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mainstay room is a beautiful two-dimensional theory in
that is often presented before people talk about uh... stokes theorem
integrations theorem
however it
turns out that he can also be present it as a quick
consequence of stamps the year and and even though it is not terribly
convention calkins courses to do that there are good reasons for delaying
greens theorem and so
b four
actually talk about dreams the right i'd like to explain
if u
off zzz reasons so
uh... there are certain reasons too
de-emphasize greens theory and this is a term that used to be quite low within
the
educational community
when i say de-emphasize i don't mean we're not going to talk about it i'm
just saying
european cities the urine
that is more powerful imp more important and that is
stokes theorem
pain cdrom is any theater talks about
two-dimensional vector fields into kirstin
two dimensional space which means
four
real applications
nobody can argue with that whether we like it like it has one
dimension
classical mechanics classical electrodynamics any kind of classical
description off
physical reality sixteen
three-dimensional space
and so that means greens theorem
uh... would always grow to be applied if we have some kind of great symmetry and
by now signs that it has advanced to the point where the three dimensional
modeling is
what people need to do
most practitioners outside of mathematics actually think of greens
identities not green cdrom
with the windy here greens whatever
in contact actually i have asked
several people about that and and they all immediately q on resented fees which
are very very important
uh... but dreams that your own when i was described greens the into a
colleague in engineering
he just said oh you mean to little stops
uh... india that's why i think he may have said it as a question but it could
also be misstatement actually
finally green's theorem actually can using the recovered from
stokes theorem by attaching artificial
constancy coordinate to everything and that's in fact what we're going to do
when we prove greens theory so these are reasons where i was a welcoming
is it is beautiful and really was a great mathematician but
there are other things that we could probably spend our time with that that
may be
worth more of our time
on the other hand
i'm not saying let's toss it out there are reasons to retain dreams theorem one
of them is that brings the year
is a good intermediate stage
when we prove stokes theorem and of course we've seen that
proof that we have to stokes theorem was
i'm not for the faint of heart this just say is that of course there is a
colorful interactive mexican brad protested grant me as a heretic at least
a certain amount of attention
which is that although we should respect proves and working understanding that
calculus simply is not the proof glass and then of course be sites
we could prove soaks theorem directly it was blood sweat and tears but we got a
dot
uh... history is another reason and that is that
even though i'm not a historian
uh... supposedly what we call stokes theorem was community beauty
communicate the stokes line or two thousand actually
stokes popularized the result and by one account i read
is uh... is actually green goo
wrote down
the first
correct proved false results so in some ways stokes was completely only quote
unquote
a program you were certainly growing that's alright
but in this particular sitting he was in some ways a broker between dollar killed
in situation and apparently than the
rock-solid
mathematical proof that
presumably came from green but that means off the three
gentleman
stokes at that he's playing to the theater and i think that in his name was
attached to it
because people aren't
off the theory into those folks
that is how history sometimes works
dreams theorem also needs to a way to make sure you read about interpreting
along the perimeter and that's actually pretty cool in that something
that we're going to look at in detail because that is something where
we potentially would even have an actual application if you have in
a strangely shaped area
you want to find out how much
the area is you can just run something along the perimeter
increased
line integral figure out what the area a sweet which sounds downright strange but
we will see a determination that it does work
at the end finally then the proof that the present here actually is also good
practice for stokes theorem
for this cooling from three dimensions to two dimensions that set
philosophical thing that i have i'd like to go from that
more general more powerful results to the sender results that as a whole lot
easier
and doing things up
from the center of stuff to that
more complicated stuff because jumping from two dimensions to three dimensions
that is something where quite a few people will take me to tell you that is
fraught with daryl things get more complicated than freedom hill space in
some ways just also behaves differently
opt for example the notion of single connectivity in three-dimensional space
that we talked about in the previous presentation
is more complicated in three-dimensional space
than in two-dimensional states
okay in terms of brings fear of what does it say yet say is that if you have
occurred in the plane that is positively oriented
piecewise smooth simple and close the window and get
technical hypotheses appeared shouldn't
self intersected ought to be closed
piecewise smoothness to many technical partner positively or you could just
read the going around the country club questions about what you patience
if he has a region in the plane that has found it by that curve
in ph u_s_ kick continues partner in this political region that contains the
same
line integrals the vector field p q and along the coast for
c
is equal choosy
integral over the region do you think
d q d_x_ minus the pty
estate
uh... as in the area and to grow and you may say well that
looks like stokes yeah sure because the two are
very closely
related to each other as this one is the simpler two-dimensional chasm
stokes
fear of so this is an also the precursor to the croats just that i'd rather have
the curl burned into my memory because that's the thing
that we will need to use all the way through physics
this one is just something that's
we can derive from that
in terms of the proof well if we prove it from stokes theorem we first note
that the standard parametrization
off a true dimensional region but i put it into three-dimensional space as
d cross zero we're just put in artificial
zero cd component on the air
that surface of nixon twice just x_y_ zero in a few
with x_y_ dvds so i'm not going to
change anything there in terms of trying to figure out of the paramus rice d
then i get that the ex parte across the white partial of this parametrization
well the explosion is the first unit victory in the white partial
is the second unit vector right d x d x is one and then you get the roast
and d y d y is one of the other components give u zero
in the cross product of the first unit lecture with the second unit director
you can just
look at that with three fingers or computer directly well you get first
component zero
second component zero third component one so this is
zero zero one
which is to get a factor in that
z direction well that means that now
integral over the curve of the vector few pty dot dec
is going to go from a to b off peel except e
five t q accepting y f t
ex-prime
white crime detainee
which of course it is the same as when i get when i take the integral of p of
except t
wyeth teach you a fix of t y f t zero
ex-prime off to you by prime of t
zero
and then
go a head
and work that out
into that is now the same as per the
integral in three-dimensional space off the vector fifty q zero turkey see
that's not the explicit
parametrization right
this thing
i can apply stokes theorem two
so this is now
societies the inca grosvenor dribble a
surface
off the corral
office vector field g_g_ editors call their thing g dot d_s_
isn't that gives us that we have to the curled up this vector field well
let's see
the farrakhan
whether the first and second component actually don't matter because they are
multiplied
in adult product with the
with the unit i was within normal vector off the surface of s just zero zero one
d_a_ where the area and it is for your domain
isn't
uh... that means we really only have to worry about third component of their
component is did you do you explain is d pty
and saw that and he was he had to go over the region d cutie xm honesty pty
dat
as it
should be
greens sierra
finishes
the proof of that result
is now we can look at an example well first let's be careful with notation
green steven is also often stated as
clueless line in a pool of p_d_a_ extra security wise
equal to
era into going over the intel's region deal d q d_x_ muncie pty times theory
element
d_a_ this location i didn't portion very much in the line in the polls because
actually z other way to santa monica integrals it refuses
physically a whole lot more intuitive as far as i can see and so
that is just something we need to be able to
rehabilitation it's going to come back to us on the computer
area with and i know
so as an example
computing author feature film ethel xli being excluded xli over the positively
or you did
boundary of the triangle with vertices at zero
zero one zero and zero one
well rather than competing three diamond across this week after the statement
left to right then right to left going up and then back from the top back down
to the lower left corner rather than doing that
we use green's theorem which means we have to replace that
line integral in an ex-wife coordinate system
from zero zero two one zero two zero one and back along with this kernigan
dislike that line
with a
with any area integral over the area off the triangle which means that the
integral of peak here at the scene is actually the integral over that
triangle d
g_q_ d_x_ lausd pty dat
the integral over the triangle is that x goes from zero to one
why then goes from zero to this line here which is one minus six or zero two
point ninety six
defuse d_a_x_ since didi except x_y_
nice-guy pty which is dee dee why
off x square d y the exit begins
the integrins is
why minus
zero so this is just you know from zero one
in from zero to one man's ex-wife t won t x twelve x easy to work out because
that's the end of the printer one
of one he has twice we heard from zero to
one ninety six which is the end of frontier integral from zero to one
of one-half one monies x squared which is the end of the from zero to one of
one half times one minus two x plastics
square
which
ends up being hey i A
higher lost
negative sign your hang on just a second units
and indeed it was in the assignment now is fixed let's go with with respect him
we have one one two six five two square which is one minus to express x where
different multiplayer with one-half
i get one half minus one half times two experts miners extras one-half
x square and of course that's easily integrated
that is an editor one-half
one-half excuse me just get the interim um... one-half minus one-half x square
it seemed to triple x
class one six six q which is the editor of the one-half x squared from zero to
one by the cat one-half minus one half plus one six is a result is
so this works very similar to
soaps fearing because as we know it is
of stokes peter
needs application is that you can actually compute ear here with the line
into qu
salman still really really funny but if it does work
because here's what we can do the gary offered region
is nothing but the double integral off the function one
all over
the region itself writes that is something that i think we also talked
about
wintry
this double integrals and traveling across the volume of region is a triple
nickel off one over that
region itself
but one key in b britain against d x d x minus
derivative of zero
with respect to y int that now can be interpreted s
achey u_n_ eighteen
uh... in ingredients the urine and that means this is the same as the
intercooler with parameter off
uh... are region team of the picture field zero x dot dec
which it's at the same as the end of may two b
aches of team by prime of t detained
if we use a parametrization or
in the notation which here is very prevalent so i'm not going to avoid here
it's the into going over the clothes perimeter of x
do you want
similarity as a corollary if we have
peter cook curve in the plane this positively oriented piecewise smooth
simple enclosed
in fifty houston region in the plane that is bounded by this
positively or you could perimeter curve p
then the area obviously equals into part of the perimeter exterior
seminary is the negative integral over the perimeter of why d_x_ that's where
you
simply
the
victor field indians and the wider if it if it's the same direction as before and
of course
if something is people to to quantities and it's also equal to the average of
these two quantities who is also sometimes witness
one half off the perimeter integral xti y
minus
why
d_x_
and that is something that then leads to any
sample computation off the area often of that so that i take a general lips
it's clear race for a class west but it would be screwed equals one
and i want to compute the year you know i came to the following
the perimeter esp around a fry spicy
standard forever curve which is
a course on t_v_ signed team blankets
polar coordinates only
different radii so it's the
parametrization of a circle with two different radii which sloppy speaking is
what and and that's the s
and that means i can write the eerie uh... is the integral
off
xvi y_-one x's equal sign t
indeed why it would be the derivative obiee sign t
so there is just not just computation because not to parameterized
i need to integrate from zero to two by kaczynski willful cleanup around the
perimeter
of any course ninety eight times as a derivative of why why prime dat which is
derive my pf t was my friend he t
so we end up with b times site time sign t
is that is nothing but eight integral soldier to play was on t_v_ sign t d t
which is eight-b_ in the grocery coupon to pry will sign square off tea and if
we work that out
with treaty identities to realize that the course of swifty is one-half one
class who signed to achieve because what happens when you square is everything
has flipped out
which is like a doubling off the frequency but at the same time it now
has on the
inadvertently one-half around behind
want around the height one equals one-half in because it starts at one
instead of one class was sent to tear out of that one line this wasn't too
cheap
which would be
sites where it either way if you just memorize decide entity it gives you the
same things of course this is now
to integrate we have eighty times the input of one half times when his wife
can't t the into derivative
that poor sign off to t
is is is one
half sign ta dis one-half gives us the one forced now keep the into going off
course signed to which he is one have signed to deal with this one have you
got
one-quarter assigned to ante it turns out that that ultimately doesn't matter
because now if you plug in to apply
to buy times one half which is highly
classes one quarter signed four five which is zero minus one half times zero
minus one fourth times sign zero which are all zero signin dot
with aid the
times pi which is indeed
the area often in this
so
aspen conclusion
for
our excursion into victor calculation in many ways as a conclusion to our
capstone of calculus
in other quit presentation like i said we definitely have earned that by
working our way through that
divergence theorem and stokes theorem
we had another presentation that allows us to hopefully
further work with the seniors and appreciate them and
as a sideline we've got it
leading me philip uh... application which gives that we can compute here yes
line integrals which feels like a paradox but as we can see here with the
area of minutes
it works rather
alright gets over progress on dreams theorem to make sure you're on top of
this one also
and other than that
congratulations if you have follow this course in the way
i've outlined it
you are done with vector calculus in sometimes in my outline packwood's
optimization museum so you may still have to do that
but i can tell you you have gotten over the hump
the issues that will
if you were fully understand
uh... stokes theorem into divergence theorem as when its consequences and
origins the coral
and degree heat operator
you pretty much
had a commando everything that calculus does because all the computations
involve the computational parts of calculus
that visualizations involve the visual part of calculus including graphing at
some point in time the one thing that is a little bit missing his optimization
which
you either still have to do or maybe you've already done it either way
congratulations