of a modern marathon,

00:07:56.286 --> 00:07:58.119
around 25 or 26 miles.

00:07:59.693 --> 00:08:01.497
The apocryphal story goes,

00:08:01.497 --> 00:08:04.515
they went to deliver the
message of the Persian attack

00:08:04.515 --> 00:08:06.758
and the success of the Athenians,

00:08:06.758 --> 00:08:09.053
and then when the messenger
gave that message,

00:08:09.053 --> 00:08:11.493
he just passed out and died.

00:08:11.493 --> 00:08:15.503
So when the modern Olympics
that we've set up now,

00:08:15.503 --> 00:08:17.758
over the last hundred and something years,

00:08:17.758 --> 00:08:19.510
they said, "Hey, let's have an event

00:08:19.510 --> 00:08:23.207
"that celebrates that run
from Marathon to Athens."

00:08:23.207 --> 00:08:26.507
That's where our modern
marathon actually comes from.

00:08:26.507 --> 00:08:29.026
Now the actual story as
best, as I can figure out,

00:08:29.026 --> 00:08:30.957
and I encourage you to
look up primary documents

00:08:30.957 --> 00:08:32.750
to figure this out for yourself,

00:08:32.750 --> 00:08:35.500
is it's mixing up a few stories.

00:08:35.500 --> 00:08:37.786
After the victory at Marathon,

00:08:37.786 --> 00:08:40.953
the Athenian army did quickly mobilize

00:08:42.012 --> 00:08:44.605
and go back to Athens
and cover that distance

00:08:44.605 --> 00:08:46.536
in order to defend Athens.

00:08:46.536 --> 00:08:48.485
Because remember it looks
like the Persian fleet

00:08:48.485 --> 00:08:52.595
were coming around this
peninsula right over here

00:08:52.595 --> 00:08:54.012
to get to Athens.

00:08:55.008 --> 00:08:57.083
The story of that runner is

00:08:57.083 --> 00:08:58.992
there was runner named, Filipides,

00:08:58.992 --> 00:09:01.268
who was sent from Athens to Sparta

00:09:01.268 --> 00:09:02.365
to convey a message.

00:09:02.365 --> 00:09:03.744
That's actually a much further distance.

00:09:03.744 --> 00:09:06.143
In a lot ways it's much more impressive,

00:09:06.143 --> 00:09:08.791
but those two stories
were mixed up together

00:09:08.791 --> 00:09:10.481
into the story of Marathon,

00:09:10.481 --> 00:09:12.537
which is where we get our modern-day

00:09:12.537 --> 00:09:13.761
distance of the marathon.

00:09:13.761 --> 00:09:15.360
Hey, let's all run roughly the distance

00:09:15.360 --> 00:09:17.360
from Marathon to Athens.

00:09:18.251 --> 00:09:22.522
In the next video, we'll see
that the Persians are not done.

00:09:22.522 --> 00:09:25.249
Darius will not be able
to seek the revenge

00:09:25.249 --> 00:09:27.513
that he wants in his lifetime,

00:09:27.513 --> 00:09:31.513
but his son Xerxes is
going to keep going at it.


Marginal distribution and conditional distribution

WEBVTT
Kind: captions
Language: en

00:00:00.195 --> 00:00:01.788
- [Instructor] Let's
say we're a professor at

00:00:01.788 --> 00:00:03.999
a university of a statistics class

00:00:03.999 --> 00:00:06.168
and we administer an exam

00:00:06.168 --> 00:00:08.462
and we're curious about the relationship

00:00:08.462 --> 00:00:11.899
between the amount of
time that students study

00:00:11.899 --> 00:00:15.321
and the percent that they
get correct on the test.

00:00:15.321 --> 00:00:17.853
So, what we do is we
grade all of the exams.

00:00:17.853 --> 00:00:19.322
We set up these buckets.

00:00:19.322 --> 00:00:21.286
The time studied, zero to 20 minutes,

00:00:21.286 --> 00:00:23.698
21 to 40 minutes, 41 to 60 minutes,

00:00:23.698 --> 00:00:25.909
or greater than 60 minutes.

00:00:25.909 --> 00:00:28.490
So, those are our buckets for
the amount of time studying.

00:00:28.490 --> 00:00:31.642
And then, we also create
buckets for the percent correct.

00:00:31.642 --> 00:00:33.444
So, zero to 19% correct,

00:00:33.444 --> 00:00:36.681
20 to 39% correct, 40 to 59% correct,

00:00:36.681 --> 00:00:40.848
60 to 79% correct, or 80 to 100% correct.

00:00:41.923 --> 00:00:43.887
And then, we figure out what percentage of

00:00:43.887 --> 00:00:45.687
our entire student population

00:00:45.687 --> 00:00:47.942
falls into each of these categories.

00:00:47.942 --> 00:00:52.109
So, for example, 2% of our
students studied 21 to 40 minutes

00:00:53.795 --> 00:00:58.249
and got between 80 and 100% on the exam.

00:00:58.249 --> 00:01:02.462
16% studied for more than an
hour, more than 60 minutes,

00:01:02.462 --> 00:01:06.638
and got between 40 and 59% on the exam.

00:01:06.638 --> 00:01:07.865
So, what I have, right over here,

00:01:07.865 --> 00:01:09.499
this is a two-way table.

00:01:09.499 --> 00:01:11.587
It's describing a joint distribution.

00:01:11.587 --> 00:01:13.960
The joint distribution between...

00:01:13.960 --> 00:01:16.250
You can view these as two variables.

00:01:16.250 --> 00:01:20.057
The time studied and the percent correct.

00:01:20.057 --> 00:01:24.355
Now, what we're gonna introduce
ourselves to in this video

00:01:24.355 --> 00:01:28.531
is two new ideas outside of
just the joint distribution.

00:01:28.531 --> 00:01:32.620
One is the idea of the
marginal distribution.

00:01:32.620 --> 00:01:34.584
I will write this in green.

00:01:34.584 --> 00:01:36.417
Marginal distribution.

00:01:38.919 --> 00:01:42.153
And this is the idea
of, "Okay, I can see...

00:01:42.153 --> 00:01:46.320
I can break down my class based
on both of these variables,

00:01:47.555 --> 00:01:51.689
but what if I only care
about the distribution

00:01:51.689 --> 00:01:53.697
of just the percent correct.

00:01:53.697 --> 00:01:56.112
I don't wanna break it
down by time studied."

00:01:56.112 --> 00:01:57.748
Well, if I wanna figure out

00:01:57.748 --> 00:02:00.042
the distribution of percent correct,

00:02:00.042 --> 00:02:02.911
I can just total up each of these rows

00:02:02.911 --> 00:02:07.078
and I would end up with this
distribution right over here.

00:02:10.692 --> 00:02:12.001
And, just to make it clear,

00:02:12.001 --> 00:02:16.168
I would see that 20% of my
students got 80 to 100% correct.

00:02:19.692 --> 00:02:21.821
I would see that 30% of my students got

00:02:21.821 --> 00:02:25.096
between 60 and 79% correct.

00:02:25.096 --> 00:02:27.799
I see that 35% of my students

00:02:27.799 --> 00:02:30.460
get between 40 and 59% correct.

00:02:30.460 --> 00:02:31.689
I think you see where this is going.

00:02:31.689 --> 00:02:36.479
10% of my students got
between 20 and 39% correct.

00:02:36.479 --> 00:02:38.605
And then, finally, 5% of my students

00:02:38.605 --> 00:02:41.772
got between zero and 19% correct.

00:02:43.550 --> 00:02:46.370
All we did is we totaled
up each of these rows.

00:02:46.370 --> 00:02:48.618
And, notice, these now add up to 100.

00:02:48.618 --> 00:02:52.785
This describes the distribution
of the scores in my class.

00:02:54.468 --> 00:02:56.885
If someone were to just
give you this column,

00:02:56.885 --> 00:03:00.523
you would say, "Okay, 20% of
my students got 80 to 100%."

00:03:00.523 --> 00:03:02.121
you don't know the breakdown by

00:03:02.121 --> 00:03:03.880
how much they actually studied.

00:03:03.880 --> 00:03:07.729
You would say 5% got between
zero and 19% on my test,

00:03:07.729 --> 00:03:09.285
but you wouldn't know what the breakdown

00:03:09.285 --> 00:03:12.967
of that 5% based on how much they studied.

00:03:12.967 --> 00:03:14.522
So, this type of distribution...

00:03:14.522 --> 00:03:16.486
It's called a marginal distribution,

00:03:16.486 --> 00:03:19.353
because you can view it as
it's written in the margin

00:03:19.353 --> 00:03:22.544
right over here that we total these rows

00:03:22.544 --> 00:03:24.266
and we write it in the margin.

00:03:24.266 --> 00:03:25.782
Now, there's another
marginal distribution.

00:03:25.782 --> 00:03:27.952
We could figure out the distribution

00:03:27.952 --> 00:03:31.594
of the amount of time
people studied in my class.

00:03:31.594 --> 00:03:34.010
And so, if we cared about that,

00:03:34.010 --> 00:03:37.940
that we would total up
each of these columns.

00:03:37.940 --> 00:03:39.615
We would total up each of the columns

00:03:39.615 --> 00:03:41.989
and look at this right over here.

00:03:41.989 --> 00:03:44.322
We would say, "Okay, 7% of my class

00:03:44.322 --> 00:03:46.901
studied between zero and 20 minutes.

00:03:46.901 --> 00:03:51.038
15% of my class studied
between 21 and 40 minutes.

00:03:51.038 --> 00:03:54.313
43% of my class studied
between 41 and 60 minutes.

00:03:54.313 --> 00:03:57.095
35% of my class studied
more than 60 minutes.

00:03:57.095 --> 00:03:58.678
More than an hour."

00:03:59.550 --> 00:04:01.926
And, if I just look at
this marginal distribution,

00:04:01.926 --> 00:04:05.076
this marginal distribution
of the time studied,

00:04:05.076 --> 00:04:08.060
I'm not able to get the
breakout of that 35%

00:04:08.060 --> 00:04:09.944
that studied more than an hour.

00:04:09.944 --> 00:04:11.951
If I just looked at that
marginal distribution...

00:04:11.951 --> 00:04:12.933
Once again, it's called that

00:04:12.933 --> 00:04:14.201
because I'm writing in the margin,

00:04:14.201 --> 00:04:16.899
in this case, below our table.

00:04:16.899 --> 00:04:19.109
If I just looked at that
marginal distribution,

00:04:19.109 --> 00:04:21.198
I would not know the breakdown

00:04:21.198 --> 00:04:23.698
by the actual percent correct.

00:04:24.642 --> 00:04:26.276
Now, there's another type of distribution

00:04:26.276 --> 00:04:28.691
that's related to these
joint distributions

00:04:28.691 --> 00:04:30.453
or you could say these two-way tables.

00:04:30.453 --> 00:04:33.603
And that's thinking about the
distribution of one variable

00:04:33.603 --> 00:04:37.285
given what bucket you fall
in for the other variable.

00:04:37.285 --> 00:04:39.002
So, let me write this down.

00:04:39.002 --> 00:04:43.169
So, if I wanna say the
distribution of scores given...

00:04:53.005 --> 00:04:53.904
All right let me write it this way.

00:04:53.904 --> 00:04:57.821
Distribution of scores
among those who studied,

00:05:03.118 --> 00:05:06.392
let's say, more than 60 minutes.

00:05:06.392 --> 00:05:07.945
So, where would I get that?

00:05:07.945 --> 00:05:08.778
Well, it's all right.

00:05:08.778 --> 00:05:10.116
I'll go to the column of the people

00:05:10.116 --> 00:05:11.558
who studied more than 60 minutes

00:05:11.558 --> 00:05:14.042
and then I'd find this
distribution of scores.

00:05:14.042 --> 00:05:15.230
And I see it right over here.

00:05:15.230 --> 00:05:18.180
Among the folks who studied
more than 60 minutes,

00:05:18.180 --> 00:05:21.097
I have this distribution of scores.

00:05:22.553 --> 00:05:24.969
So, in that group, 10% got 80 to 100,

00:05:24.969 --> 00:05:28.652
5% got 60 to 79, 16& got 40 to 59,

00:05:28.652 --> 00:05:32.819
none of them, 0% got 20 to
39, and 4% got 0 to 19%.

00:05:34.499 --> 00:05:36.992
And so, this distribution of one variable

00:05:36.992 --> 00:05:40.594
given a bucket that you're
falling into another variable.

00:05:40.594 --> 00:05:43.173
This is called a conditional distribution.

00:05:43.173 --> 00:05:45.303
'Cause you're getting a
distribution conditioned

00:05:45.303 --> 00:05:47.595
on a value of another variable.

00:05:47.595 --> 00:05:51.762
So, this right over here is
a conditional distribution.

00:05:54.961 --> 00:05:57.288
So, the big idea here, you
have this two-way table.

00:05:57.288 --> 00:05:59.865
We're trying to relate
how two variables...

00:05:59.865 --> 00:06:01.827
We're trying to study how two variables

00:06:01.827 --> 00:06:03.506
relate to each other.

00:06:03.506 --> 00:06:05.062
If we care about just the distribution

00:06:05.062 --> 00:06:08.214
of one of the variables, for
example, the time studied,

00:06:08.214 --> 00:06:10.320
we can sum up the columns here

00:06:10.320 --> 00:06:12.691
and we would get this
marginal distribution.

00:06:12.691 --> 00:06:15.144
If we cared about the
distribution of percent correct,

00:06:15.144 --> 00:06:17.723
we could sum up the rows
and get that distribution.

00:06:17.723 --> 00:06:20.919
And, if we wanted, in the
case that I just talked about,

00:06:20.919 --> 00:06:23.906
the distribution of one variable...

00:06:23.906 --> 00:06:25.298
The distribution of one variable,

00:06:25.298 --> 00:06:27.303
in this case, the distribution of scores,

00:06:27.303 --> 00:06:29.061
the distribution of percent correct,

00:06:29.061 --> 00:06:31.430
given a certain value conditioned on

00:06:31.430 --> 00:06:32.657
a value of another variable.

00:06:32.657 --> 00:06:35.873
Well, that's going to be a
conditional distribution.


Lagrange multiplier example, part 2

WEBVTT
Kind: captions
Language: en

00:00:00.369 --> 00:00:02.160
- [Instructor] So where we
left off we have these two

00:00:02.160 --> 00:00:04.529
different equations that we wanna solve

00:00:04.529 --> 00:00:06.000
and there's three unknowns.

00:00:06.000 --> 00:00:08.299
There's s, the tons of
steel that you're using,

00:00:08.299 --> 00:00:10.599
h the hours of labor, and then lambda,

00:00:10.599 --> 00:00:12.501
this Lagrange Multiplier we introduced

00:00:12.501 --> 00:00:15.387
that's basically a
proportionality constant

00:00:15.387 --> 00:00:18.251
between the gradient vectors
of the revenue function

00:00:18.251 --> 00:00:19.897
and the constraint function.

00:00:19.897 --> 00:00:23.380
And always the third equation
that we're dealing with here

00:00:23.380 --> 00:00:25.912
to solve this, is the constraint itself.

00:00:25.912 --> 00:00:27.627
That gives us another equation
that'll help solve for

00:00:27.627 --> 00:00:30.198
h and s, and ultimately lambda,

00:00:30.198 --> 00:00:32.302
if that's something that you want as well.

00:00:32.302 --> 00:00:33.511
So it's kind of a first pass here,

00:00:33.511 --> 00:00:35.104
I'm gonna do a little more simplifying

00:00:35.104 --> 00:00:36.354
but I'm gonna make a substitution

00:00:36.354 --> 00:00:37.879
that'll make this easier for us.

00:00:37.879 --> 00:00:41.075
So I see s over h here, and
they're both to the same power,

00:00:41.075 --> 00:00:42.764
so I feel it might be a little bit easier

00:00:42.764 --> 00:00:46.514
if I just substitute u
in for s divided by h.

00:00:47.606 --> 00:00:50.054
And what that'll let me do, is rewrite

00:00:50.054 --> 00:00:53.304
this first equation here as 200 thirds,

00:00:54.406 --> 00:00:58.156
200 thirds times u to
the power of one third.

00:00:59.100 --> 00:01:02.906
And that's equal to 20 times lambda.

00:01:02.906 --> 00:01:06.346
And then likewise, what
that means for this guy,

00:01:06.346 --> 00:01:09.773
is, well, this is h over s, not s over h,

00:01:09.773 --> 00:01:12.426
so that one's gonna be 100 thirds,

00:01:12.426 --> 00:01:14.777
not times u to the power of two thirds,

00:01:14.777 --> 00:01:17.944
but times u to the negative two thirds

00:01:19.292 --> 00:01:21.281
because we swapped the h and s here.

00:01:21.281 --> 00:01:23.645
So that's u to the negative two thirds

00:01:23.645 --> 00:01:27.754
and this is just to make it a
little bit cleaner, I think.

00:01:27.754 --> 00:01:31.921
We kind of want to treat h
and s in the same package.

00:01:32.877 --> 00:01:34.661
Now I'm gonna go ahead and
put all the constants together

00:01:34.661 --> 00:01:36.067
and then I'm gonna take this guy

00:01:36.067 --> 00:01:39.714
and multiply it by three divided by 200,

00:01:39.714 --> 00:01:41.736
multiply both sides of
that just to cancel out

00:01:41.736 --> 00:01:44.110
what's on the left side here.

00:01:44.110 --> 00:01:46.360
And what that's gonna give
me, and I'll go ahead and

00:01:46.360 --> 00:01:49.323
write it over here, kind
of all over the place,

00:01:49.323 --> 00:01:51.754
u to the one third is equal to,

00:01:51.754 --> 00:01:54.090
let's see three over 200, so that 20

00:01:54.090 --> 00:01:56.876
almost cancels out with the
200, it just leaves a 10,

00:01:56.876 --> 00:01:59.876
so that's gonna give me three tenths

00:02:02.591 --> 00:02:03.424
of lambda.

00:02:04.846 --> 00:02:06.124
And then similarly over here,

00:02:06.124 --> 00:02:08.197
I'm gonna take this full equation

00:02:08.197 --> 00:02:10.947
and multiply it by three over 100

00:02:12.448 --> 00:02:14.366
and what that's gonna leave me with

00:02:14.366 --> 00:02:17.612
is that u to the negative two thirds,

00:02:17.612 --> 00:02:19.567
u to the negative two thirds is equal to,

00:02:19.567 --> 00:02:22.036
let's see this 2,000
when we divide it by 100

00:02:22.036 --> 00:02:25.342
becomes 20, and that 20 times three is 60,

00:02:25.342 --> 00:02:27.842
so that'll be 60 times lambda.

00:02:29.522 --> 00:02:32.852
Alright, so now I want a way
to simplify each of these,

00:02:32.852 --> 00:02:35.381
and what I notice is they look
pretty similar on each side,

00:02:35.381 --> 00:02:38.191
you know it's something
related to u equal to lambda,

00:02:38.191 --> 00:02:39.433
so if I can get this in a form

00:02:39.433 --> 00:02:42.817
where I'm really isolating
u, that would be great.

00:02:42.817 --> 00:02:44.648
The way I'm gonna do this
is I'm gonna multiply

00:02:44.648 --> 00:02:47.566
each one of them by u to the two thirds,

00:02:47.566 --> 00:02:49.329
so I'm gonna multiply it into this guy

00:02:49.329 --> 00:02:51.298
and I'm gonna multiply it into that guy,

00:02:51.298 --> 00:02:54.555
because on the top, it's
gonna turn this into just u,

00:02:54.555 --> 00:02:56.357
which will be nice, and on the bottom

00:02:56.357 --> 00:02:58.111
it'll cancel out that u entirely.

00:02:58.111 --> 00:02:59.838
So it feels like it'll
make both of these nicer,

00:02:59.838 --> 00:03:02.278
even though it might make the
right side a little uglier,

00:03:02.278 --> 00:03:04.016
those right sides will
still kind of be the same,

00:03:04.016 --> 00:03:05.721
and we'll take advantage of that.

00:03:05.721 --> 00:03:07.751
So, when I do that to the top part,

00:03:07.751 --> 00:03:09.047
like I said, that u to the one third

00:03:09.047 --> 00:03:12.109
times u to the two thirds ends up being u,

00:03:12.109 --> 00:03:14.106
and then on the right side we have

00:03:14.106 --> 00:03:16.106
our three tenths lambda,

00:03:17.222 --> 00:03:19.555
but now u to the two thirds.

00:03:20.498 --> 00:03:23.122
And then on the bottom here, we,

00:03:23.122 --> 00:03:25.135
when I multiply it by u to the two thirds,

00:03:25.135 --> 00:03:27.305
the right side becomes
one, 'cause it cancels out

00:03:27.305 --> 00:03:29.376
with u to the negative two thirds,

00:03:29.376 --> 00:03:32.459
and the right side is 60 times lambda

00:03:33.542 --> 00:03:35.709
times u to the two thirds.

00:03:38.077 --> 00:03:39.658
Now these right sides look very similar,

00:03:39.658 --> 00:03:41.155
and the left sides are quite simple.

00:03:41.155 --> 00:03:42.628
So I'm gonna multiply this top one

00:03:42.628 --> 00:03:44.509
by whatever it takes
to make it look exactly

00:03:44.509 --> 00:03:45.575
like that right side.

00:03:45.575 --> 00:03:48.581
So in this case I'm gonna
multiply that top by 10,

00:03:48.581 --> 00:03:51.249
which will get it to three,
and then by another 20

00:03:51.249 --> 00:03:53.716
to make that constant 60,
so I'm gonna multiply this

00:03:53.716 --> 00:03:56.134
entire top equation by 200,

00:03:56.134 --> 00:03:59.634
and what that gives me is that 200 times u

00:04:00.972 --> 00:04:03.222
is equal to 60 times lambda

00:04:04.707 --> 00:04:06.874
times u to the two thirds.

00:04:07.806 --> 00:04:11.839
And now these two equations,
these two equations

00:04:11.839 --> 00:04:13.368
have the same right side.

00:04:13.368 --> 00:04:15.205
So this is the same as saying,

00:04:15.205 --> 00:04:18.122
200 times u is equal to, well, one.

00:04:20.175 --> 00:04:21.478
Because each one of those expressions

00:04:21.478 --> 00:04:24.125
equals the same complicated thing.

00:04:24.125 --> 00:04:27.554
And now 200 times u, well
that's s divided by h.

00:04:27.554 --> 00:04:30.948
So this is the same thing as saying 200

00:04:30.948 --> 00:04:33.115
times s over h equals one,

00:04:34.409 --> 00:04:37.742
which we can write much more simply as h

00:04:38.825 --> 00:04:40.075
is equal to 200

00:04:41.226 --> 00:04:42.059
times s.

00:04:43.975 --> 00:04:45.522
Great, so I'm gonna go
ahead and circle that,

00:04:45.522 --> 00:04:48.267
h is equal to 200 times s.

00:04:48.267 --> 00:04:50.700
And now what we apply that to

00:04:50.700 --> 00:04:52.789
is the constraint, is the 200 times h

00:04:52.789 --> 00:04:55.940
plus 2,000 times s equals our budget.

00:04:55.940 --> 00:04:57.649
We'll go ahead and kind
of write that down again.

00:04:57.649 --> 00:05:02.118
That are 20 times h, I think,
20 times the hours of labor

00:05:02.118 --> 00:05:04.451
plus $2,000 per ton of steel

00:05:06.320 --> 00:05:07.917
is equal to our budget

00:05:07.917 --> 00:05:11.601
of $20,000, and now we
can just substitute in.

00:05:11.601 --> 00:05:16.599
Instead of h I'm gonna
write 200 s, so that's 200,

00:05:16.599 --> 00:05:18.432
sorry, 20 times 200 s,

00:05:20.590 --> 00:05:21.423
200 s,

00:05:23.529 --> 00:05:24.362
plus 2,000

00:05:25.692 --> 00:05:27.156
times s

00:05:27.156 --> 00:05:28.739
is equal to 20,000.

00:05:30.288 --> 00:05:34.910
And now this right side, 20
times 200 is equal to 4,000,

00:05:34.910 --> 00:05:36.426
and I'm just gonna go
ahead and kind of write

00:05:36.426 --> 00:05:38.027
so this here

00:05:38.027 --> 00:05:42.194
is 4,000 s, so the entire
right side of the equation

00:05:43.514 --> 00:05:45.181
simplifies to 6,000,

00:05:46.646 --> 00:05:47.729
6,000 times s

00:05:49.274 --> 00:05:50.774
is equal to 20,000

00:05:53.648 --> 00:05:55.070
and when those cancel
out, what that gives us

00:05:55.070 --> 00:05:57.999
is s is equal to 20 divided by six,

00:05:57.999 --> 00:06:01.416
which is the same as 10 divided by three.

00:06:02.938 --> 00:06:05.276
So that's how many tons
of steel we should get.

00:06:05.276 --> 00:06:07.562
S is 10 over three,
then when we apply that

00:06:07.562 --> 00:06:10.039
to the fact that h is 200 times s,

00:06:10.039 --> 00:06:12.271
that's gonna mean that h is equal

00:06:12.271 --> 00:06:14.271
to 200 times that value,

00:06:16.257 --> 00:06:18.924
10 over three, which is equal to

00:06:20.701 --> 00:06:21.534
2,000,

00:06:22.832 --> 00:06:23.915
2,000 thirds,

00:06:25.125 --> 00:06:29.410
2,000 thirds, that's how
many hours of labor we want.

00:06:29.410 --> 00:06:32.049
So, evidently in our original problem,

00:06:32.049 --> 00:06:35.127
where we have this model
for our revenue function

00:06:35.127 --> 00:06:37.949
for our Widgets with
$20 per hour of labor,

00:06:37.949 --> 00:06:40.458
and $2,000 per ton of steel,

00:06:40.458 --> 00:06:43.224
with a budget of $20,000, the allocation

00:06:43.224 --> 00:06:47.341
that you should make is to buy
10 thirds of a ton of steel

00:06:47.341 --> 00:06:50.091
and 2,000 thirds hours of labor.


Lagrange multiplier example, part 1

WEBVTT
Kind: captions
Language: en

00:00:00.674 --> 00:00:01.507
- [Instructor] So let's say you're running

00:00:01.507 --> 00:00:04.504
some kind of company and
you guys produce widgets.

00:00:04.504 --> 00:00:08.369
You produce some little trinket
that people enjoy buying

00:00:08.369 --> 00:00:12.313
and the main costs that
you have are labor,

00:00:12.313 --> 00:00:16.480
the workers that you have
creating these and steel.

00:00:17.857 --> 00:00:19.804
And let's just say that your labor costs

00:00:19.804 --> 00:00:22.387
are $20 per hour, $20 each hour

00:00:24.650 --> 00:00:27.733
and then your steel costs are $2,000.

00:00:29.452 --> 00:00:31.275
Keep the numbers kind of
related to each other.

00:00:31.275 --> 00:00:33.775
$2,000 for every ton of steel.

00:00:35.043 --> 00:00:37.277
And then you've had your
analysts work a little bit

00:00:37.277 --> 00:00:38.940
on trying to model the
revenues you can make

00:00:38.940 --> 00:00:41.056
with your widgets as a function of

00:00:41.056 --> 00:00:43.400
hours of labor and tons of steel.

00:00:43.400 --> 00:00:45.790
Now let's say the revenue model
that they've come up with.

00:00:45.790 --> 00:00:49.153
The revenue as a function
of hours of labor

00:00:49.153 --> 00:00:51.820
and then S for steel, let's say,

00:00:52.917 --> 00:00:57.141
is equal to about 100
times the hours of labor

00:00:57.141 --> 00:00:59.817
to the power of 2/3 multiplied by

00:00:59.817 --> 00:01:04.137
the tons of steel to the power of 1/3.

00:01:04.137 --> 00:01:05.573
If you put in a given amount of labor

00:01:05.573 --> 00:01:08.213
and a given amount of steel,
this is about how much money

00:01:08.213 --> 00:01:10.274
you're gonna expect to earn.

00:01:10.274 --> 00:01:12.146
And of course you wanna
earn as much as you can

00:01:12.146 --> 00:01:13.461
plus you actually have
a budget for how much

00:01:13.461 --> 00:01:15.455
you're able to spend on all these things.

00:01:15.455 --> 00:01:17.705
And your budget is $20,000.

00:01:23.288 --> 00:01:25.367
You're willing to spend $20,000

00:01:25.367 --> 00:01:27.355
and you wanna make as
much money as you can,

00:01:27.355 --> 00:01:30.057
according to this model based on that.

00:01:30.057 --> 00:01:32.366
Now this is exactly the kind of problem

00:01:32.366 --> 00:01:34.910
that the Lagrange multiplier
technique is made for.

00:01:34.910 --> 00:01:37.055
We're trying to maximize
some kind of function

00:01:37.055 --> 00:01:38.583
and we have a constraint.

00:01:38.583 --> 00:01:41.156
Now right now the constraint
isn't written as a formula,

00:01:41.156 --> 00:01:43.438
but we can pretty easily
write as a formula.

00:01:43.438 --> 00:01:45.292
Because what makes up our budget?

00:01:45.292 --> 00:01:46.481
Well, it's gonna be the number of

00:01:46.481 --> 00:01:49.129
hours of labor multiplied by 20,

00:01:49.129 --> 00:01:51.712
so that's gonna be $20 per hour

00:01:52.755 --> 00:01:55.427
multiplied by the number
of hours you put in

00:01:55.427 --> 00:01:58.260
plus $2000 per tons of steel times

00:02:00.902 --> 00:02:02.867
the tons of steel that you put in.

00:02:02.867 --> 00:02:05.784
So the constraint is basically that

00:02:06.898 --> 00:02:10.305
you have to have these guys equal $20,000.

00:02:10.305 --> 00:02:11.495
I mean you could say less than,

00:02:11.495 --> 00:02:13.657
you could say you're not willing
to go any more than that,

00:02:13.657 --> 00:02:15.883
but intuitively and in reality,

00:02:15.883 --> 00:02:17.257
it's gonna be the case that in order

00:02:17.257 --> 00:02:19.437
to maximize your revenues, you
should squeeze every dollar

00:02:19.437 --> 00:02:22.690
that you have available and
actually hit this constraint.

00:02:22.690 --> 00:02:24.928
So this right here is the
constraint of our problem.

00:02:24.928 --> 00:02:26.810
And let's go ahead and
give this guy a name,

00:02:26.810 --> 00:02:29.387
the function that we're
dealing with a name.

00:02:29.387 --> 00:02:33.554
And I'm gonna call it g of h,
s, which is gonna be that guy.

00:02:35.638 --> 00:02:37.563
And now if you'll remember
in the last few videos,

00:02:37.563 --> 00:02:39.769
the way we visualize something like this,

00:02:39.769 --> 00:02:43.206
is to think about the set
of all possible input,

00:02:43.206 --> 00:02:47.819
so in this case, you might be
thinking about the h s plane,

00:02:47.819 --> 00:02:49.843
the number of hours of labor on one axis,

00:02:49.843 --> 00:02:52.028
the number of tons of steel on another.

00:02:52.028 --> 00:02:53.765
And this constraint, well, in this case,

00:02:53.765 --> 00:02:56.265
it's a linear function, so this
constraint is gonna give us

00:02:56.265 --> 00:03:00.519
some kind of line that tells
us which pairs of s and h

00:03:00.519 --> 00:03:02.229
are gonna achieve that constraint.

00:03:02.229 --> 00:03:03.367
And then the revenue function that

00:03:03.367 --> 00:03:06.311
we're dealing with will
have certain contours.

00:03:06.311 --> 00:03:09.506
Maybe revenues of $10,000 have

00:03:09.506 --> 00:03:11.308
a certain contour that looks like this

00:03:11.308 --> 00:03:13.706
and revenues of $100,000
have a certain contour

00:03:13.706 --> 00:03:14.641
that looks like this.

00:03:14.641 --> 00:03:16.573
But what we want is to find which value

00:03:16.573 --> 00:03:20.140
is barely touching the constraint curve.

00:03:20.140 --> 00:03:22.174
Just tangent to it at a given point

00:03:22.174 --> 00:03:23.888
cause that's gonna be the contour line,

00:03:23.888 --> 00:03:26.422
where if you up the value
by just a little bit,

00:03:26.422 --> 00:03:28.302
it would no longer
intersect with that curve,

00:03:28.302 --> 00:03:29.508
there would no longer be values

00:03:29.508 --> 00:03:33.560
of h and s that satisfy this constraint.

00:03:33.560 --> 00:03:35.918
And the way to think about
finding that tangency

00:03:35.918 --> 00:03:39.631
is to consider the vector perpendicular

00:03:39.631 --> 00:03:41.835
to the tangent line to
the curve at that point,

00:03:41.835 --> 00:03:44.552
which fortunately is
represented by... (mumbles)

00:03:44.552 --> 00:03:46.682
Make some room for myself here.

00:03:46.682 --> 00:03:51.589
Represented by the gradient,
the gradient of our r function,

00:03:51.589 --> 00:03:54.214
the function whose contour's
this is, the revenue.

00:03:54.214 --> 00:03:55.658
And what it means for this to be

00:03:55.658 --> 00:03:57.413
tangent to the constraint line

00:03:57.413 --> 00:04:01.181
is that there's gonna be another vector,

00:04:01.181 --> 00:04:05.204
the gradient of g, of
our constraint function,

00:04:05.204 --> 00:04:06.802
that points in the same direction,

00:04:06.802 --> 00:04:08.088
that's proportional to that.

00:04:08.088 --> 00:04:09.561
And typically the way you write this

00:04:09.561 --> 00:04:13.036
is to say that the
gradient of this function

00:04:13.036 --> 00:04:16.036
is proportional to the gradient of g

00:04:19.902 --> 00:04:21.313
and this proportionality constant

00:04:21.313 --> 00:04:24.329
is called our Lagrange multiplier.

00:04:24.329 --> 00:04:27.329
It's called the Lagrange multiplier.

00:04:33.849 --> 00:04:35.569
So, let's go ahead and
start working it out.

00:04:35.569 --> 00:04:38.497
Let's first compute the gradient of r.

00:04:38.497 --> 00:04:42.236
So the gradient of r is gonna
be the partial derivative of r

00:04:42.236 --> 00:04:44.829
with respect to its first
variable, which is h,

00:04:44.829 --> 00:04:47.054
so partial derivative with respect to h.

00:04:47.054 --> 00:04:49.788
And the second component
is its partial derivative

00:04:49.788 --> 00:04:53.955
with respect to that second
variable, s, with respect to s.

00:04:56.330 --> 00:05:00.478
And in this case, that
first partial derivative,

00:05:00.478 --> 00:05:03.217
if we treat h as a variable
and s as a constant,

00:05:03.217 --> 00:05:05.211
then that 2/3 gets brought down,

00:05:05.211 --> 00:05:09.378
so that'll be 100 times 2/3
times h to the power of,

00:05:12.734 --> 00:05:14.920
well, we've gotta subtract one from 2/3

00:05:14.920 --> 00:05:18.576
when we bring it down, so
that'll be negative 1/3

00:05:18.576 --> 00:05:21.743
multiplied by s to the 1/3.

00:05:23.001 --> 00:05:24.713
And then the second component here,

00:05:24.713 --> 00:05:26.899
the partial derivative with respect to s,

00:05:26.899 --> 00:05:29.211
is gonna be 100 times, well now,

00:05:29.211 --> 00:05:30.851
by treating s as the variable,

00:05:30.851 --> 00:05:34.304
we take down that 1/3, so that's 1/3,

00:05:34.304 --> 00:05:36.713
h to the 2/3 just looks like a constant

00:05:36.713 --> 00:05:39.439
as far as s is concerned
and then we take s

00:05:39.439 --> 00:05:43.606
to the 1/3 minus 1, which is negative 2/3.

00:05:46.746 --> 00:05:48.486
Great, so that's the gradient of R.

00:05:48.486 --> 00:05:51.214
And now we need the gradient of g.

00:05:51.214 --> 00:05:53.050
And that one's a lot easier, actually,

00:05:53.050 --> 00:05:54.972
cause g is just a linear function.

00:05:54.972 --> 00:05:57.805
So when we take the gradient of g,

00:05:58.967 --> 00:06:01.509
which is its partial derivative,

00:06:01.509 --> 00:06:05.852
with respect to h, partial
h, and its partial derivative

00:06:05.852 --> 00:06:08.269
with respect to s, partial s.

00:06:10.604 --> 00:06:13.017
Well, the partial with
respect to h is just 20.

00:06:13.017 --> 00:06:15.451
The function looks like
20 times h plus something

00:06:15.451 --> 00:06:19.547
that's a constant so
that ends up being 20.

00:06:19.547 --> 00:06:21.108
And then the partial with respect to s,

00:06:21.108 --> 00:06:24.371
likewise is just 2,000,
cause it's just some constant

00:06:24.371 --> 00:06:26.795
multiplied by s plus
a bunch of other stuff

00:06:26.795 --> 00:06:28.531
that looks like constants.

00:06:28.531 --> 00:06:30.742
So that's great, and this means

00:06:30.742 --> 00:06:34.125
when we set the gradient of
r equal to the gradient of g,

00:06:34.125 --> 00:06:36.627
the pair of equations that we get,

00:06:36.627 --> 00:06:38.362
here let me just write it all out again,

00:06:38.362 --> 00:06:42.529
is we have this top one,
which I'll call 200/3 times,

00:06:43.992 --> 00:06:45.418
and let's go ahead and
do a little simplifying

00:06:45.418 --> 00:06:46.904
while I'm rewriting things here.

00:06:46.904 --> 00:06:51.071
So h to the 1/3 is really one
over, h to the negative 1/3,

00:06:52.985 --> 00:06:57.176
sorry, is one over h to the
1/3, that's s to the 1/3.

00:06:57.176 --> 00:06:58.807
So all of this, that first component,

00:06:58.807 --> 00:07:00.949
is being set equal to the first component

00:07:00.949 --> 00:07:04.199
of the gradient of g, which is 20

00:07:04.199 --> 00:07:07.024
times lambda, times this
Lagrange multiplier,

00:07:07.024 --> 00:07:08.916
cause we're not setting the
gradients equal to each other,

00:07:08.916 --> 00:07:11.510
we're just setting them
proportional to each other.

00:07:11.510 --> 00:07:13.360
So that's the first equation
and then the second one,

00:07:13.360 --> 00:07:14.784
I'll go ahead and do some simplifying

00:07:14.784 --> 00:07:16.603
while I rewrite that one also.

00:07:16.603 --> 00:07:20.770
That's gonna be 100/3
and then h to the 2/3

00:07:21.607 --> 00:07:25.774
so times h to the 2/3
divided by s to the 2/3

00:07:27.084 --> 00:07:28.918
cause s to the negative 2/3 is the same

00:07:28.918 --> 00:07:31.262
as 1 over s to the 2/3.

00:07:31.262 --> 00:07:34.845
All of that is equal
to 2,000 times lambda.

00:07:38.119 --> 00:07:39.876
And the important thing
it's that same lambda,

00:07:39.876 --> 00:07:43.385
because the entire vector
has to be proportional.

00:07:43.385 --> 00:07:45.161
And I think right here's
probably a pretty good point

00:07:45.161 --> 00:07:46.405
to stop and in the next video,

00:07:46.405 --> 00:07:47.470
I'll go ahead and work through the details

00:07:47.470 --> 00:07:49.573
and we'll land on a solution.


Finishing the intro lagrange multiplier example

WEBVTT
Kind: captions
Language: en

00:00:00.547 --> 00:00:01.589
- [Instructor] So in the last two videos

00:00:01.589 --> 00:00:04.576
we were talking about this
constrained optimization problem

00:00:04.576 --> 00:00:07.170
where we want to maximize
a certain function

00:00:07.170 --> 00:00:09.820
on a certain set, the
set of all points x, y

00:00:09.820 --> 00:00:12.152
where x squared plus y squared equals one.

00:00:12.152 --> 00:00:13.792
And we ended up working
out through some nice,

00:00:13.792 --> 00:00:15.683
geometrical reasoning
that we need to solve

00:00:15.683 --> 00:00:17.360
this system of equations.

00:00:17.360 --> 00:00:19.533
So there's nothing left
to do but to just solve

00:00:19.533 --> 00:00:21.021
this system of equations.

00:00:21.021 --> 00:00:22.524
I will start with this
first one at the top,

00:00:22.524 --> 00:00:23.927
see what we can simplify.

00:00:23.927 --> 00:00:26.311
We notice there's an x term in each one

00:00:26.311 --> 00:00:28.603
so we'll go ahead and cancel those out,

00:00:28.603 --> 00:00:30.333
which is basically a way
of saying we're assuming

00:00:30.333 --> 00:00:33.014
that x is not zero, and we
can kind of return to that

00:00:33.014 --> 00:00:35.695
to see if x equals zero
could be as you wish.

00:00:35.695 --> 00:00:36.921
And so maybe we'll kind
of write that down.

00:00:36.921 --> 00:00:39.500
We're assuming x is not equal to zero

00:00:39.500 --> 00:00:40.501
in order to cancel out.

00:00:40.501 --> 00:00:42.213
And we kind of can revisit whether that

00:00:42.213 --> 00:00:44.297
could give us another
possible solution later,

00:00:44.297 --> 00:00:48.399
but that will be two times y
is equal to lambda times two,

00:00:48.399 --> 00:00:51.231
and from here, the twos can cancel out,

00:00:51.231 --> 00:00:53.732
no worries about two equaling zero,

00:00:53.732 --> 00:00:56.002
and we know that y equals lambda.

00:00:56.002 --> 00:00:59.565
So that's a nice, simplified
form for this equation.

00:00:59.565 --> 00:01:01.446
And for this next
equation, though we can use

00:01:01.446 --> 00:01:03.821
what we just found, that
y is equal to lambda

00:01:03.821 --> 00:01:07.091
to replace the lambda
that we see, and instead,

00:01:07.091 --> 00:01:10.198
if I replace this with
a y, what I'm gonna get

00:01:10.198 --> 00:01:14.050
is that x squared is equal
to y times two times y.

00:01:14.050 --> 00:01:16.825
So that's two times y squared.

00:01:16.825 --> 00:01:19.106
And I'll leave it in
that form because I see

00:01:19.106 --> 00:01:20.849
that in the next equation,
I see an x squared,

00:01:20.849 --> 00:01:22.251
I see a y squared, so it might be nice

00:01:22.251 --> 00:01:24.784
to be able to plug this guy right into it.

00:01:24.784 --> 00:01:27.455
So in that next equation, x squared,

00:01:27.455 --> 00:01:30.099
I'm gonna go ahead and
replace that with y squared,

00:01:30.099 --> 00:01:32.016
so that's two y squared

00:01:33.391 --> 00:01:35.558
plus y squared equals one.

00:01:36.848 --> 00:01:41.015
And then from there, simplifies
to three times y squared

00:01:42.120 --> 00:01:45.537
equals one, which in turn means y squared

00:01:46.981 --> 00:01:49.400
is equal to one third,

00:01:49.400 --> 00:01:52.233
and so y is equal to plus or minus

00:01:53.111 --> 00:01:56.293
the square root of one third.

00:01:56.293 --> 00:01:57.126
Great.

00:01:57.126 --> 00:01:59.585
So this gives us y, and
I'll go ahead and put a box

00:01:59.585 --> 00:02:03.513
around that, where we
have found what y must be.

00:02:03.513 --> 00:02:05.811
Now if y squared is equal to one third,

00:02:05.811 --> 00:02:07.375
then we look up here and say, mhm,

00:02:07.375 --> 00:02:10.428
two times y squared, that's
gonna be the same thing

00:02:10.428 --> 00:02:14.345
as two times one third,
so two times one third,

00:02:15.505 --> 00:02:18.770
so if x squared is equal to two thirds,

00:02:18.770 --> 00:02:21.208
what that implies is that x

00:02:21.208 --> 00:02:24.625
is equal to plus or minus the square root

00:02:25.564 --> 00:02:26.731
of two thirds.

00:02:29.699 --> 00:02:31.329
And then there we go, that's another,

00:02:31.329 --> 00:02:32.603
one of the solutions.

00:02:32.603 --> 00:02:34.371
And I could write down
what lambda is, right,

00:02:34.371 --> 00:02:37.279
I mean, in this case it's
easy, 'cause y equals lambda,

00:02:37.279 --> 00:02:39.605
but all we really want in
their final form are x and y,

00:02:39.605 --> 00:02:40.991
since that's gonna give us the answer

00:02:40.991 --> 00:02:42.958
to the original constraint problem.

00:02:42.958 --> 00:02:44.728
So this, this gives us what we want,

00:02:44.728 --> 00:02:47.598
and we just have that
pesky, little possibility

00:02:47.598 --> 00:02:49.480
that x equals zero to address,

00:02:49.480 --> 00:02:50.999
and for that we can take a look and say,

00:02:50.999 --> 00:02:53.007
if x equaled zero, you
know let's go through

00:02:53.007 --> 00:02:55.435
the possibility that
maybe that's one of the

00:02:55.435 --> 00:02:57.308
constrained solutions.

00:02:57.308 --> 00:02:59.572
Well, in this equation
that would make sense,

00:02:59.572 --> 00:03:01.690
since two times zero would equal zero.

00:03:01.690 --> 00:03:03.758
In this equation, that would
mean that we're setting zero

00:03:03.758 --> 00:03:06.888
equal to lambda times two times y.

00:03:06.888 --> 00:03:08.946
Well, since lambda
equals y, that would mean

00:03:08.946 --> 00:03:10.985
that for this side to equal zero,

00:03:10.985 --> 00:03:13.759
y would have to equal zero, so evidently,

00:03:13.759 --> 00:03:16.463
you know, if it was the
case that x equals zero,

00:03:16.463 --> 00:03:17.790
that would have to imply,

00:03:17.790 --> 00:03:20.535
from the second equation
that y equals zero,

00:03:20.535 --> 00:03:22.160
but if x and y both equal zero,

00:03:22.160 --> 00:03:24.358
this constraint can't be satisfied.

00:03:24.358 --> 00:03:26.102
So none of this is
possible, so we never even

00:03:26.102 --> 00:03:27.828
had to worry about this to start with,

00:03:27.828 --> 00:03:29.238
but it's something you do need to check,

00:03:29.238 --> 00:03:31.249
just every time you're
dividing by a variable,

00:03:31.249 --> 00:03:35.564
you're basically assuming
that it's not equal to zero.

00:03:35.564 --> 00:03:39.040
So, this right here gives
us four possible solutions,

00:03:39.040 --> 00:03:41.788
four possible values for x and y

00:03:41.788 --> 00:03:45.385
that satisfy this constraint,
and which potentially

00:03:45.385 --> 00:03:47.199
maximize this, and remember, when I say

00:03:47.199 --> 00:03:49.138
potentially maximize, the whole idea

00:03:49.138 --> 00:03:51.290
of this Lagrange Multiplier,
is that we were looking

00:03:51.290 --> 00:03:53.897
for where there's a point of tangency

00:03:53.897 --> 00:03:56.353
between the contour lines.

00:03:56.353 --> 00:03:58.289
So just to make it
explicit, the four points

00:03:58.289 --> 00:04:00.444
that we're dealing with, here
I'll right them all here.

00:04:00.444 --> 00:04:04.444
So, x could be the square
root of two thirds,

00:04:06.532 --> 00:04:07.444
square root of two thirds,

00:04:07.444 --> 00:04:11.949
and y could be the positive
square root of one third,

00:04:11.949 --> 00:04:13.493
and then we can basically just toggle,

00:04:13.493 --> 00:04:17.535
you know maybe x is the negative
square root of two thirds

00:04:17.535 --> 00:04:21.146
and y is still the positive
square root of one third,

00:04:21.146 --> 00:04:25.046
or maybe x is the positive square root of

00:04:25.046 --> 00:04:30.019
two thirds, and y is the negative
square root of one third,

00:04:30.019 --> 00:04:31.302
kind of monotonous, but just getting

00:04:31.302 --> 00:04:34.042
all of the different
possibilities on the table here.

00:04:34.042 --> 00:04:36.811
X is negative square root of two thirds,

00:04:36.811 --> 00:04:41.718
and then y is positive, no,
negative, that's the last one,

00:04:41.718 --> 00:04:43.805
square root of one third.

00:04:43.805 --> 00:04:46.640
So these are the four points
where the contour lines

00:04:46.640 --> 00:04:49.286
are tangent, and to
find which one of these

00:04:49.286 --> 00:04:50.549
maximizes our function,

00:04:50.549 --> 00:04:52.316
here let's go ahead and write
down our function again,

00:04:52.316 --> 00:04:53.587
it gets easy to forget.

00:04:53.587 --> 00:04:55.338
So the whole thing we're
doing is maximizing

00:04:55.338 --> 00:04:58.347
f of x, y, equals x squared times y.

00:04:58.347 --> 00:05:01.006
So let me just put that down again,

00:05:01.006 --> 00:05:03.173
we're looking at f of x, y

00:05:04.829 --> 00:05:07.579
is equal to x squared times y.

00:05:08.774 --> 00:05:11.327
So we could just plug these values in

00:05:11.327 --> 00:05:13.458
and see which one of them
is actually greatest.

00:05:13.458 --> 00:05:14.845
And the first thing to observe is,

00:05:14.845 --> 00:05:16.903
x squared is always gonna be positive,

00:05:16.903 --> 00:05:19.048
so if I plug in a negative value for y,

00:05:19.048 --> 00:05:22.190
right, if I plug in either
this guy here or this guy here,

00:05:22.190 --> 00:05:23.858
where the value for y is negative,

00:05:23.858 --> 00:05:25.924
the entire function would be negative.

00:05:25.924 --> 00:05:27.617
So I'm just gonna say
that neither of these

00:05:27.617 --> 00:05:30.453
can be the maximum because
it'll be some positive number,

00:05:30.453 --> 00:05:32.900
some x squared times a negative.

00:05:32.900 --> 00:05:34.459
Whereas I know that these
guys are gonna produce

00:05:34.459 --> 00:05:36.352
a positive number, and specifically,

00:05:36.352 --> 00:05:39.252
now if we plug in, if we
plug in f of let's say

00:05:39.252 --> 00:05:42.585
this top one, square root of two thirds,

00:05:43.678 --> 00:05:46.261
square root of one third, well,

00:05:47.152 --> 00:05:49.616
x squared is gonna be two thirds,

00:05:49.616 --> 00:05:53.172
and then y is square root of one third.

00:05:53.172 --> 00:05:54.416
And in fact that's gonna be the same

00:05:54.416 --> 00:05:56.396
as what we get plugging
in this other value.

00:05:56.396 --> 00:05:58.850
So either one of these
maximizes the function.

00:05:58.850 --> 00:06:00.977
It's got two different maximizing points

00:06:00.977 --> 00:06:03.898
and each one of them has a maximum value

00:06:03.898 --> 00:06:08.363
of two thirds times the
square root of one third.

00:06:08.363 --> 00:06:09.876
And that's the final answer.

00:06:09.876 --> 00:06:12.180
But I do want to emphasize,
that the takeaway here

00:06:12.180 --> 00:06:14.449
is not the specific
algebra that you work out

00:06:14.449 --> 00:06:16.882
going towards the end, but it's the whole,

00:06:16.882 --> 00:06:19.261
the whole idea of this
Lagrange Multiplier technique

00:06:19.261 --> 00:06:21.421
to find the gradient of one function,

00:06:21.421 --> 00:06:23.954
find the gradient of the
constraining function,

00:06:23.954 --> 00:06:25.762
and then set them
proportional to each other.

00:06:25.762 --> 00:06:28.862
That's the key takeaway, and
then the rest of it is just,

00:06:28.862 --> 00:06:30.875
you know making sure
that we check our work

00:06:30.875 --> 00:06:32.713
and go through the minute details,

00:06:32.713 --> 00:06:35.340
which is important, it has its place.

00:06:35.340 --> 00:06:38.125
And coming up, I'll go
through a few more examples.