Phylogenetic trees

WEBVTT
Kind: captions
Language: en

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- [Voiceover] When we look at
all of the living diversity

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around us,

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then a natural question is,

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well, how related are the
difference species to each other?

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And if you put that into
an evolutionary context,

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relatedness should be tied to

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how recent did two species
share a common ancestor.

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And what we're going to
try to do in this video

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is construct a tree for
showing how different species

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evolved from common ancestors,

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and we're gonna do it

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based on some of these
observable traits that we see.

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But this is going to be a
huge oversimplification.

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I'm only doing it with five species

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and five very simple traits.

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As we'll see, or as we'll
talk about in future videos,

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this can be done in a
much more complex way,

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and that's what biologists would do.

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They would look at much
more than five traits,

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and they would look at molecular evidence,

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molecular evidence in terms
of protein differences,

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in terms of DNA differences,

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to really start to build out

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what we call a phylogenetic tree.

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So let me write this down.

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That's what we're going to create.

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Phylogenetic,

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genetic,

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tree.

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Phylo comes from the Greek
for group or kind or tribe,

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and then genetic comes, you know,

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related to the word "genesis."

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How do these things come about?

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How do the different groups or tribes,

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or in this case, how do the
different species come about?

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Well, when you're trying
to make one of these trees,

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it's important to realize
that this is a hypothesis,

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but you're, like always,

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trying to come up with
the simplest hypothesis

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that can explain the observations
that you actually see.

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And when we look at these,

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at least the species
that we have listed here,

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it looks like there's one
that is more different

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than all the other ones.

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The lamprey here does not
have any of these five traits

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that we are observing.

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So this, we would call the out group.

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The lamprey is the outgroup.

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And a lot of times

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when you need to construct
a phylogenetic tree,

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they might provide you something,

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with something that is
clearly an outgroup.

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Here it doesn't have any
of these observable traits.

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And sometimes, if we're
looking at genetic differences,

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it might have the largest
number of genetic differences

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relative to everything else.

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And so it makes sense,

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the simplest hypothesis
is its common ancestor

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is most distant into the
past with everything else.

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And so let me start to draw this tree.

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So I'm going to put deep into the past,

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so deep into the past, there
is a branching out point

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where you have the common
ancestor of the lamprey

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and everything else we see here.

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So eventually, you have
that common ancestor,

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and there's many, many
species along the way,

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and eventually, we get,

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we get a lamprey in present time.

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In present time.

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And so the next thing to
think about is, alright,

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well how did everything
else end up branching?

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Well, what's common about everything else

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that maybe wasn't common
about the lamprey?

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Well, one common thing is
we see that everything else,

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at least that we have
listed here, have jaws.

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And so it's reasonable to say, alright,

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we have this common ancestor,

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between the lamprey and everything else

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at this branching point right over here,

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and then it branched off
into multiple species,

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and one of those species
must have evolved jaws.

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So let me put jaws right over here.

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So jaws right over there.

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And jaws, that's called,

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jaws are considered a derived trait.

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This ancestral species at
this root did not have jaws,

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we're assuming,

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but at some point, they
evolved, and they stuck around

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because they proved to be
favorable in certain environments.

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Or it could have even been
things like genetic drift,

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who knows, but I'm guessing
that it was favorable

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in certain environments.

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So let's see, let's see if we
can classify everyone else.

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So now, out of the four, so we've,

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let's actually cross out the lamprey

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just for simplification

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since we've already
classified that character.

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Now of everyone else,

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we've already thought
about everyone's got jaws,

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so now let's go to the
next most common trait.

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So, and actually, let me
cross out the jaws too,

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just for, keep things simple.

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So we can do that a little bit cleaner.

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So I'm gonna cross out the jaws.

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And now, let's see,

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the next most common trait are the lungs,

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but not every species that
we have left has lungs.

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The sea bass does not have lungs.

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It does not breathe air
the way that animals

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that live outside of
the water breathe air.

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And so the next point of divergence

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must be between the sea bass

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and everything that we have left over.

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So let me draw that.

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So,

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now once again, I said, "must be."

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This is a hypothesis.

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I think it's a reasonable hypothesis.

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So let me draw that.

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So

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this is the sea,

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sea bass.

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And there's a common
ancestor between the sea bass

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and everything else, and the antelope,

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the bald eagle, and the alligator.

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And at some point, that
common ancestor diverged

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into multiple species,

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and one of those child species
must have evolved lungs.

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So lungs must have evolved at some point,

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but we're assuming that
that wasn't on this lineage

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for the sea bass.

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And once again,

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I'm just trying to find
the simplest explanation.

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There might have been some situation where

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maybe lungs evolved and then
went away at some point.

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You reverted to an ancestral form.

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But we like to go with
the simplest explanation.

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This is a property that
biologists will also often call

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parsimony, and actually,
let me write this down.

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Parsimony, which in everyday
language, means cheap.

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When someone tells you
that you're parsimonious,

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it's a nice sounding word, but
it means that you are cheap.

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But parsimony, in this context,

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say hey, we're trying to
be cheap with complexity.

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We're trying to be as simple as possible

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in our explanation of what's going on.

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But anyway, let's go back
to what we were doing.

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So we've already put into consideration,

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we have already talked
about the sea bass here,

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and we have already talked about lungs.

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Alright, so what do we have left?

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So we have to talk about the antelope,

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the bald eagle, the alligator,
and gizzards and fur.

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Alright, it looks like the
bald eagle and alligator

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have a gizzard.

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The antelope has fur,

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oh and actually, we haven't talked about

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the bald eagle and
feathers as yet, either.

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Alright, so it is possible.

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So let's make the next thing between,

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well, we could do it this way.

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And once again, I'm trying
to do this in real time,

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something that seems,

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so let's make a branch here.

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And let's say that that is the branch for,

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now let's say that's the
branch for the bald eagle.

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Say, "B. Eagle."

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That's the branch for the bald eagle,

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and let's see if I can construct one

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that will explain the differences
between the bald eagle,

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the antelope, and the alligator.

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Well, the bald eagle and the alligator

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have something in common.

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They have a gizzard in common.

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So let me make a branching point.

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Make them a little bit closer

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than the bald eagle is to the antelope.

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So let me do that.

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So let me put the alligator there,

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and then I'm gonna talk about

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when we get these derived traits.

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So that is

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the alligator.

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And obviously, I could
have written the alligator

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on this side and the
bald eagle on that side,

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or I could've rotated,

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I could rotate at any one
of these branching points.

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And then what we would
have left is the antelope.

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Now let's see if I can account

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for all of these derived traits.

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Antelope.

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Alright.

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So we have the common
ancestor of the sea bass,

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the bald eagle, the alligator,
and antelope right over here.

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We have a branching point.

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At some point, the lungs,

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we're hypothesizing
evolved in this branch.

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And then this branch,

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well let's say that this branch,

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this is the common ancestor

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between the antelope,
alligator, and bald eagle.

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And a common ancestor of the
bald eagle and alligator,

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they have to get the gizzard.

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So let's put the gizzard
down right over here.

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This is where the gizzard,

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this is our hypothesis.

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Doing the same color as,

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so that's the gizzard.

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Gizzard

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right over there.

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And so everything that
descended from that ancestor

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that had the gizzard,

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well, they're going to have gizzards.

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That's what we're assuming,

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but once again, that can be lost.

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This is a hypothesis.

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And so we have accounted for the gizzard.

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Let me,

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let me cross that out.

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So we have accounted for the gizzard.

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And so let's see,

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we have to account for the feathers,

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and the bald eagle is the
only one that has feathers.

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So let me put that here.

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So at some point, you
have a common ancestor

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of an alligator and a bald eagle.

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It branches off into multiple species,

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one of which gets a
feather or gets feathers.

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And once again, you know,
that could have branched off

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into many, many things,

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'cause we know that the bald eagle

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isn't the only species with feathers.

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But the bald eagle, for sure,

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is a species that has feathers.

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And let's see, so we've
accounted for the feathers now.

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Feathers.

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And now, we just have
to account for the fur,

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the fur of the antelope.

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And so, we don't know where
this could have happened.

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We might wanna look for more evidence

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to come up with a good hypothesis.

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But some place along this right branch,

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we could put,

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we could put the fur.

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And so there you have it.

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This is actually a
reasonable phylogenetic tree.

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I practiced the practice of parsimony

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to come up with the simplest explanation,

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but there are more
complicated explanations.

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And we don't know,

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some of those more
complicated explanations

00:09:54.343 --> 00:09:56.343
could very well be true.

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But from this,

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we have a very quick and
easy graphical representation

00:10:01.535 --> 00:10:04.905
of how related different species could be

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and where they share common ancestors.

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So bald eagle and an alligator,

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based on this phylogenetic tree,

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we would say are more related

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than a bald eagle is to an antelope.

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They have a more, the bald
eagle and the alligator,

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have a more common common ancestor,

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or more recent common ancestor right there

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than both of their common
ancestors with the antelope,

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and that would make them more related.

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And if we were doing this for real,

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we would wanna look at genetic evidence

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and look at the various proteins

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and say, okay, does that back this up?

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Are the bald eagle and alligator's DNA,

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do they have more in
common with each other

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than they do with the
DNA, say, of an antelope?

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And many, for especially
once you get complex,

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there could be many
different explanations,

00:10:49.402 --> 00:10:51.813
and we just wanna get more
and more and more evidence

00:10:51.813 --> 00:10:55.146
to keep refining our phylogenetic trees.


RC natural response example (3 of 3)

WEBVTT
Kind: captions
Language: en

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- [Voiceover] We just
derived what the current is

00:00:01.960 --> 00:00:02.949
and the voltage.

00:00:02.949 --> 00:00:06.382
These are both the natural
response of the RC.

00:00:06.382 --> 00:00:10.145
Now what I did is I went
ahead and I plotted out this

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using a computer, just using Excel,

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to plot out what these
two expressions look like,

00:00:15.510 --> 00:00:17.593
and let me show you that.

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So let's do a real quick example here

00:00:25.667 --> 00:00:29.834
with real values just to
see how this equation works.

00:00:31.981 --> 00:00:34.663
We'll say that this is 1,000 ohms.

00:00:34.663 --> 00:00:36.330
That's our resistor.

00:00:37.265 --> 00:00:40.098
And we'll say C is one microfarad.

00:00:42.187 --> 00:00:45.947
And what we wanna work out is R times C

00:00:45.947 --> 00:00:47.780
equals 10 to the third

00:00:50.385 --> 00:00:52.468
times 10 to the minus six

00:00:53.538 --> 00:00:57.288
and that equals 10 to
the minus three seconds

00:00:59.555 --> 00:01:01.138
or one millisecond.

00:01:04.823 --> 00:01:07.985
That's the product of R and C.

00:01:07.985 --> 00:01:09.130
I forgot V-naught.

00:01:09.130 --> 00:01:13.501
Let's say we put two
volts on this capacitor

00:01:13.501 --> 00:01:15.584
to start with, like that.

00:01:19.415 --> 00:01:23.415
And now we can say V of
t equals V-naught two volts

00:01:27.418 --> 00:01:31.085
times e to the minus t
over one millisecond.

00:01:35.292 --> 00:01:38.933
And that's our natural response
for this particular circuit.

00:01:38.933 --> 00:01:42.350
Now let me show you what that looks like.

00:01:44.587 --> 00:01:46.920
This is V of t on this side.

00:01:48.611 --> 00:01:52.694
Equals two e to the minus
t over one millisecond.

00:01:54.795 --> 00:01:57.038
And you can see it starts at two volts

00:01:57.038 --> 00:01:59.856
and then sags down as we predicted,

00:01:59.856 --> 00:02:02.661
and that's an exponential curve.

00:02:02.661 --> 00:02:04.734
And then over here on this side,

00:02:04.734 --> 00:02:07.347
i, as we said before, starts out at zero

00:02:07.347 --> 00:02:10.471
in the capacitor, the current
in the capacitor is zero,

00:02:10.471 --> 00:02:12.881
and as soon as we throw open that switch,

00:02:12.881 --> 00:02:17.281
the charge charges over
through the resistor

00:02:17.281 --> 00:02:20.364
and this is the equation here, i of t

00:02:22.257 --> 00:02:24.507
equals two volts over 1,000

00:02:27.438 --> 00:02:29.688
e to the minus t over RC or

00:02:36.021 --> 00:02:37.938
t over one millisecond.

00:02:43.425 --> 00:02:45.706
So this is what we call
the natural response

00:02:45.706 --> 00:02:48.257
of an RC circuit, and you'll run into this

00:02:48.257 --> 00:02:51.507
in almost every circuit you ever build.


Impedance

WEBVTT
Kind: captions
Language: en

00:00:00.652 --> 00:00:01.527
- [Voiceover] Now we're gonna talk about

00:00:01.527 --> 00:00:02.571
the idea of impedance.

00:00:02.571 --> 00:00:05.007
This is a really important
idea in electronics

00:00:05.007 --> 00:00:09.413
and it's something that comes
from the study of AC analysis.

00:00:09.413 --> 00:00:12.931
And AC analysis is where we
limit ourselves to inputs

00:00:12.931 --> 00:00:17.354
to our circuits that look like
sinusoids, cosines or sines.

00:00:17.354 --> 00:00:19.848
And of all the signals
that we could possibly have

00:00:19.848 --> 00:00:22.284
in the entire universe,
we're gonna limit ourselves

00:00:22.284 --> 00:00:24.770
just for the moment, to sine waves.

00:00:24.770 --> 00:00:26.469
And there's some great simplifications

00:00:26.469 --> 00:00:28.740
that emerge from this.

00:00:28.740 --> 00:00:30.232
So in this video, we're gonna look at,

00:00:30.232 --> 00:00:33.436
we're gonna develop,
basically the i-v equations

00:00:33.436 --> 00:00:36.507
for our three favorite passive components,

00:00:36.507 --> 00:00:39.704
resistor, inductor, and capacitor.

00:00:39.704 --> 00:00:43.871
And we're gonna look at those
when the input is a sinusoid.

00:00:47.247 --> 00:00:50.207
So that means that i or v,
the voltage or the current

00:00:50.207 --> 00:00:51.780
is in the shape of a sinusoid.

00:00:51.780 --> 00:00:54.382
And we're gonna see what that
means for the i-v equations

00:00:54.382 --> 00:00:56.465
for our favorite devices.

00:00:58.073 --> 00:01:01.251
So as we look at i-v equations
with sinusoid inputs,

00:01:01.251 --> 00:01:03.514
we're actually gonna break down sinusoids

00:01:03.514 --> 00:01:05.681
into complex exponentials.

00:01:07.381 --> 00:01:09.536
And when we studied
sinusoids, we found out

00:01:09.536 --> 00:01:11.424
that we could disassemble sinusoids

00:01:11.424 --> 00:01:15.084
into complex exponentials
using Euler's equation.

00:01:15.084 --> 00:01:17.956
So, for example, if we have a cosine wave,

00:01:17.956 --> 00:01:20.376
if we have a cosine as a function of time,

00:01:20.376 --> 00:01:22.537
omega t, we can express that

00:01:22.537 --> 00:01:26.008
in terms of complex
exponentials like this:

00:01:26.008 --> 00:01:29.175
1/2 e to the plus j omega t

00:01:32.327 --> 00:01:34.827
plus e to the minus j omega t.

00:01:38.972 --> 00:01:39.805
Like that.

00:01:40.813 --> 00:01:42.470
And what I'm gonna do
now is I'm gonna say,

00:01:42.470 --> 00:01:44.520
let's look at what happens

00:01:44.520 --> 00:01:46.761
when we use this as an input signal.

00:01:46.761 --> 00:01:48.131
This is not a real input signal,

00:01:48.131 --> 00:01:51.108
it's an imaginary, rotating vector.

00:01:51.108 --> 00:01:52.598
But if I have two of these,

00:01:52.598 --> 00:01:55.592
I can reassemble them into a cosine wave.

00:01:55.592 --> 00:01:57.372
And we like to use these exponentials,

00:01:57.372 --> 00:01:59.577
because they go through
the differential equations

00:01:59.577 --> 00:02:01.388
of a circuit really easily.

00:02:01.388 --> 00:02:04.055
These are the inputs
that we know how to solve

00:02:04.055 --> 00:02:06.773
when we do differential equations.

00:02:06.773 --> 00:02:09.724
So what I'm gonna do is
develop the i-v equations

00:02:09.724 --> 00:02:12.334
for the resistor, inductor, and capacitor

00:02:12.334 --> 00:02:14.137
in terms of this kind of an input,

00:02:14.137 --> 00:02:16.359
when the voltage or the
current looks like this,

00:02:16.359 --> 00:02:19.224
what do those equations look like?

00:02:19.224 --> 00:02:23.391
So we're gonna start with,
we'll start with the resistor.

00:02:25.059 --> 00:02:27.378
Here's a resistor.

00:02:27.378 --> 00:02:30.625
And we know from that
that just Ohm's law is

00:02:30.625 --> 00:02:32.208
v equals i times R.

00:02:34.177 --> 00:02:36.933
And just for the moment I'm
gonna assume that the current,

00:02:36.933 --> 00:02:41.100
let's assume that i equals
e to the plus j omega t.

00:02:46.153 --> 00:02:49.918
So if this is i, what
is v for our resistor?

00:02:49.918 --> 00:02:51.691
Well, we just plug i in here

00:02:51.691 --> 00:02:53.274
and we get v equals

00:02:55.080 --> 00:02:57.747
R times e to the plus j omega t.

00:03:02.144 --> 00:03:02.977
All right.

00:03:05.742 --> 00:03:07.277
Now I'm gonna do something that looks like

00:03:07.277 --> 00:03:08.591
it's a little too simple,

00:03:08.591 --> 00:03:10.985
but it's gonna get interesting soon.

00:03:10.985 --> 00:03:14.758
I wanna look at the ratio
of voltage to current.

00:03:14.758 --> 00:03:16.503
In this situation where we're driving

00:03:16.503 --> 00:03:19.439
with this complex exponential.

00:03:19.439 --> 00:03:23.606
So voltage turned out to be
R e to the plus j omega t

00:03:26.892 --> 00:03:30.303
and that's the voltage
I get if I put current

00:03:30.303 --> 00:03:34.220
though the resistor of
e to the plus j omega t.

00:03:35.879 --> 00:03:37.327
And what does that equal to?

00:03:37.327 --> 00:03:40.280
Well, these two are the
same so they cancel,

00:03:40.280 --> 00:03:42.680
and I get the ratio of voltage to current

00:03:42.680 --> 00:03:44.347
is R for a resistor.

00:03:46.592 --> 00:03:49.425
So for our resistor we just proved

00:03:50.691 --> 00:03:52.608
that v over i equals R.

00:03:59.159 --> 00:04:01.354
So this isn't news, we
didn't make a discovery here,

00:04:01.354 --> 00:04:03.130
this is just Ohm's law.

00:04:03.130 --> 00:04:05.917
And for a resistor, the
voltage over the current

00:04:05.917 --> 00:04:07.875
is always equal to the resistance.

00:04:07.875 --> 00:04:08.708
All right.

00:04:08.708 --> 00:04:10.341
This is gonna get more
interesting now as we go do

00:04:10.341 --> 00:04:12.100
inductors and capcitors.

00:04:12.100 --> 00:04:14.100
So let's do an inductor.

00:04:15.515 --> 00:04:17.812
It has a value of L henries.

00:04:17.812 --> 00:04:21.979
And for an inductor we know
that voltage equals L di dt.

00:04:27.760 --> 00:04:29.284
All right, and let's do
the same thing again.

00:04:29.284 --> 00:04:32.284
Let i equal e to the plus j omega t.

00:04:34.887 --> 00:04:37.451
So it's a complex exponential current

00:04:37.451 --> 00:04:40.245
that we're forcing through our inductor.

00:04:40.245 --> 00:04:42.245
And let's go ahead and work out what v is.

00:04:42.245 --> 00:04:44.245
So v equals L times d dt

00:04:48.187 --> 00:04:51.854
of this value here, e
to the plus j omega t.

00:04:54.737 --> 00:04:58.662
Or v equals, now we take the derivative

00:04:58.662 --> 00:05:02.495
and the j omega term
comes down to multiply L,

00:05:04.069 --> 00:05:06.652
so we get j omega L times what?

00:05:09.220 --> 00:05:13.431
Times the same thing, e
to the plus j omega t,

00:05:13.431 --> 00:05:15.565
this is the beautiful
thing about exponentials,

00:05:15.565 --> 00:05:17.982
they give us back themselves.

00:05:19.014 --> 00:05:20.657
All right, now let's do this.

00:05:20.657 --> 00:05:22.439
Let's take, once more,

00:05:22.439 --> 00:05:26.207
what's the ratio of voltage to current?

00:05:26.207 --> 00:05:29.114
And that equals, here's the voltage,

00:05:29.114 --> 00:05:32.447
j omega L times e to the plus j omega t,

00:05:34.351 --> 00:05:39.141
and let's divide that by i
which is i is right here,

00:05:39.141 --> 00:05:41.558
i is e to the plus j omega t.

00:05:44.039 --> 00:05:45.456
So, those cancel.

00:05:47.456 --> 00:05:48.289
And we get

00:05:49.962 --> 00:05:52.129
v over i equals j omega L.

00:06:01.910 --> 00:06:05.774
So now we have an equation
for v over i for an inductor.

00:06:05.774 --> 00:06:07.008
And this is interesting,

00:06:07.008 --> 00:06:09.738
this time we have the inductance value

00:06:09.738 --> 00:06:13.519
which we expected, and
there's also this omega,

00:06:13.519 --> 00:06:15.573
j omega term that comes in.

00:06:15.573 --> 00:06:18.429
So this tells us this is
frequency, omega is frequency.

00:06:18.429 --> 00:06:22.765
So this tells us that the
ratio of v to i for an inductor

00:06:22.765 --> 00:06:24.932
is dependent on frequency.

00:06:29.743 --> 00:06:31.532
Now we'll do the same
thing for a capacitor.

00:06:31.532 --> 00:06:33.365
So here's a capacitor.

00:06:35.521 --> 00:06:38.585
And there's it's
capacitance value in farads.

00:06:38.585 --> 00:06:42.752
And for the capacitor, we know
that i equals C times dv dt.

00:06:48.697 --> 00:06:51.427
And this time let's let v of t,

00:06:51.427 --> 00:06:54.927
let's let v equal e to the plus j omega t.

00:06:57.186 --> 00:06:58.587
So this time we're gonna force

00:06:58.587 --> 00:07:00.597
a voltage across our capacitor

00:07:00.597 --> 00:07:05.458
that is this imaginary,
this complex exponential.

00:07:05.458 --> 00:07:07.332
And that gives us, let's
plug that into here,

00:07:07.332 --> 00:07:09.082
i equals C times d dt

00:07:13.659 --> 00:07:15.909
of e to the plus j omega t.

00:07:20.255 --> 00:07:22.755
Now let's take that derivative

00:07:23.763 --> 00:07:26.379
i equals, same thing happens,

00:07:26.379 --> 00:07:29.585
j omega comes down to
multiply out in front with C,

00:07:29.585 --> 00:07:31.006
and we get the same thing over here.

00:07:31.006 --> 00:07:33.089
So we get j omega C times

00:07:35.816 --> 00:07:37.816
e to the plus j omega t.

00:07:39.387 --> 00:07:41.165
And now we'll ask the same question again

00:07:41.165 --> 00:07:45.332
that we did before, which is
what is v over i for capacitor?

00:07:47.386 --> 00:07:48.338
And we can fill this in,

00:07:48.338 --> 00:07:52.987
v is sitting right here,
v is e to the j omega t,

00:07:52.987 --> 00:07:54.904
e to the plus j omega t

00:07:57.652 --> 00:08:01.588
and the current is, we
worked that out down here,

00:08:01.588 --> 00:08:02.921
that's j omega C

00:08:05.779 --> 00:08:08.279
times e to the plus j omega t.

00:08:11.099 --> 00:08:13.340
And once again, we get this
nice cancellation here,

00:08:13.340 --> 00:08:17.708
this cancels with this,
and for capacitor we get

00:08:17.708 --> 00:08:18.958
v over i equals

00:08:21.212 --> 00:08:22.795
one over j omega C.

00:08:26.944 --> 00:08:30.027
And we put a box around that one too.

00:08:34.803 --> 00:08:37.108
So this says that the
ratio of voltage to current

00:08:37.108 --> 00:08:39.792
in a capacitor, it depends on
the value of the capacitor,

00:08:39.792 --> 00:08:43.435
of course, and it also
depends on frequency.

00:08:43.435 --> 00:08:45.126
So just like over with the inductor,

00:08:45.126 --> 00:08:48.364
we have a frequency term in here.

00:08:48.364 --> 00:08:52.250
So now we're gonna give this
ratio of voltage to current

00:08:52.250 --> 00:08:54.613
in all three cases, we're
gonna give a special name

00:08:54.613 --> 00:08:56.863
and that name is impedance.

00:09:02.058 --> 00:09:06.141
And the symbol we often
use for impedance is a Z.

00:09:07.471 --> 00:09:09.386
So this word impedance
is the general notion

00:09:09.386 --> 00:09:12.419
of the ratio of voltage to current.

00:09:12.419 --> 00:09:16.331
And we can do that for all
three of our passive components.

00:09:16.331 --> 00:09:20.498
For a resistor, the impedance
is its resistance, R.

00:09:21.918 --> 00:09:24.789
So the word impedance is
like the word resistance,

00:09:24.789 --> 00:09:26.488
except it's a more general concept.

00:09:26.488 --> 00:09:29.752
It's the general concept of
voltage divided by current.

00:09:29.752 --> 00:09:33.737
For a resistor, the
impedance is the resistance.

00:09:33.737 --> 00:09:37.904
For an inductor, the impedance,
v over i is j omega L.

00:09:40.796 --> 00:09:43.088
And down here for
capacitor, the impedance is

00:09:43.088 --> 00:09:46.581
one over j omega C, for capacitance.

00:09:46.581 --> 00:09:49.634
So this is where the idea of
impedance, this word impedance,

00:09:49.634 --> 00:09:51.934
this is where it comes from.

00:09:51.934 --> 00:09:55.383
And the idea includes both
the values of the components

00:09:55.383 --> 00:09:58.420
and the effect that frequency has

00:09:58.420 --> 00:10:01.597
on the voltage to current ratio.

00:10:01.597 --> 00:10:05.764
So both of those things
are combined into one idea.

00:10:10.451 --> 00:10:12.803
So just a quick summary of impedance,

00:10:12.803 --> 00:10:16.970
if we say the impedance of
a resistor, that equals R.

00:10:17.960 --> 00:10:21.793
The impedance of an
inductor equals j omega L.

00:10:24.888 --> 00:10:28.259
And the impedance of a capacitor

00:10:28.259 --> 00:10:30.426
equals one over j omega C.

00:10:35.821 --> 00:10:37.930
And as a reminder of
the assumptions we made,

00:10:37.930 --> 00:10:41.302
we said that we're only gonna
consider sinusoidal inputs.

00:10:41.302 --> 00:10:44.392
And what we did is we
broke up our sinusoid,

00:10:44.392 --> 00:10:46.495
our cosine wave, into these,

00:10:46.495 --> 00:10:48.336
we looked at how these
rotating vectors passed

00:10:48.336 --> 00:10:52.503
through our components in the
form of voltages and currents.

00:10:53.422 --> 00:10:54.517
So there's no new physics here.

00:10:54.517 --> 00:10:56.861
What happened is we took these j omegas

00:10:56.861 --> 00:10:59.197
that came out of the exponentials

00:10:59.197 --> 00:11:00.520
when we took the derivatives,

00:11:00.520 --> 00:11:04.436
we associated those with
the component itself,

00:11:04.436 --> 00:11:06.457
we did that here and we did that here,

00:11:06.457 --> 00:11:07.585
and you can see it here.

00:11:07.585 --> 00:11:08.820
So we sort of done,

00:11:08.820 --> 00:11:11.630
it's somewhat of a notational trick.

00:11:11.630 --> 00:11:13.767
We associate this frequency dependence,

00:11:13.767 --> 00:11:16.995
not with the inputs and
the voltages and currents,

00:11:16.995 --> 00:11:19.912
but with the components themselves.

00:11:21.426 --> 00:11:22.677
And this is something we call,

00:11:22.677 --> 00:11:25.677
this is referred to as transforming.

00:11:30.198 --> 00:11:32.392
We transformed our components.

00:11:32.392 --> 00:11:34.377
That's the phrase that's used there.

00:11:34.377 --> 00:11:36.520
But from this comes the idea of impedance

00:11:36.520 --> 00:11:40.100
and in the general sense of
the voltage to current ratio.


Definite integrals of product of sines

WEBVTT
Kind: captions
Language: en

00:00:00.586 --> 00:00:02.625
- [Voiceover] So we've already established

00:00:02.625 --> 00:00:04.077
that these three definitive roles

00:00:04.077 --> 00:00:05.622
are going to be equal to zero

00:00:05.622 --> 00:00:07.253
over slightly different conditions.

00:00:07.253 --> 00:00:08.401
Let's keep on going,

00:00:08.401 --> 00:00:10.350
and remember the goal here
is to make it simple for us

00:00:10.350 --> 00:00:14.105
to find our Fourier coefficients
in a few videos from now.

00:00:14.105 --> 00:00:16.960
So now let's think about
the definitive role

00:00:16.960 --> 00:00:20.017
from zero to two pie if we're taking

00:00:20.017 --> 00:00:23.850
the product of sine of
mt times sine of nt dt,

00:00:28.703 --> 00:00:31.451
and we're gonna think about
it in two different scenarios.

00:00:31.451 --> 00:00:35.368
We're gonna think about
when m does not equal n

00:00:37.374 --> 00:00:39.920
or m does not equal negative n,

00:00:39.920 --> 00:00:42.249
and really we care more
about this scenario

00:00:42.249 --> 00:00:43.881
because in the Fourier series

00:00:43.881 --> 00:00:46.148
we're gonna be dealing
with positive integers,

00:00:46.148 --> 00:00:48.710
or non negative integers
I guess we could say,

00:00:48.710 --> 00:00:51.287
but this is going to be
true for both of them

00:00:51.287 --> 00:00:53.138
and we're gonna see when this is true

00:00:53.138 --> 00:00:56.374
this integral is going to
be zero when that's true,

00:00:56.374 --> 00:00:59.291
and it's gonna be equal to pie when

00:01:01.498 --> 00:01:03.642
m is equal to n,

00:01:03.642 --> 00:01:05.414
or another way we could write this.

00:01:05.414 --> 00:01:07.357
Actually let me write it this way.

00:01:07.357 --> 00:01:09.857
This integral is equal to zero

00:01:11.667 --> 00:01:13.584
when m does not equal n

00:01:16.106 --> 00:01:18.606
or m does not equal negative n

00:01:20.514 --> 00:01:23.658
and we're gonna assume integers.

00:01:23.658 --> 00:01:26.216
When integers m does not equal n

00:01:26.216 --> 00:01:28.181
or m does not equal negative n,

00:01:28.181 --> 00:01:31.449
and then if n is equal to m or visa versa.

00:01:31.449 --> 00:01:34.819
We could say that the
integral from zero to two pie.

00:01:34.819 --> 00:01:37.058
Let me do this in a different color.

00:01:37.058 --> 00:01:39.662
Just 'cause I think it will look nice.

00:01:39.662 --> 00:01:42.524
So the integral from zero to two pie.

00:01:42.524 --> 00:01:44.554
If m is equal to n we could write that as

00:01:44.554 --> 00:01:46.685
sine of mt times sine of mt

00:01:46.685 --> 00:01:50.586
which is gonna be sine squared of mt dt.

00:01:50.586 --> 00:01:53.109
We're going to establish
that this is going

00:01:53.109 --> 00:01:55.360
to be equal to pie.

00:01:55.360 --> 00:01:57.499
Now how are we going to do that?

00:01:57.499 --> 00:02:01.752
Well we can rewrite this
original thing right over here.

00:02:01.752 --> 00:02:04.094
We can rewrite this original thing

00:02:04.094 --> 00:02:07.128
using a product to sum formula.

00:02:07.128 --> 00:02:09.618
So let's actually do that.

00:02:09.618 --> 00:02:11.454
So rewriting this integral.

00:02:11.454 --> 00:02:14.224
It's the integral from zero to two pie

00:02:14.224 --> 00:02:17.084
and I can rewrite, and this
is just a trig identity

00:02:17.084 --> 00:02:18.871
that I'm going to be using.

00:02:18.871 --> 00:02:23.038
This is the same thing as
one half times times cosine

00:02:26.648 --> 00:02:27.898
of mt minus nt.

00:02:32.372 --> 00:02:35.455
So we could write that at m minus n t

00:02:36.669 --> 00:02:39.669
and then minus cosine of mt plus nt,

00:02:41.204 --> 00:02:44.287
or we could write that as m plus n t,

00:02:45.280 --> 00:02:47.103
and actually let me put parentheses around

00:02:47.103 --> 00:02:50.779
these whole things so it's
more clear what's going on.

00:02:50.779 --> 00:02:53.744
This right over here is just
the product to sum formula.

00:02:53.744 --> 00:02:55.876
When you're taking the product of the sine

00:02:55.876 --> 00:02:58.010
of two different things here,

00:02:58.010 --> 00:03:01.196
and of course this is going to be dt.

00:03:01.196 --> 00:03:04.847
Now if we use some of our
integration properties

00:03:04.847 --> 00:03:06.632
we can rewrite all of this

00:03:06.632 --> 00:03:10.215
as being the integral
from zero to two pie.

00:03:11.809 --> 00:03:14.649
In fact we could put that
one half out front twice.

00:03:14.649 --> 00:03:16.727
I'm gonna write this is,

00:03:16.727 --> 00:03:18.913
I guess you could say the
difference of two integrals.

00:03:18.913 --> 00:03:20.208
Two definite integrals.

00:03:20.208 --> 00:03:22.791
So we could have this one half.

00:03:23.948 --> 00:03:26.814
Let me write it in that same color.

00:03:26.814 --> 00:03:31.410
That one half right over there
times the definite integral

00:03:31.410 --> 00:03:35.577
from zero two pie of cosine
of m minus n t dt minus,

00:03:41.572 --> 00:03:43.928
and once again I'm just
distributing this one half,

00:03:43.928 --> 00:03:46.825
so minus, and taking it
out of the integrals.

00:03:46.825 --> 00:03:50.575
So minus one half times
the definite integral

00:03:52.377 --> 00:03:54.127
from zero to two pie,

00:03:55.190 --> 00:03:56.964
and I'm gonna have the dt out here,

00:03:56.964 --> 00:03:59.995
so let me just write the
dt out there in magenta,

00:03:59.995 --> 00:04:02.412
of cosine of m plus n, t, dt,

00:04:09.196 --> 00:04:11.308
and this is a slightly
different shade of blue

00:04:11.308 --> 00:04:13.087
so let me color that in as well.

00:04:13.087 --> 00:04:14.941
So once again we get from that to that

00:04:14.941 --> 00:04:16.954
just using our integration properties.

00:04:16.954 --> 00:04:18.610
You can distribute the one half

00:04:18.610 --> 00:04:20.596
and say okay the integral
of the difference

00:04:20.596 --> 00:04:22.987
is the same thing as the
difference of the integrals.

00:04:22.987 --> 00:04:25.205
You could take the one half
out of the integration sine

00:04:25.205 --> 00:04:27.343
and we get to this right over here.

00:04:27.343 --> 00:04:29.728
Now what do we know about this

00:04:29.728 --> 00:04:32.895
in the case where m and n are integers

00:04:34.086 --> 00:04:36.681
that either don't equal
each other or don't equal

00:04:36.681 --> 00:04:38.420
the negatives of each other?

00:04:38.420 --> 00:04:39.851
Well they don't equal each other

00:04:39.851 --> 00:04:41.886
or don't equal to the
negatives of each other.

00:04:41.886 --> 00:04:45.679
Then this right over here is
going to be a non-zero integer.

00:04:45.679 --> 00:04:49.429
Non-zero integer and
this also is going to be

00:04:51.087 --> 00:04:52.670
a non-zero integer,

00:04:56.194 --> 00:04:58.893
and so we already know
that if we're taking

00:04:58.893 --> 00:05:02.109
the definite integral of cosine of mt dt

00:05:02.109 --> 00:05:03.557
from zero to two pie.

00:05:03.557 --> 00:05:05.585
Where m is some non-zero integer.

00:05:05.585 --> 00:05:07.955
Where the coefficient on
t is a non-zero integer

00:05:07.955 --> 00:05:10.368
then that integral's
gonna evaluate it to zero,

00:05:10.368 --> 00:05:12.022
but that's what we're doing over here,

00:05:12.022 --> 00:05:13.472
and it doesn't look exactly the same

00:05:13.472 --> 00:05:15.600
but this is a non-zero
integer coefficient.

00:05:15.600 --> 00:05:18.738
This is a non-zero integer coefficient.

00:05:18.738 --> 00:05:20.717
So this whole integral's gonna be zero,

00:05:20.717 --> 00:05:22.372
this whole integral's gonna be zero,

00:05:22.372 --> 00:05:23.987
and even if you multiply it by one half

00:05:23.987 --> 00:05:25.742
you still get a bunch of zeros.

00:05:25.742 --> 00:05:27.813
So hopefully that makes you feel good

00:05:27.813 --> 00:05:29.648
about that statement right over here.

00:05:29.648 --> 00:05:33.177
When m does not equal n or
m does not equal negative n,

00:05:33.177 --> 00:05:35.463
this whole thing is just
going to evaluate it to zero.

00:05:35.463 --> 00:05:38.769
That's gonna be zero and
that is going to be zero.

00:05:38.769 --> 00:05:42.879
Now let's think about the
situation where m does equal n,

00:05:42.879 --> 00:05:45.235
and actually let me just
delete this a little bit

00:05:45.235 --> 00:05:47.931
so that we have some space to work with.

00:05:47.931 --> 00:05:49.605
So if we assume.

00:05:49.605 --> 00:05:51.997
Actually let me keep that part there.

00:05:51.997 --> 00:05:54.747
So let me keep all of that there.

00:05:56.463 --> 00:05:58.941
Alright, now if m is,

00:05:58.941 --> 00:06:01.866
and actually I even colored.

00:06:01.866 --> 00:06:06.033
So if we assume in the
case of sine squared of mt.

00:06:09.351 --> 00:06:13.518
If we assume if m is equal
to n then what happens here?

00:06:15.467 --> 00:06:17.660
Remember, the thing I wrote in green

00:06:17.660 --> 00:06:21.094
is just the top one where m is equal to n.

00:06:21.094 --> 00:06:22.962
If m is equal to n.

00:06:22.962 --> 00:06:27.384
Well what happens is this
second integral right over here.

00:06:27.384 --> 00:06:31.551
This is still going to be a
non-zero integer coefficient.

00:06:33.725 --> 00:06:36.642
So that one is going to still be zero

00:06:36.642 --> 00:06:38.284
based on what we just argued,

00:06:38.284 --> 00:06:39.634
but this right over here,

00:06:39.634 --> 00:06:41.997
m minus n, that is going to be zero.

00:06:41.997 --> 00:06:43.944
You're going to have a
zero coefficient there.

00:06:43.944 --> 00:06:45.919
So it's going to be cosine of zero t.

00:06:45.919 --> 00:06:48.493
Well cosine of zero t,
that's cosine of zero

00:06:48.493 --> 00:06:49.872
regardless of what t you have,

00:06:49.872 --> 00:06:52.231
and cosine of zero is
going to be equal to one,

00:06:52.231 --> 00:06:55.425
and so everything in the
case where m is equal to n.

00:06:55.425 --> 00:06:58.862
This whole thing
simplifies to the one half

00:06:58.862 --> 00:07:02.948
times the integral from zero to two pie,

00:07:02.948 --> 00:07:06.865
and I could write of one
dt if we like, one dt.

00:07:10.548 --> 00:07:13.103
Remember this is the case
where m is equal to n,

00:07:13.103 --> 00:07:15.797
and so what is that going to be equal to?

00:07:15.797 --> 00:07:19.901
Well that's going to be
equal to one half times.

00:07:19.901 --> 00:07:23.117
Well the anti derivative of one is just t.

00:07:23.117 --> 00:07:27.284
So it's just going to be t
evaluated at two pie and zero,

00:07:30.141 --> 00:07:34.182
and so that is going to
be equal to one half.

00:07:34.182 --> 00:07:36.869
I'm doing the same color so
we can keep track of things.

00:07:36.869 --> 00:07:39.425
So that is going to be
equal to one half times

00:07:39.425 --> 00:07:41.446
two pie minus zero.

00:07:41.446 --> 00:07:43.029
Two pie minus zero.

00:07:43.886 --> 00:07:46.049
Well two pie minus zero is just two pie.

00:07:46.049 --> 00:07:49.021
So I could just write that's
one half times two pie.

00:07:49.021 --> 00:07:51.819
Which of course is just
going to be equal to pie.

00:07:51.819 --> 00:07:54.982
So we just said, look
when m is equal to n.

00:07:54.982 --> 00:07:58.296
Well if m is equal to n
the magenta expression

00:07:58.296 --> 00:08:00.506
is the same thing as
what I wrote in green,

00:08:00.506 --> 00:08:02.940
and so when m is equal
to n at this integral.

00:08:02.940 --> 00:08:06.185
That integral, when m is
equal to n is going to be,

00:08:06.185 --> 00:08:08.413
and that's the same thing
as sine squared of mt,

00:08:08.413 --> 00:08:11.246
that is going to be equal to pie.


Finding Fourier coefficients for square wave

WEBVTT
Kind: captions
Language: en

00:00:00.285 --> 00:00:03.064
- [Voiceover] So this could
very well be an exciting video

00:00:03.064 --> 00:00:05.878
because we start with this
idea of a Fourier series

00:00:05.878 --> 00:00:07.378
that we could take a periodic function

00:00:07.378 --> 00:00:08.991
and represent it as an infinite sum

00:00:08.991 --> 00:00:11.138
of weighted cosines and sines

00:00:11.138 --> 00:00:14.889
and we use that idea to say,
"Well can we find formulas

00:00:14.889 --> 00:00:15.973
"for those coefficients?"

00:00:15.973 --> 00:00:20.087
And we were able to do that
using the powers of calculus.

00:00:20.087 --> 00:00:21.964
And now we can actually apply it

00:00:21.964 --> 00:00:24.580
for this particular square wave.

00:00:24.580 --> 00:00:27.732
And I picked a square wave
that has a period of two pi

00:00:27.732 --> 00:00:28.690
and that's where, actually,

00:00:28.690 --> 00:00:30.628
a lot of these two pis came out from,

00:00:30.628 --> 00:00:32.134
and that's also why we started here

00:00:32.134 --> 00:00:34.237
at cosine t and sine of t.

00:00:34.237 --> 00:00:36.419
They both have a frequency
of one over two pi,

00:00:36.419 --> 00:00:38.645
which is the frequency of
our original square wave.

00:00:38.645 --> 00:00:39.860
And then the other terms

00:00:39.860 --> 00:00:42.272
have frequencies that
are multiples of that.

00:00:42.272 --> 00:00:44.300
If we had a different period
then all of that would change.

00:00:44.300 --> 00:00:45.645
But I picked this period

00:00:45.645 --> 00:00:47.129
to just make the math a little bit simpler

00:00:47.129 --> 00:00:49.194
and we will generalize in the future.

00:00:49.194 --> 00:00:51.141
But now, let's actually evaluate

00:00:51.141 --> 00:00:53.120
a-sub-zero, a-sub-n, and b-sub-n

00:00:53.120 --> 00:00:55.787
for this particular square wave.

00:00:56.638 --> 00:00:58.573
So, they key to realize

00:00:58.573 --> 00:01:02.023
is that our square wave
between zero and pi,

00:01:02.023 --> 00:01:03.349
'cause we're gonna keep
taking the definite integral

00:01:03.349 --> 00:01:05.305
from zero to two pi, from zero to two pi,

00:01:05.305 --> 00:01:06.521
from zero to two pi.

00:01:06.521 --> 00:01:09.897
The key realization
here is from zero to pi

00:01:09.897 --> 00:01:13.144
our function is equal to three,

00:01:13.144 --> 00:01:16.029
and from pi to two pi, our
function is equal to zero.

00:01:16.029 --> 00:01:18.263
So all of these definite integrals.

00:01:18.263 --> 00:01:20.561
So this definite integral, for example,

00:01:20.561 --> 00:01:22.770
is going to be the same thing as,

00:01:22.770 --> 00:01:25.769
and I'll do it once and
then we'll see that trend.

00:01:25.769 --> 00:01:29.590
This is the same thing
as one over two pi times

00:01:29.590 --> 00:01:33.097
the definite integral from zero to pi

00:01:33.097 --> 00:01:34.781
of f-of-t dt

00:01:34.781 --> 00:01:38.260
plus the integral from pi to two pi

00:01:38.260 --> 00:01:40.382
of f-of-t dt.

00:01:40.382 --> 00:01:43.749
Now f-of-t, between zero and pi,

00:01:43.749 --> 00:01:45.808
we just said it's equal to three.

00:01:45.808 --> 00:01:47.967
And f-of-t between pi and two pi,

00:01:47.967 --> 00:01:50.169
well it's going to be equal to zero.

00:01:50.169 --> 00:01:53.003
And that's actually the case
for these other situations.

00:01:53.003 --> 00:01:55.421
That f-of-t's gonna be
zero from pi to two pi

00:01:55.421 --> 00:01:57.550
and zero times anything is gonna be zero,

00:01:57.550 --> 00:02:00.282
so the integrals, the
definite integrals over this,

00:02:00.282 --> 00:02:01.620
the second part of the interval,

00:02:01.620 --> 00:02:03.489
are always going to be zero.

00:02:03.489 --> 00:02:05.110
And so it boils down to this.

00:02:05.110 --> 00:02:07.315
And so what's the definite
integral from zero to pi

00:02:07.315 --> 00:02:08.778
of three dt?

00:02:08.778 --> 00:02:11.461
Well this is going to be equal to

00:02:11.461 --> 00:02:13.294
one over two pi times,

00:02:15.391 --> 00:02:16.968
so if you evaluate this,

00:02:16.968 --> 00:02:19.622
the anti-derivative of
three, it'd be three t,

00:02:19.622 --> 00:02:23.474
evaluated from zero to pi,

00:02:23.474 --> 00:02:25.755
and so it'd be three pi
minus three times zero

00:02:25.755 --> 00:02:27.200
which is just three pi.

00:02:27.200 --> 00:02:30.442
So it's equal to three pi over the two pi

00:02:30.442 --> 00:02:33.449
that we had already, over the two pi,

00:02:33.449 --> 00:02:36.953
and so this is going to
be equal to three halves.

00:02:36.953 --> 00:02:37.970
Three halves.

00:02:37.970 --> 00:02:40.150
And that makes a lot of sense

00:02:40.150 --> 00:02:42.060
because a-sub-zero we
said you could view that

00:02:42.060 --> 00:02:44.538
as the average value of the
function over that interval.

00:02:44.538 --> 00:02:46.059
And the average value of that function

00:02:46.059 --> 00:02:48.082
is indeed, if it's three half the time,

00:02:48.082 --> 00:02:49.692
and zero the other half of the time,

00:02:49.692 --> 00:02:51.754
well then the average is
going to be one and a half,

00:02:51.754 --> 00:02:52.877
or three halves.

00:02:52.877 --> 00:02:55.044
So that is our a-sub-zero.

00:02:55.981 --> 00:02:58.423
So now let's figure
out the general a-sub-n

00:02:58.423 --> 00:03:00.950
where n is not equal zero.

00:03:00.950 --> 00:03:03.510
So, a-sub-n,

00:03:03.510 --> 00:03:07.287
well we are just gonna take,
it's gonna be one over pi,

00:03:07.287 --> 00:03:09.252
one over pi, the definite integral,

00:03:09.252 --> 00:03:10.974
I could go from zero to two pi,

00:03:10.974 --> 00:03:12.436
but instead I'm just
gonna cut to the chase.

00:03:12.436 --> 00:03:14.430
I'm gonna go just from zero to pi

00:03:14.430 --> 00:03:17.442
'cause the integral from pi to
two pi is just gonna be zero

00:03:17.442 --> 00:03:18.609
'cause the function's equal to zero.

00:03:18.609 --> 00:03:21.351
It's gonna be zero times cosine
nt; it's just gonna be zero.

00:03:21.351 --> 00:03:22.885
So for this particular square wave,

00:03:22.885 --> 00:03:25.127
I can just worry about from zero to pi.

00:03:25.127 --> 00:03:28.159
From zero to pi, f-of-t is three.

00:03:28.159 --> 00:03:30.768
So it's gonna be three cosine-nt.

00:03:30.768 --> 00:03:33.076
Three, let me do that color,

00:03:33.076 --> 00:03:38.036
three cosine-of-nt,

00:03:38.036 --> 00:03:40.619
cosine-nt dt.

00:03:41.516 --> 00:03:43.607
Now what is this going to be equal to?

00:03:43.607 --> 00:03:45.861
Well we can do a few things.

00:03:45.861 --> 00:03:48.804
We could take our three
and bring it out front.

00:03:48.804 --> 00:03:50.852
So let me do that.

00:03:50.852 --> 00:03:54.184
So we could take that three
and bring it out front.

00:03:54.184 --> 00:03:57.092
So, just putting the three out here.

00:03:57.092 --> 00:03:59.434
So take that three, put it out front.

00:03:59.434 --> 00:04:01.589
And we know that the
derivative of sine-of-nt

00:04:01.589 --> 00:04:03.681
is n cosine-of-nt.

00:04:03.681 --> 00:04:05.313
So let's multiple the inside and the out,

00:04:05.313 --> 00:04:07.322
or let's multiple the inside by n,

00:04:07.322 --> 00:04:08.672
and then we could also divide by n

00:04:08.672 --> 00:04:10.265
but we could put that outside.

00:04:10.265 --> 00:04:11.473
They would, it'd be,

00:04:11.473 --> 00:04:14.221
n divided by n, we
haven't changed the value.

00:04:14.221 --> 00:04:16.606
And so this is going to be equal to

00:04:16.606 --> 00:04:20.644
three over n pi, that's
just what we had out here,

00:04:20.644 --> 00:04:24.844
times, well the anti-derivative
of this business

00:04:24.844 --> 00:04:28.168
is going to be sine-of-nt,

00:04:28.168 --> 00:04:30.963
sine-of-nt.

00:04:30.963 --> 00:04:33.728
And we're gonna evaluate from zero to pi.

00:04:33.728 --> 00:04:37.347
So this is going to be
equal to three over n pi

00:04:37.347 --> 00:04:42.032
times sine-of-n pi, well
that's gonna be zero for any n,

00:04:42.032 --> 00:04:44.063
minus sine-of-zero times n.

00:04:44.063 --> 00:04:46.048
Well that's gonna be zero for any n.

00:04:46.048 --> 00:04:48.469
So all of this is just going to be zero.

00:04:48.469 --> 00:04:51.851
So a-sub-n is going to be equal to zero.

00:04:51.851 --> 00:04:54.879
The coefficient on any, for any,

00:04:54.879 --> 00:04:57.328
of a-sub-ns for any n not equal zero,

00:04:57.328 --> 00:04:58.161
it's going to be zero.

00:04:58.161 --> 00:04:59.994
So actually we're not gonna have

00:04:59.994 --> 00:05:03.625
any of these cosines show
up in the Fourier expansion.

00:05:03.625 --> 00:05:05.892
Let's think about our b-sub-n.

00:05:05.892 --> 00:05:07.332
When you look at the
shape of the square wave,

00:05:07.332 --> 00:05:08.819
it actually makes a lot of sense.

00:05:08.819 --> 00:05:11.835
But let's now tackle our b-sub-ns.

00:05:11.835 --> 00:05:15.252
So our b-sub-ns, get a little space here.

00:05:16.190 --> 00:05:19.117
So b-sub-n.

00:05:19.117 --> 00:05:20.325
Same thing,

00:05:20.325 --> 00:05:23.134
we could just worry about
the interval from zero to pi

00:05:23.134 --> 00:05:24.713
because from pi to two pi

00:05:24.713 --> 00:05:28.184
our f-of-t is going to be equal to zero.

00:05:28.184 --> 00:05:29.965
So this is going to be equal to

00:05:29.965 --> 00:05:33.882
one over pi times the definite integral,

00:05:34.720 --> 00:05:37.696
once again I'm only gonna
worry about from zero to pi,

00:05:37.696 --> 00:05:40.941
zero to pi dt.

00:05:40.941 --> 00:05:43.626
Now the value of the
function from zero to pi

00:05:43.626 --> 00:05:45.642
is three, we've seen that before,

00:05:45.642 --> 00:05:48.544
I could put it here, but just
to get a little bit simpler,

00:05:48.544 --> 00:05:50.073
let's just stick it right over here.

00:05:50.073 --> 00:05:51.215
Just gonna be three,

00:05:51.215 --> 00:05:52.824
actually I don't wanna
skip too many steps.

00:05:52.824 --> 00:05:56.545
It's gonna be three times sine-of-nt.

00:05:56.545 --> 00:06:00.712
Three sine-of-nt.

00:06:01.589 --> 00:06:03.205
Now let's see.
How can I reconstruct this

00:06:03.205 --> 00:06:04.765
so it's easy to take the anti-derivative?

00:06:04.765 --> 00:06:05.815
Well this is going to be,

00:06:05.815 --> 00:06:09.884
we take that three out front,
it's gonna be three over pi.

00:06:09.884 --> 00:06:14.160
And then you have your definite
integral from zero to pi

00:06:14.160 --> 00:06:18.243
of sine-of, we'll do that same color,

00:06:18.243 --> 00:06:19.160
sine-of-nt.

00:06:20.635 --> 00:06:24.802
Sine-of-nt dt.

00:06:26.253 --> 00:06:28.098
And we know the derivative of cosine-nt

00:06:28.098 --> 00:06:30.268
is negative n sine-of-nt,

00:06:30.268 --> 00:06:32.736
so let's throw a negative n in here.

00:06:32.736 --> 00:06:34.110
So negative n.

00:06:34.110 --> 00:06:36.561
Let's also divide by negative n.

00:06:36.561 --> 00:06:37.513
Just like that.

00:06:37.513 --> 00:06:39.640
If you have negative n
divided by negative n,

00:06:39.640 --> 00:06:40.473
we haven't changed the value.

00:06:40.473 --> 00:06:41.719
That's just one.

00:06:41.719 --> 00:06:43.560
And so this is going to be equal to

00:06:43.560 --> 00:06:47.060
negative three over n pi times,

00:06:47.922 --> 00:06:49.613
we're going to take the
anti-derivative here,

00:06:49.613 --> 00:06:52.696
so this is gonna be cosine-of-nt

00:06:53.606 --> 00:06:57.049
evaluated from zero

00:06:57.049 --> 00:06:59.436
zero to pi.

00:06:59.436 --> 00:07:00.269
And so, what is this?

00:07:00.269 --> 00:07:02.306
This is going to be equal to,

00:07:02.306 --> 00:07:05.908
this is equal to negative three over n pi.

00:07:05.908 --> 00:07:09.158
Cosine-of-zero n, that's
gonna be one for any n,

00:07:09.158 --> 00:07:12.580
minus cosine-of

00:07:12.580 --> 00:07:13.413
Oh sorry.

00:07:13.413 --> 00:07:14.496
Why did I start at zero?

00:07:14.496 --> 00:07:16.194
It's gonna be cosine-of-n pi,

00:07:16.194 --> 00:07:18.277
cosine-of-n pi

00:07:19.195 --> 00:07:21.890
minus cosine-of-n times zero.

00:07:21.890 --> 00:07:23.382
So minus cosine-of-zero,

00:07:23.382 --> 00:07:28.024
cosine-of-zero is just
gonna be one for any n.

00:07:28.024 --> 00:07:28.919
And so there you have it.

00:07:28.919 --> 00:07:29.798
We have a general

00:07:29.798 --> 00:07:32.381
And you might say, "Well, cosine-of-n pi,

00:07:32.381 --> 00:07:35.000
"is that positive one,
or is that negative one?"

00:07:35.000 --> 00:07:36.248
Well, it depends.

00:07:36.248 --> 00:07:39.928
If n is, if n is even,

00:07:39.928 --> 00:07:42.676
this is going to be positive one.

00:07:42.676 --> 00:07:46.139
If n is odd, this is
going to be negative one.

00:07:46.139 --> 00:07:46.972
So it depends.

00:07:46.972 --> 00:07:48.824
And actually let's just write that out.

00:07:48.824 --> 00:07:50.324
So b-sub-n,

00:07:51.523 --> 00:07:54.481
b-sub-n is equal to,

00:07:54.481 --> 00:07:56.151
let's write those two cases.

00:07:56.151 --> 00:07:59.060
So if n is

00:07:59.060 --> 00:08:02.556
if n is even, and, another one,

00:08:02.556 --> 00:08:05.112
if n is odd.

00:08:05.112 --> 00:08:08.232
So if n is even, you're gonna have

00:08:08.232 --> 00:08:11.571
negative three over n pi times

00:08:11.571 --> 00:08:13.980
Actually let me just do it
over here so we have the space.

00:08:13.980 --> 00:08:17.014
So what would this thing
evaluate to if n is even?

00:08:17.014 --> 00:08:18.879
So this n, if it's even it'd be like

00:08:18.879 --> 00:08:22.355
cosine-of-two pi, cosine-of-four
pi, cosine-of-six pi.

00:08:22.355 --> 00:08:23.703
Well in that situation,

00:08:23.703 --> 00:08:25.745
this is going to evaluate to one.

00:08:25.745 --> 00:08:27.868
You're gonna have one
minus one, which is zero,

00:08:27.868 --> 00:08:31.010
so the whole thing is
gonna evaluate to zero.

00:08:31.010 --> 00:08:32.289
And if n is odd,

00:08:32.289 --> 00:08:37.144
cosine-of-pi, cosine-of-three
pi, cosine-of-five pi,

00:08:37.144 --> 00:08:39.630
well those are gonna
evaluate to negative one.

00:08:39.630 --> 00:08:41.862
Then it's gonna be
negative one if n is odd.

00:08:41.862 --> 00:08:44.274
Negative one minus one is negative two,

00:08:44.274 --> 00:08:47.011
and so this is all gonna be negative two.

00:08:47.011 --> 00:08:50.372
Negative two times
negative three over n pi

00:08:50.372 --> 00:08:52.244
is going to be six.

00:08:52.244 --> 00:08:56.440
Six over n pi.

00:08:56.440 --> 00:08:57.273
So there you have it.

00:08:57.273 --> 00:08:59.617
We have been able to figure out

00:08:59.617 --> 00:09:01.552
our Fourier expansion.

00:09:01.552 --> 00:09:03.721
It is going to be, our square wave,

00:09:03.721 --> 00:09:05.532
and we definitely deserve a drumroll,

00:09:05.532 --> 00:09:07.551
this is many videos in the making,

00:09:07.551 --> 00:09:11.431
f-of-t is going to be equal to a-sub-zero,

00:09:11.431 --> 00:09:15.039
we figured out in this video
is equal to three halves.

00:09:15.039 --> 00:09:17.336
So it's gonna be three halves.

00:09:17.336 --> 00:09:18.800
Let me write this.

00:09:18.800 --> 00:09:20.844
Well I'm just write it all in yellow.

00:09:20.844 --> 00:09:22.177
So three halves.

00:09:22.177 --> 00:09:24.656
Now we don't have any a-sub-ns.

00:09:24.656 --> 00:09:25.489
We figured that out.

00:09:25.489 --> 00:09:27.726
All of the a-sub-ns are going to be zero.

00:09:27.726 --> 00:09:30.084
And then the b, the only b-sub-ns we have

00:09:30.084 --> 00:09:33.311
is when n is odd.

00:09:33.311 --> 00:09:34.372
When n is odd.

00:09:34.372 --> 00:09:37.699
So, we're gonna have b-sub-one.

00:09:37.699 --> 00:09:40.149
So that's gonna be six over one pi.

00:09:40.149 --> 00:09:41.560
Six over one pi.

00:09:41.560 --> 00:09:44.579
Sine-of-one times t.

00:09:44.579 --> 00:09:46.210
So sine-of-t.

00:09:46.210 --> 00:09:49.115
Plus, now we're not
gonna have a b-sub-two,

00:09:49.115 --> 00:09:50.657
we're gonna have a b-sub-three.

00:09:50.657 --> 00:09:54.225
So six over three pi

00:09:54.225 --> 00:09:57.942
times sine-of-three t,

00:09:57.942 --> 00:10:00.635
and of course this is the same thing as,

00:10:00.635 --> 00:10:05.217
this is the same thing as two
over pi times sine of three t,

00:10:05.217 --> 00:10:08.385
plus, now we're not
gonna have a b-sub-four,

00:10:08.385 --> 00:10:09.906
we're gonna have a b-sub-five.

00:10:09.906 --> 00:10:10.739
So that's going to be

00:10:10.739 --> 00:10:14.406
six over five pi.

00:10:15.539 --> 00:10:17.375
Actually I liked writing,

00:10:17.375 --> 00:10:19.747
it's nicer to actually not simplify here

00:10:19.747 --> 00:10:21.176
because you can see the pattern.

00:10:21.176 --> 00:10:25.153
So it's gonna be plus six over five pi

00:10:25.153 --> 00:10:28.999
times sine-of-five t.

00:10:28.999 --> 00:10:33.015
And we're just gonna go on and on and on.

00:10:33.015 --> 00:10:33.848
And there you have it.

00:10:33.848 --> 00:10:35.374
We have our Fourier expansion.

00:10:35.374 --> 00:10:38.870
In the next video, we're
actually going to visualize this.


Fourier series coefficients for cosine terms

WEBVTT
Kind: captions
Language: en

00:00:00.298 --> 00:00:01.849
- [Voiceover] So we've
been spending some time now

00:00:01.849 --> 00:00:03.556
thinking about the idea
of a Fourier series,

00:00:03.556 --> 00:00:06.156
taking a periodic function
and representing it

00:00:06.156 --> 00:00:09.329
as the sum of weighted cosines and sines,

00:00:09.329 --> 00:00:11.107
and some of you might say,
well, how is this constant

00:00:11.107 --> 00:00:12.559
a weighted cosine or sine?

00:00:12.559 --> 00:00:14.997
Well, you can view a
sub zero as a sub zero

00:00:14.997 --> 00:00:16.914
times cosine of zero t,

00:00:18.653 --> 00:00:20.831
and of course cosine of zero t is just 1,

00:00:20.831 --> 00:00:21.664
so this,

00:00:21.664 --> 00:00:23.142
it'll just end up being a sub zero,

00:00:23.142 --> 00:00:26.039
but that could be a weighted
cosine, if you want to view it

00:00:26.039 --> 00:00:27.710
that way, and in the last video,

00:00:27.710 --> 00:00:30.601
we started leveraging
some of the integrals,

00:00:30.601 --> 00:00:32.888
definite integrals of sines and cosines

00:00:32.888 --> 00:00:35.459
to establish a formula for a sub zero

00:00:35.459 --> 00:00:39.059
which intuitively ended up
being the average value of

00:00:39.059 --> 00:00:40.770
our function over,

00:00:40.770 --> 00:00:42.356
over 1 of its period,

00:00:42.356 --> 00:00:44.194
or over zero to 2 pi

00:00:44.194 --> 00:00:47.279
which is the interval
that we are caring about.

00:00:47.279 --> 00:00:50.228
What I want to do now is
find a general expression for

00:00:50.228 --> 00:00:53.554
a sub n, where n is not equal to zero.

00:00:53.554 --> 00:00:56.143
So for n is greater than zero,

00:00:56.143 --> 00:00:57.761
for n is an integer greater than zero

00:00:57.761 --> 00:01:00.311
and I'm going to use a very similar,

00:01:00.311 --> 00:01:02.157
a very similar technique.

00:01:02.157 --> 00:01:03.996
To figure out a sub zero, I just multiply,

00:01:03.996 --> 00:01:06.225
I just took the integral of both sides.

00:01:06.225 --> 00:01:10.156
What I'm going to do now is
I'm going to multiply both

00:01:10.156 --> 00:01:13.406
sides of this equation by cosine of nt.

00:01:14.730 --> 00:01:15.793
So let me do that,

00:01:15.793 --> 00:01:17.854
so on the left-hand side,
I'm going to multiply by

00:01:17.854 --> 00:01:21.012
cosine of nt, and on the right-hand side,

00:01:21.012 --> 00:01:22.502
multiplying it by cosine of nt,

00:01:22.502 --> 00:01:25.029
if I distribute that cosine
of nt, I'm going to multiply

00:01:25.029 --> 00:01:27.128
every term by cosine nt.

00:01:27.128 --> 00:01:30.457
So it's going to be cosine of nt,

00:01:30.457 --> 00:01:32.358
cosine of nt,

00:01:32.358 --> 00:01:33.707
you might see where this is going,

00:01:33.707 --> 00:01:36.382
especially 'cause we took
all that trouble to take,

00:01:36.382 --> 00:01:38.926
to figure out the properties of,

00:01:38.926 --> 00:01:40.265
of definite integrals.

00:01:40.265 --> 00:01:41.683
Cosine of nt,

00:01:41.683 --> 00:01:42.766
cosine of nt,

00:01:44.388 --> 00:01:46.127
cosine of nt,

00:01:46.127 --> 00:01:47.355
or the properties of definite integrals

00:01:47.355 --> 00:01:49.665
of products of cosines and sines.

00:01:49.665 --> 00:01:51.546
Cosine of nt.

00:01:51.546 --> 00:01:53.328
And this is going to be
true for all of the terms.

00:01:53.328 --> 00:01:54.704
There's an infinite number of terms here.

00:01:54.704 --> 00:01:56.318
That's what that dot,
dot, dot is representing.

00:01:56.318 --> 00:01:58.936
We're just going to keep
on going on and on and on.

00:01:58.936 --> 00:02:01.320
And now, let's take the
definite integral of both sides

00:02:01.320 --> 00:02:03.005
from zero to 2 pi.

00:02:03.005 --> 00:02:06.126
So the definite integral
from zero to 2 pi,

00:02:06.126 --> 00:02:09.704
dt, well that's going to
be the integral of this,

00:02:09.704 --> 00:02:11.287
from zero to 2 pi, and
actually, let me just take

00:02:11.287 --> 00:02:13.103
that coefficient out of the integral,

00:02:13.103 --> 00:02:14.770
so zero to 2 pi, dt,

00:02:16.999 --> 00:02:18.666
so zero to 2 pi, dt,

00:02:20.204 --> 00:02:22.491
a little bit monotonous but it's worth it,

00:02:22.491 --> 00:02:23.908
zero to 2 pi, dt,

00:02:25.616 --> 00:02:27.033
zero to 2 pi, dt,

00:02:28.807 --> 00:02:30.849
we might as well have fun while we do it,

00:02:30.849 --> 00:02:32.266
zero to 2 pi, dt,

00:02:34.869 --> 00:02:35.797
all right.

00:02:35.797 --> 00:02:36.819
Zero, my hand is hurting,

00:02:36.819 --> 00:02:38.236
zero to 2 pi, dt,

00:02:39.307 --> 00:02:40.724
zero to 2 pi, dt,

00:02:42.669 --> 00:02:44.405
once again, just integration property.

00:02:44.405 --> 00:02:45.898
I'm taking the integral from zero,

00:02:45.898 --> 00:02:47.390
the definite integral from zero to 2 pi

00:02:47.390 --> 00:02:50.065
of both sides, and I'm just
saying, hey, the integral

00:02:50.065 --> 00:02:52.574
of this infinite sum is
equal to the infinite sum

00:02:52.574 --> 00:02:54.402
of the different integrals.

00:02:54.402 --> 00:02:56.846
Now, what is this going to be equal to?

00:02:56.846 --> 00:02:57.970
Well, we know,

00:02:57.970 --> 00:02:59.853
we know from before that if I just take

00:02:59.853 --> 00:03:02.971
the definite integral from
zero to 2 pi of cosine nt,

00:03:02.971 --> 00:03:05.641
where n is some non-zero integer,

00:03:05.641 --> 00:03:06.831
this is just going to be zero,

00:03:06.831 --> 00:03:08.803
so this whole thing is going to be zero.

00:03:08.803 --> 00:03:11.624
We also know, and we've
established it before,

00:03:11.624 --> 00:03:13.170
if we take,

00:03:13.170 --> 00:03:14.985
so I just used this property,

00:03:14.985 --> 00:03:18.331
we also know if we take
cosine nt times sine of mt

00:03:18.331 --> 00:03:21.599
for any integers mt over the
interval from zero to 2 pi,

00:03:21.599 --> 00:03:23.494
that's going to be zero,

00:03:23.494 --> 00:03:25.262
and we also know,

00:03:25.262 --> 00:03:28.718
we also know that if we take
the integral of cosine mt,

00:03:28.718 --> 00:03:30.662
cosine nt, where m does not equal n,

00:03:30.662 --> 00:03:32.745
where m does not equal n,

00:03:34.018 --> 00:03:36.518
then that is going to be zero.

00:03:38.035 --> 00:03:39.368
And so, we know,

00:03:40.431 --> 00:03:41.951
well, here, we have,

00:03:41.951 --> 00:03:43.345
we're assuming that,

00:03:43.345 --> 00:03:46.427
we're assuming here that
n is not equal to 1,

00:03:46.427 --> 00:03:49.528
the coefficient here, so that
is going to be equal to zero.

00:03:49.528 --> 00:03:52.344
This is sines and cosines with
the different coefficients,

00:03:52.344 --> 00:03:54.445
and frankly, if they had
the same coefficient,

00:03:54.445 --> 00:03:55.333
that's going to be zero.

00:03:55.333 --> 00:03:57.474
We established that in
the last few videos.

00:03:57.474 --> 00:03:58.606
This is going to be zero,

00:03:58.606 --> 00:03:59.439
same argument.

00:03:59.439 --> 00:04:00.272
This is going to be zero,

00:04:00.272 --> 00:04:01.105
this is going to be zero.

00:04:01.105 --> 00:04:03.076
Everything's going to be
zero except for this thing

00:04:03.076 --> 00:04:04.489
right over here,

00:04:04.489 --> 00:04:05.859
but what is this thing?

00:04:05.859 --> 00:04:07.162
This is the definite,

00:04:07.162 --> 00:04:08.297
what we're taking the integral of,

00:04:08.297 --> 00:04:12.214
this is the same thing
as cosine-squared of nt,

00:04:13.422 --> 00:04:15.559
dt, the definite integral
from zero to 2 pi

00:04:15.559 --> 00:04:17.863
of cosine-squared nt, dt.

00:04:17.863 --> 00:04:19.535
Well, what is that?

00:04:19.535 --> 00:04:22.273
Well, we established that for any m,

00:04:22.273 --> 00:04:25.920
non-zero m, this is
going to be equal to pi.

00:04:25.920 --> 00:04:27.825
So all our work is paying off.

00:04:27.825 --> 00:04:30.327
So, this whole integral,

00:04:30.327 --> 00:04:32.585
this whole integral right
over here is going to evaluate

00:04:32.585 --> 00:04:33.418
to pi.

00:04:33.418 --> 00:04:34.603
We did that in another video

00:04:34.603 --> 00:04:37.539
and so now we can start
solving for a sub n.

00:04:37.539 --> 00:04:40.822
We know that a sub n times pi,

00:04:40.822 --> 00:04:42.239
a sub n times pi,

00:04:44.061 --> 00:04:45.811
times pi is equal to,

00:04:47.206 --> 00:04:49.946
is equal to this 'cause everything
else ended up being zero

00:04:49.946 --> 00:04:52.060
which is very nice.

00:04:52.060 --> 00:04:56.087
It's equal to the definite
integral from zero to 2 pi

00:04:56.087 --> 00:04:56.920
of f of t,

00:04:58.074 --> 00:05:00.241
of f of t times cosine nt,

00:05:02.388 --> 00:05:03.805
cosine of nt, dt.

00:05:06.857 --> 00:05:10.851
Cosine of nt, dt, and so we
can now solve for a sub n.

00:05:10.851 --> 00:05:13.382
We just divide both sides by,

00:05:13.382 --> 00:05:15.382
we just divide both sides by pi,

00:05:15.382 --> 00:05:17.753
and we get a sub n is equal to,

00:05:17.753 --> 00:05:19.611
I think we deserve a
little bit of a drum roll,

00:05:19.611 --> 00:05:22.920
a sub n is equal to 1 over pi,

00:05:22.920 --> 00:05:24.602
times all of this business,

00:05:24.602 --> 00:05:28.435
times the definite
integral from zero to 2 pi,

00:05:29.797 --> 00:05:30.630
f of t,

00:05:31.558 --> 00:05:32.391
f of t,

00:05:33.636 --> 00:05:34.469
cosine nt,

00:05:36.119 --> 00:05:36.952
cosine nt,

00:05:40.181 --> 00:05:41.014
dt.

00:05:41.914 --> 00:05:43.577
Dt.

00:05:43.577 --> 00:05:46.709
So if you want to find the
nth coefficient for one of our

00:05:46.709 --> 00:05:48.491
cosines, a sub n,

00:05:48.491 --> 00:05:50.377
well, you take your
function, multiply it times

00:05:50.377 --> 00:05:54.122
cosine of nt, and then
take that definite integral

00:05:54.122 --> 00:05:57.511
from zero to 2 pi, and then divide by pi,

00:05:57.511 --> 00:05:59.364
so pretty intuitive, and what's cool is

00:05:59.364 --> 00:06:01.151
that the math kind of works out this way,

00:06:01.151 --> 00:06:03.648
that you can actually do this,

00:06:03.648 --> 00:06:06.871
so hopefully you enjoyed
that as much as I did.


Integral of product of cosines

WEBVTT
Kind: captions
Language: en

00:00:00.393 --> 00:00:01.936
- [Voiceover] We've been
doing several videos now

00:00:01.936 --> 00:00:05.907
to establish a bunch of
truths of definite integrals

00:00:05.907 --> 00:00:09.588
of various combinations
of trigonometric functions

00:00:09.588 --> 00:00:12.585
so that we will have a really
strong mathematical basis

00:00:12.585 --> 00:00:15.353
for actually finding
the fourier coefficients

00:00:15.353 --> 00:00:17.469
and I think we only have
one more video to go.

00:00:17.469 --> 00:00:19.878
In the last video, we said hey,
if you take any combinations

00:00:19.878 --> 00:00:24.061
of signs where M and N
are integers that either

00:00:24.061 --> 00:00:25.527
don't equal each other or don't equal

00:00:25.527 --> 00:00:27.203
the negative each other,
then you're gonna get that,

00:00:27.203 --> 00:00:28.932
integrals gonna be equal to zero

00:00:28.932 --> 00:00:30.425
and then if they did equal to each other,

00:00:30.425 --> 00:00:32.931
well it's just gonna be the
same thing as sine squared

00:00:32.931 --> 00:00:36.082
of some multiple of T
and then that actually

00:00:36.082 --> 00:00:38.177
over the interval from zero to two pi

00:00:38.177 --> 00:00:40.137
is going to be equal to pi.

00:00:40.137 --> 00:00:43.196
Just to be clear, I wasn't
as clear as I should've been

00:00:43.196 --> 00:00:45.077
in the last video, this
is going to be true

00:00:45.077 --> 00:00:48.232
where M is a non zero integer.

00:00:48.232 --> 00:00:50.953
If M was zero, then the
inside of this integral

00:00:50.953 --> 00:00:52.942
would just simplify the zero

00:00:52.942 --> 00:00:54.477
and then the integral would be zero.

00:00:54.477 --> 00:00:56.060
So M has to be

00:00:56.060 --> 00:00:57.560
a non-zero integer

00:00:59.251 --> 00:01:02.270
for this right over here to be true.

00:01:02.270 --> 00:01:03.492
Now what we want to do in this video

00:01:03.492 --> 00:01:05.667
is do the same thing we
did in the last video

00:01:05.667 --> 00:01:09.657
but now do it for cosines but
the product of two cosines

00:01:09.657 --> 00:01:12.742
where M and N are different integers

00:01:12.742 --> 00:01:14.759
or they're not the negative of each other,

00:01:14.759 --> 00:01:18.392
that's going to be zero, but
they are the same integer

00:01:18.392 --> 00:01:21.336
and they are not zero
so that will boil down

00:01:21.336 --> 00:01:24.795
to cosine square of MT then
that is going to be equaled,

00:01:24.795 --> 00:01:28.005
this definite integral, is
going to be equal to pi.

00:01:28.005 --> 00:01:29.020
We're gonna do it the same way that

00:01:29.020 --> 00:01:31.519
we did it with the signs
we are going to use

00:01:31.519 --> 00:01:33.150
some of our trigonometric,

00:01:33.150 --> 00:01:35.565
some of our trigonometric identities

00:01:35.565 --> 00:01:39.217
and so let's rewrite, let's
rewrite this right over here.

00:01:39.217 --> 00:01:42.158
What we're trying to take the integral of

00:01:42.158 --> 00:01:46.325
and so this is going to be the
integral from zero to two pi

00:01:47.633 --> 00:01:49.966
so cosine MT times cosine NT

00:01:51.355 --> 00:01:54.068
using a product to sum trig identity.

00:01:54.068 --> 00:01:55.160
Now if this is unfamiliar,

00:01:55.160 --> 00:01:57.064
you can review it on Khan Academy,

00:01:57.064 --> 00:02:00.192
that is going to be one half

00:02:00.192 --> 00:02:03.942
times cosine of the
difference of MT minus NT

00:02:05.738 --> 00:02:08.350
so I can write that as M minus MT,

00:02:08.350 --> 00:02:09.183
M minus NT

00:02:11.724 --> 00:02:12.641
plus cosine

00:02:14.780 --> 00:02:18.690
plus cosine of MT plus
NT, which I can write as

00:02:18.690 --> 00:02:19.523
M plus NT

00:02:22.481 --> 00:02:23.314
DT

00:02:24.249 --> 00:02:25.479
D

00:02:25.479 --> 00:02:26.542
DT

00:02:26.542 --> 00:02:28.730
So let's think about two situations.

00:02:28.730 --> 00:02:30.713
Let's think about the first situation.

00:02:30.713 --> 00:02:34.098
When, let me redo that DT in blue.

00:02:34.098 --> 00:02:34.931
So DT.

00:02:35.909 --> 00:02:40.060
So when, actually let me now
use some integration properties

00:02:40.060 --> 00:02:41.569
to expand this out a little bit.

00:02:41.569 --> 00:02:43.860
This is going to be equal to,

00:02:43.860 --> 00:02:46.053
so I'm a write this as
two different integrals.

00:02:46.053 --> 00:02:48.964
So one integral from zero to two pi

00:02:48.964 --> 00:02:51.438
and we'll put the DT right over here

00:02:51.438 --> 00:02:55.220
and then have another
integral from zero to two pi

00:02:55.220 --> 00:02:58.400
then I'm a throw that DT out here

00:02:58.400 --> 00:03:00.617
and so, just using some
integration properties,

00:03:00.617 --> 00:03:03.146
we're gonna do one half
times this integral

00:03:03.146 --> 00:03:04.146
of cosine of

00:03:06.241 --> 00:03:07.324
M minus NT DT

00:03:09.467 --> 00:03:11.550
and then plus, I'm just
distributing the one half

00:03:11.550 --> 00:03:14.041
and using some integration
properties, one half,

00:03:14.041 --> 00:03:15.576
and now this integral
is going to be cosine

00:03:15.576 --> 00:03:16.576
of M plus N.

00:03:18.145 --> 00:03:18.978
M plus NT

00:03:20.817 --> 00:03:22.163
DT

00:03:22.163 --> 00:03:23.558
Now let's think about it.

00:03:23.558 --> 00:03:28.132
When M and N are integers
that don't equal each other,

00:03:28.132 --> 00:03:29.990
don't equal their negative,

00:03:29.990 --> 00:03:33.401
so let's think about M not equaling N

00:03:33.401 --> 00:03:38.121
or M not equaling negative
N and we're always assuming

00:03:38.121 --> 00:03:41.247
that these things are
going to be integers.

00:03:41.247 --> 00:03:44.009
M and N, well in that situation

00:03:44.009 --> 00:03:48.648
this right over here is going
to be a non zero integer

00:03:48.648 --> 00:03:52.369
and this right over here
is going a non zero integer

00:03:52.369 --> 00:03:55.775
and we've already established,
we've already established

00:03:55.775 --> 00:03:57.456
that if you have a non
zero coefficient here

00:03:57.456 --> 00:03:59.342
that this definite integral
is going to be equal to zero.

00:03:59.342 --> 00:04:01.586
The definite integral from zero to two pi

00:04:01.586 --> 00:04:05.415
of cosine of some non
zero integer times T DT.

00:04:05.415 --> 00:04:09.282
Well that's exactly what
both of these integrals are.

00:04:09.282 --> 00:04:11.504
This is the integral from
zero to two pie of cosine

00:04:11.504 --> 00:04:13.946
times some non zero integer T

00:04:13.946 --> 00:04:16.903
or non zero integer times T DT.

00:04:16.903 --> 00:04:19.578
So in this case where M and N are integers

00:04:19.578 --> 00:04:20.589
that don't equal each other,

00:04:20.589 --> 00:04:22.289
don't equal the negatives of each other,

00:04:22.289 --> 00:04:24.041
both of these integrals
are going to be zero.

00:04:24.041 --> 00:04:25.675
Then you're going to
multiply that times one half,

00:04:25.675 --> 00:04:27.620
one half times zero zero,
one half times zero zero,

00:04:27.620 --> 00:04:30.312
it's all gonna end up being zero.

00:04:30.312 --> 00:04:32.312
So that should hopefully

00:04:33.156 --> 00:04:37.970
make you feel pretty good
about the case, this first case

00:04:37.970 --> 00:04:40.114
and now let's think about the second case

00:04:40.114 --> 00:04:42.564
where M is a non zero
integer, or we can say,

00:04:42.564 --> 00:04:46.508
were M is equal to N, so in that situation

00:04:46.508 --> 00:04:50.384
N and M are the same and
they are not equal to zero.

00:04:50.384 --> 00:04:52.279
So let's just take that situation,

00:04:52.279 --> 00:04:53.283
especially because when we're looking

00:04:53.283 --> 00:04:55.602
at fourier coefficients, we care about

00:04:55.602 --> 00:04:59.312
the non-negative
coefficients, at least the way

00:04:59.312 --> 00:05:03.434
we have defined so let's just
assume that M is equal to N

00:05:03.434 --> 00:05:06.017
and that M is not equal to zero

00:05:07.787 --> 00:05:11.176
and that would just resolve,
that would take this integral

00:05:11.176 --> 00:05:13.017
and turn it into that integral.

00:05:13.017 --> 00:05:16.594
Well in that situation,
what's gonna happen?

00:05:16.594 --> 00:05:19.459
Well, this first integral right over here,

00:05:19.459 --> 00:05:23.190
if M is equal to N and
M is not equal to zero,

00:05:23.190 --> 00:05:25.997
well it's going to be M minus N,

00:05:25.997 --> 00:05:28.748
this is gonna be zero
T so this whole thing

00:05:28.748 --> 00:05:31.970
is going to simplify to one
and then this right over here

00:05:31.970 --> 00:05:36.707
gonna have M plus N, that's
going to simplify to two M.

00:05:36.707 --> 00:05:39.134
So let's rewrite the integrals here.

00:05:39.134 --> 00:05:41.378
This is going to be equal to one half

00:05:41.378 --> 00:05:45.378
times the definite integral
from zero to two pi.

00:05:47.698 --> 00:05:51.884
Zero to two pi of one times,
I'll write that one here,

00:05:51.884 --> 00:05:52.717
one DT

00:05:55.527 --> 00:05:57.944
plus one half, plus one half,

00:05:59.918 --> 00:06:02.736
that's a new color, times the integral

00:06:02.736 --> 00:06:05.153
from zero to two pi of cosine

00:06:07.676 --> 00:06:09.676
let me do that the same,

00:06:10.651 --> 00:06:11.651
of cosine of

00:06:13.687 --> 00:06:14.520
two MT DT.

00:06:16.989 --> 00:06:17.822
Two MT D

00:06:19.365 --> 00:06:20.198
DT

00:06:21.727 --> 00:06:23.131
DT

00:06:23.131 --> 00:06:25.835
So once again, we're assuming
M is not equal to zero,

00:06:25.835 --> 00:06:27.290
this is the definitive integral

00:06:27.290 --> 00:06:29.417
from zero to two pi of cosine times

00:06:29.417 --> 00:06:32.250
some non zero coefficient times T.

00:06:33.122 --> 00:06:35.579
Well once again, we have
established multiple times

00:06:35.579 --> 00:06:38.183
that this is going to be zero
so this whole second term

00:06:38.183 --> 00:06:41.079
is going to be zero and this first one

00:06:41.079 --> 00:06:43.409
is going to be equal to,
let's go to neutral color,

00:06:43.409 --> 00:06:46.247
it's going to be one half times
the anti derivative of one,

00:06:46.247 --> 00:06:50.293
now that just T evaluated
from zero to two pi

00:06:50.293 --> 00:06:52.654
so that's going to be equal to one half

00:06:52.654 --> 00:06:55.386
times two pi minus zero

00:06:55.386 --> 00:06:58.960
two pi minus zero, well
that's just one half two pi

00:06:58.960 --> 00:07:00.595
which is equal to pi.

00:07:00.595 --> 00:07:03.680
So we have now established
this one as well

00:07:03.680 --> 00:07:06.239
and now we have a full toolkit.

00:07:06.239 --> 00:07:07.909
We now have a full toolkit

00:07:07.909 --> 00:07:10.314
for evaluating the fourier coefficients

00:07:10.314 --> 00:07:12.303
which we will now do in the next video

00:07:12.303 --> 00:07:13.837
which is very exciting.


Definite integrals of sin(mx) and cos(mx)

WEBVTT
Kind: captions
Language: en

00:00:00.347 --> 00:00:02.556
- [Voiceover] In the last
video we introduced the idea

00:00:02.556 --> 00:00:05.740
that we could represent any
arbitrary, periodic function

00:00:05.740 --> 00:00:09.240
by a series of weighted cosines and sines.

00:00:10.689 --> 00:00:12.861
And what I'm gonna start
doing in this video,

00:00:12.861 --> 00:00:16.143
is starting to establish
our mathematical foundation.

00:00:16.143 --> 00:00:19.376
So it'll be pretty
straightforward for us to find

00:00:19.376 --> 00:00:23.444
these coefficients that
give us that function.

00:00:23.444 --> 00:00:26.233
So the first thing I
wanna do, the first thing

00:00:26.233 --> 00:00:30.199
I'm gonna do is establish
some truths using,

00:00:30.199 --> 00:00:32.584
or some truths with
the definite integrals.

00:00:32.584 --> 00:00:35.069
I'm gonna focus over
the intervals zero to pi

00:00:35.069 --> 00:00:37.312
over this video and the next few videos

00:00:37.312 --> 00:00:39.332
because the function we're approximating

00:00:39.332 --> 00:00:41.472
has a period of two
pi, completes one cycle

00:00:41.472 --> 00:00:43.709
from zero to two pi, we
could've done it over

00:00:43.709 --> 00:00:46.712
other intervals of length two pi,

00:00:46.712 --> 00:00:49.168
and if this period was other than two pi,

00:00:49.168 --> 00:00:51.256
we would've done it over
intervals of that period,

00:00:51.256 --> 00:00:53.344
but I'm focusing on two pi
because it makes the math

00:00:53.344 --> 00:00:55.009
a little bit cleaner,
a little bit simpler,

00:00:55.009 --> 00:00:57.545
and then we can generalize in the future.

00:00:57.545 --> 00:01:00.647
So let's just establish some things about

00:01:00.647 --> 00:01:03.781
definite integrals of trig functions.

00:01:03.781 --> 00:01:06.094
So the first I wanna establish,

00:01:06.094 --> 00:01:08.430
I wanna establish that
the definite integral

00:01:08.430 --> 00:01:11.597
from zero to two pi of sine of mx, dx,

00:01:15.532 --> 00:01:17.787
actually let me stay in t, since our

00:01:17.787 --> 00:01:20.302
original function is in terms of t.

00:01:20.302 --> 00:01:24.469
Sine of mt, dt, I wanna establish
that that is equal to zero

00:01:25.681 --> 00:01:27.931
for any non-zero integer m.

00:01:29.140 --> 00:01:32.640
For non-zero, non-zero integer, integer m,

00:01:39.176 --> 00:01:41.467
and I also wanna establish
that the integral

00:01:41.467 --> 00:01:45.634
from zero to two pi of cosine
of mt, dt, is equal to zero

00:01:49.583 --> 00:01:51.833
for any non-zero integer m.

00:01:52.902 --> 00:01:55.453
And you might already
take this for granted

00:01:55.453 --> 00:01:57.195
or you feel good about it,
or you've already proven it

00:01:57.195 --> 00:02:00.391
to yourself, and if so, you
could actually skip this video.

00:02:00.391 --> 00:02:02.261
But let's work through
it, 'cause it's actually

00:02:02.261 --> 00:02:05.689
a good review of some
integral calculus here.

00:02:05.689 --> 00:02:07.809
So let's first do this top one,

00:02:07.809 --> 00:02:09.382
so let me just re-write the integral.

00:02:09.382 --> 00:02:12.392
So we're gonna take the
integral from zero to two pi

00:02:12.392 --> 00:02:15.809
of sine mt, dt, now we know we won't take

00:02:19.665 --> 00:02:21.721
the antiderivative of sine of mt,

00:02:21.721 --> 00:02:24.933
so we know that the derivative

00:02:24.933 --> 00:02:28.600
with respect to t of
cosine mt, is equal to,

00:02:32.616 --> 00:02:35.710
what is this, this is
going to be equal to m.

00:02:35.710 --> 00:02:39.170
The derivative of mt with respect to t

00:02:39.170 --> 00:02:42.990
times the derivative of
cosine mt with respect to mt.

00:02:42.990 --> 00:02:46.907
So times negative sine
of mt, or we could write

00:02:49.208 --> 00:02:51.371
this is gonna be equal to negative m

00:02:51.371 --> 00:02:55.538
sine of mt, and I could put a
parentheses there if I like.

00:02:57.062 --> 00:02:59.696
And so I almost have
negative m sine of mt,

00:02:59.696 --> 00:03:01.343
I just wanna have a negative m here.

00:03:01.343 --> 00:03:03.213
So what if I put a negative m there,

00:03:03.213 --> 00:03:05.004
but I can't just do that,
that would change the value

00:03:05.004 --> 00:03:08.531
of the expression, but
I could also multiply

00:03:08.531 --> 00:03:10.655
by negative one over m.

00:03:10.655 --> 00:03:12.824
Now these two would,
if we take the product

00:03:12.824 --> 00:03:13.852
they're gonna cancel out,

00:03:13.852 --> 00:03:15.587
we're gonna get our original expression.

00:03:15.587 --> 00:03:17.288
But this is useful because now we can say

00:03:17.288 --> 00:03:20.141
this is equal to negative one over m

00:03:20.141 --> 00:03:23.189
and now the antiderivative
of this business

00:03:23.189 --> 00:03:25.941
right over here, we know is cosine mt.

00:03:25.941 --> 00:03:29.108
So it's gonna be cosine mt, cosine mt,

00:03:30.348 --> 00:03:34.098
evaluated at two pi and
zero, two pi and zero

00:03:36.348 --> 00:03:39.207
and so this is going to be equal to,

00:03:39.207 --> 00:03:42.207
this is equal to negative one over m

00:03:44.077 --> 00:03:47.994
times cosine of m times
two pi, right this way,

00:03:50.033 --> 00:03:52.700
cosine, cosine of m times two pi

00:03:56.800 --> 00:04:00.883
minus cosine of, well it's
gonna be m times zero,

00:04:02.118 --> 00:04:05.455
which is just gonna be, we
can just write that as zero.

00:04:05.455 --> 00:04:09.998
And so let's see, cosine
of any multiple of two pi,

00:04:09.998 --> 00:04:12.078
well that's just gonna be one,

00:04:12.078 --> 00:04:14.398
and cosine of zero, well that's also one.

00:04:14.398 --> 00:04:16.174
So you have one minus one is zero,

00:04:16.174 --> 00:04:19.528
times negative one over
m, well this is all

00:04:19.528 --> 00:04:23.533
going to evaluate to the result we wanted,

00:04:23.533 --> 00:04:25.996
this is all going to evaluate to zero.

00:04:25.996 --> 00:04:29.440
So we have just proven
that first statement,

00:04:29.440 --> 00:04:30.741
so now let's prove the second one.

00:04:30.741 --> 00:04:34.674
It's gonna be a very similar
argument, so let's re-write it.

00:04:34.674 --> 00:04:38.200
We're gonna get the
integral from zero to two pi

00:04:38.200 --> 00:04:41.117
of cosine of mt, dt, and now let me

00:04:45.484 --> 00:04:47.396
engineer this a little
bit, we know that the

00:04:47.396 --> 00:04:51.500
derivative of sine of mt is m cosine mt,

00:04:51.500 --> 00:04:54.372
so let me multiply and divide by m,

00:04:54.372 --> 00:04:56.783
and we multiply by an
m and divide by an m,

00:04:56.783 --> 00:04:58.565
not changing the actual value,

00:04:58.565 --> 00:05:01.595
and so this is going to
be equal to one over m,

00:05:01.595 --> 00:05:04.178
and then the antiderivative of,

00:05:05.147 --> 00:05:06.111
let me find a nice color here,

00:05:06.111 --> 00:05:09.115
the antiderivative of
that right over there,

00:05:09.115 --> 00:05:11.652
that should say, the
antiderivative of this

00:05:11.652 --> 00:05:15.402
right over here is sine
of mt, so sine of mt.

00:05:19.374 --> 00:05:22.865
Notice the derivative of
sine of mt is m cosine mt,

00:05:22.865 --> 00:05:25.907
and we're going to evaluate that at

00:05:25.907 --> 00:05:29.334
from zero to two pi, so this
is going to be equal to,

00:05:29.334 --> 00:05:31.747
we still have our one over m out front,

00:05:31.747 --> 00:05:34.914
one over m, and so this is going to be

00:05:35.877 --> 00:05:39.960
sine of m times two pi,
or we could say two pi m,

00:05:41.931 --> 00:05:45.815
two pi m, minus sine of m times zero,

00:05:45.815 --> 00:05:49.692
so sine, we can just write
that as sine of zero,

00:05:49.692 --> 00:05:51.760
and what's the sine of
any multiple of two pi.

00:05:51.760 --> 00:05:53.663
Remember m is a non-zero integer,

00:05:53.663 --> 00:05:56.187
so any multiple that's gonna
be a multiple of two pi here,

00:05:56.187 --> 00:05:57.769
well that's just going to be zero,

00:05:57.769 --> 00:05:59.682
and sine of zero is just going to be zero,

00:05:59.682 --> 00:06:01.411
so this whole thing, this whole thing

00:06:01.411 --> 00:06:03.993
is just going to be zero,
and so we have established

00:06:03.993 --> 00:06:05.590
our second statement there.

00:06:05.590 --> 00:06:08.879
So this is going to be a
nice base to build from,

00:06:08.879 --> 00:06:13.208
and now we're going to do
slightly more complex integrals

00:06:13.208 --> 00:06:14.732
in the next few videos,
so there's gonna be

00:06:14.732 --> 00:06:19.235
hopefully pretty straightforward
to find our Fourier coefficients

00:06:19.235 --> 00:06:23.402
with a little bit of calculus
and algebraic manipulation.