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now that we have stokes theorem into diversions theorem aslan aslan
recollections org
uh... differentiation off
function system of variables
are familiar with the three ways in the kidnapper operator
kian be applied to other entities the gradient is where you multiply now
operator with scared
uh... divergence is for you to use the dot product off the mountain operator
with a vector field into curled as we take the cross product
off the map no orbiter with it that you feel
is uh...
the cars we already have stokes into risks the room for you also know
the meaning of these operations these operations are just make an ankle
formal operations we know what the graded means we know what their fortunes
means we know what the cronies and so now it is time to see
how these operators also
relate to each other because if i can for example computer-aided often skater
idea que picture feel and
for that victor field i can compute something like
the divergence
the coral
this is something that use in many courses done before you actually has a
divergence theorem in stocks theory might prefer
to have this
asked that we have had the fear into nafta we have thoroughly familiarized
ourselves with
how the operators were in what the operators mean because then
we can't really talk about the scientific underline
uh... principles
that that governor
how these operators behave in that way
in which they describe nature sulk that us quickly review with the gradient
of eighty skater function that is
victor field at points in the direction of steves asset which means that
physical quantities that people are often interested in is the negative
great he pitches to correctional steepest
did sandwiches to flow direction for example even a
potential feel
let diversions of a vector field is something that measures
source strength that is something that we can explain
with the divergence theorem because that's the divergence since positive at
a point and it'll be
if it is good if the veterans continues in this continuous three that is this
week
typically
uh... it seemed to have an application synthetic turf field vendor
diversions will be positive in a little neighborhood off
that point which means that uh... volume in tupelo over that little neighborhood
ofthe versions will be positive and by the divergence theorem that means
that the vector feel it would have to be
outfield from that regions at that
region and that means that the region acts
like any source
in similarly the crop
more measures to avoid his appeal of victor field and so if the
curl
not zero in
spent some point in space than again assuming the right continuity in
different ability conditions
crew will be
uh... no one zero in a small neighborhood around that point and there
will be
certain small curves
so that if i
integrate over a surface as bound by the curve of a different occur over that i
will get a non-zero integral
and by stokes theorem that translates into a non-zero
layin' in to post a vector field
around that curve and that means
the vector fields
has known zero in tupelo clothes cars which means there is some kind of a
vortex
valley area
so
these are the meeting suspect talked about first of course we can also
observe yeah this is
if we
pretend that matt and i really is a vector work to do with the victory
multiplied by the number
i can take the doctor richter's another victory ninety and that can take the
cross track there's another factor well great i mean that's
that's the computational sciences
so now the question becomes what else can we say
is certainly make an extinct
we
kian combine these things in various ways
uh... but it turns out
that
the
case of
curly gradient diversions and credit at croft
crew of the gradient divergence of a gradient and divergence of a coral there
is a bit of physical sense to that and so we're going to look at the
combinations of any two of these operators with each other
since enso
bill not look expedia gradient of a divergence even though that is something
that can be computed and that is because
i am not sure there is much of a good physical interpretation and if there is
one i apologize i just
is uh... essentially beyond
my indication
uh... however
free combinations that i just mentioned
certainly have uh... physical violence in some ways akita versions of across
just that
quickie at the end
into their eyes and certain other combinations of remaining combinations
are simply
possible such yes for example i
viviana at least the way the operators defined i cannot take the gradient of a
coral because degraded applies to scale it right
that doesn't stop people from doing that component wise potentially but we're not
going to go
that far
those first take a look at
the relationship between curl ingredient and one thing that is
easy to prove is that gradient feels
fields have zero karo
and we have to assume that we have continuous a compartment earlier so that
they can apply
kero syrup that goes with the same
philosophy that
permeates a lot of good applications which is that if you need something to
be continues det
continuously differentiable
you assume it is
okay so the corroborated we setup its
nabil operator across nabil operator times yes and so
now they have fears of course the f_d_a_ exp f_t_ why the f_t_c_ now this
cross product
***
d if derive easy minus the f
dvd whyy x component in your really realize that that's going to be zero
then salinity
uh... and i i think for too many s anderson is getting his d_f_ deride easy
d f d v d y
that's going to be zero because you can't make the party in any order
so that is actually candy to live de white easy monies due to f dvd one
in the
single ponent i can't
that i apply the x_ partial to the z partially because the part of the x_
partial
insp dot again hideaway ends up with d_-two
infra d_a_x_ easy for both in the first track those i get zero and for the third
component i take the expulsion of the white partial minus the white part of
the exposure which
islam as we have clear rose fear and
turns out
back to this past zero
now the question is
how about the controversy cost this would be
really nice computational criterion
that says hey if i can just computer crow into croat ends up being zero
media gradient field uh... beat with great in fields of course having good
physical relevance as we have seen in the presentation
one is a fundamental theorem for line integrals
that would be a nice computational criterion that also that
verifies this condition whether something is a great infield
so does zero curl implied that the field as a gradient field the answer is
it's sick
to be cause let's look at the following zero pro alone is not going to be good
enough
consider the essence of the magnetic field that is generated by a current
along the z axis and i say the essence of a
mean
that has the mistakes component negative light will break script plus my spirit
and its playing component
uh... leading exporter
it's clear class was great
that is
eight vector fields that rotate com rotates counterclockwise around is the
access
int d case
uh... here's the strength of win over the peace the scarf
point will re-evaluate from does the axis which means
it decays exactly like
the magnetic field generated by a coral
d_k_
as we go away from
the current that goes straight about this the axis in this visualization
i lift out the scaling factor c_i_a_ over
too high in all the other things that's or i were to find you zero in all those
other things
that are only dot that they are effected also by what you mean system beaches
so this is the mathematical essence off that that you feel difficult cute
the curl
off that thing well i have to inflict mein ab operator across the vector field
can't be
that back because i get for the x_ component dee dee why don't you wrote
mine is did you feel something that doesn't dependencies of the x_ component
to zero
idea dee dee x of zero
minus dat ze old something that doesn't depend on zee and under extremely aside
from the sea component from from the middle component doesn't even matter
because a kid zero for that also in dizzy component i get d_d_ except this
one
minus dee dee why have that one so i end up with zero zero
d_x_ or xlr expect us by scared 'cause pty over
what times
uh... off
why
overexpressed
class was great so that certainly is
not half as bad as for example the divergence off the electrostatic field
of a point chart right
so we can work out and it's a product ap
truck political trouble short
derivative of the numerator which is one
times denominator bridges expect us why square
minus
numerator which is x times derivative author denominator the x_-rated
is two weeks
divided by the denominator
in for this one well derivative of the numerator which is one
times denominator which is expected sway spirit
minus
numerator which is why
times of the wider riveted both the denominator which is to why
times uh... divided by the denominator square which expect us my spirit
quantity
square if i've simplified i i i end up with x squared plus was great minds to
expect so that's why scrutinize exclude overdosed on her
right here
if it was a bit by this when i get expect us twice that much to west grits
with the expected scrutinized by spread over that denominator
and you can see when you add up you kids
zero
*** well
we know
that the magnetic field generated by a car along the z axis is not any
gradient field and i speak with if you take a current
that goes
in the political direction becker and i
then back around in
generates
circular magnetic field around the wire which goes along the z axis so currents
are surrounded by secular magnetic fields
circular magnetic fields in particular and
fieldset have closed
felines in general
fields that can be great if it's because the defining quality for a great if you
notice
clo was
line integrals are
so
even though the current off the field of this field is two zero
do not have that this thing it's a great detail of the problem is that
we have a singularity and onto c-axis because
in the moral
the current goes exactly along this the access to the access does not have any
physical extend
so we have infinite current density in a wire that has
which of course
physically
possible but nonetheless that is that
mata mata kal
problem here and when you go to stuff where this is active physical possible
that
q description change is a bit
is
you mathematically that means we've got any
so you know if you don't have the abstract we divide by zero things go
well
int
if we tried to apply stokes theorem then the interpolating surface
but i'd apply stokes theorem along ini off these
lines of these closed doors here
if i put in interpolating surface into
this armed
if i put in interpolating surface inside
coastline around
the uh... z axis using the interpolated surfaced
interpreter putting surface would contain the singularity and that means
technical akbar disease of stokes theorem
apply and therefore stokes theorem cannot be applied and things don't work
out
conclusion of the cruelest a
norcal measure off which is to do so as long as i don't
uh... as i do include the singularity it's perfectly fine to see that here
there is no coral
and as long as they don't fix it does region the field
acts like a great if you are the problem is when i run around singularity
so this is
what we have to somehow
take into account in a new technical
gee you metric hypothesis and that is what these as to this
idea this definition of the simply connected regions so
when looking at a new region
aviation kin have colds and their eventual kinds of course there awards
like the answer like in a
swiss cheese or so we're
even if i have aid
uh... clue was curve that
so rough
contains these goals there is a way to put insert
in interpolating surface
into
the region so that the surface stays inside the region doesn't get the walls
and interpolate the surface
just find this is a situation where even though i've got a hold of these walls
might contain singularities here
i can apply stokes theorem
because
surface and using two interpolate
my
uh... walk around these holes
is contained inside the region
simply connected means that closed curves can be interpolated with surfaces
that stay inside the region and that's the important part
this definition of simple
connectivity
you find out where to talk about
uh... a different kind of cool so i take the same ball that now
i drill a hole through this thing because all the way sort of like
inland coffee not war
calls that you have in a bowling ball only that this time we draw them all the
way so that
the finger that goes in you can
touch the finger that goes in here if you choose to do so
then all of us and we have a problem because if i take a similar clueless
cerf
i cannot find and surface-to-air in terrible it's that kare in such a way
the ice-t_
inside this region because no matter how interpolate this curve
out always somehow have to cut through this whole
and will have something like
this right here where i have to do you've
the region
so in a region that is not simply connected some careers
cannot be interpolated with surfaces with if i have a curve here of course
that can be interpolate
but some cannot
investing sake
what now leads us to how we can actually make a connection between current
gradient that that is
a connected region three dimensional space is quote simply connected
if they don't be afraid very simple closed curtsy contained in the region
he is a service in cities also contained in the region
in which case the years it's about three
which means
that i can apply stokes theorem with that surface and that service has all
the properties
that you ki and get out of the region
so now the theory is that a vector field yet
with continuous equip our children to test on a simply connected region
so now we're back now
simply connected avoids this problem that we had with the magnetic field wife
right
uh... that is a gradient feels
youth and over the years
the crew of this field is equal to zero
in the proof now is
we already have proved that the cooperating to the zero so if this thing
is arabian field we know that map across s yes
zero
that's perfectly fine
conversely
because the region is simply connected stokes theorem can apply to kurds inside
the region
brief a surface that also is inside the region so with a surface
on which
the grading is equal to zero
in so that means it's a simple consequence of stokes theorem that zero
curl
when he said he connected region implies that in the course of a close curves are
zero
you take a close curry inside the region
you in trouble at the close curve with a surface that is inside the region
you know that the surface and to put off the car over that surface is zero and
that means that the line and to go along
closed curve also it's zero in fact
tells us that indeed and this is something that
physicists then certainly also apply people just
use
frequently
uh... were at least i have a great fears are concerned
z
connecting to deter d
condition to determine whether if you does it make you feel dizzy computer
corrupt
if the cruelest zero it's a great if you look if not
as long as you pass simply connected regions but
singularity such as currents going through things and things like that
that's the kind of stuff that when you look at the plaques in there you will go
to exceed
so now that's not correct
the divergence and degraded we want to talk about the divergence off the
gradient
and dot actually needs to
the way ke transfers that needs to be
kiki question
so
when i want us to consider is a small bowl senator kennedy center are with
radius aliant surface dis
so here is that ball
and if we were to assume that this paul was harder than its surroundings
then of course he energy would transfer out off the ball through the surface and
so what do we know what he energy well first of all we know that heat flux is
proportional to the may give a grain of you will
where you know it's
i mean if you were having zaki would have to be the heat energy density
if you use the u_s_ the temperature
then it's at least a proportional we are not assuming any kinds of complications
where uh... different parts of the region have different heat conduction
coefficients it just supposed to be a homogeneous
into this comes from the fact that
if you've got a region that's hot
if you've got a region that is called
then the thermal flux
doesn't go
back from here on around somewhere the from a flat schools directly
from hot
if way then look at the net heat transfer through the surface off the
region per time unit so think about that ball again
that heat transfer is proportional to the surface integral off the negative
gradient
over the surface and that is because
if you have a large gradient that means you have at large
temperature differential in fact means
you have allot
of energy transfer
through that surface
that is why eight is
visi
keat a house
uh... early in the fall when it's uncomfortable outside maybe somewhere in
the fifties
eternally here it doesn't stay on very long the inside of your house is fine
and states find for quite a while
where s
if you are in that
middle of a bitter cold kansas type winter it's zero degrees outside
amma fahrenheit scale right now
uh... if it's zero degrees outside you did your house it feels warm for a lot
of cools down pretty quick that's because you have a
huge temperature differential
in speaking with the best of insulation you've got to be credited you've got
more
g transfers and if the temperatures
locate so
that he transfers proportional to the end of the great you know with the
surface which they also takes into account that large surface area means
you've got more energy lost
than small surface area and things like that
on the other hand the net heat transfer through the surface per time unit is the
rate of change affinity content
off this uh... little ball b
for time get it because
doesn't just jump from the center of the ball to the outside only transfers to
the surface right
uh... if you go inside you feel cold your your skin gets cold
your core body temperature as long as you're healthy and don't stay outside
very long accord climate
will not change significantly even though
you probably won't feel too well after a while and if you take measures
to stay warm such as warm clothing and moving around
now this
when rate of change as indeed he content is simply
the derivative off the trickle in to go off the heat energy density
or a disproportionately we'd use u_s_ a temperature is proportional
diri time derivative off the triple into going off the temperature
we have to think about this negative sign real quick remember that
and bridges the room
judy
orientation is positive which means out quick twenty normal victories and so
if the integral of the negative gradient over the surface with oakwood pointing
electorates positive that's being that transfer out
weariness
this part of the relative off the
into put off the temperature of the inside when that part of the root of
without a negative sign so just this thing when that's positive
fifteen eight heat transfer in that means things warm up
which means we need is extremely assigned to make sure that that we have
a proportionality
with the closing of proportionality is positive
uh... which makes things much easier
to read
all kids so that means
that's that
negative surface into a lot of making a few
is equal to some number times in negative part of your e-ticket
often traveling to global view
over the volumes were were you with us
the temperature
well now we know by the divergence theorem the triple interval of the
vectra feels over a surface these jess
of peace or now
first and apply the divergence the indirectly and first
politically inside
pulling za
negative signs away i'm canceling the negative side here and here and that is
pretty easy to do
the part
that can cause mathematicians a bit of a stomach ache rightly so
is that again as applied people say under the right circumstances under the
right circumstances whatever they are and that's where
the receptor brick a date
this part of it can be pulled into
the triple integral now this is something
results just invigorated of moving inside
but it does turn out that there are situations in which activity this
interchange cannot be made it requires for applications to the game
invoked his philosophy that that light people rightly shoes to have which he is
if we need so too many different ability conditions to make this work
if there are these conditions that allow that
then we just assume that these conditions are
satisfied at these
for the moment
because typically the mathematical pathologies don't of course not much
applied scenarios
that has simplify things a bit now the left side is that the serbs samples the
right side is a triple integral now
we're going to apply the divergence here and i just have to puke here
double integral of the victor feel over the
surface of a solid is equal to the
soldered integral off the divergence off that vector field over the inside so
that means that the left side is to trickle in to post backwards as a third
reading of you
dvd
the right side statistic for political of key
d humidity
that it's now something
this works
for
any balls off any thoughts
in so that means i can now state that limit
is the radius goes to zero well if i take the limits of these integrals as
the radius goes to zero
it'll just goes to zero
but if i divide these integrals
by the volume off the ball then basically agate
however much
diverges off the gradient the ball contains divided by the volume of
football
so i A
i divide things out than i did something like a density
i do the same thing on the right side
ins interest
powell
these then works out in generally give assuming that everything is continuous
at this ball is really small
than the divergence of the great you know if you all over the small boys
jessica versions of the gradient
at the center
similarity
judy humidity of the ball is sufficiently small and everything is
nice and continuous will be about
cagey judy tea at the center
which means as i take this deem it as a goes to zero this turns into the
divergence of the grading at the center at time t is equal to k times
uh... the part of their evidence with respect to time at the center at time t
in that is
indeed
the he questioned the huge question since
kate times apart under the pocket temperatures equal to the divergence of
the gradient
off your temperature and then that things can be but mister famous law
class operator because of the versions of the rating there's nothing else but
second partial with respect to explicitly parker with respect to why
trust second partial with respect
so he's is
weirdly he question comes from
what it basically see issues
the source strength
off your phil field
off your potential
is equal to the partial derivative if people too
k times the partial derivative
the time derivative
or fewer
potential here
if we want to visualize that we know that the triple integrals are equal and
limiting processes
that they shrank in that paul an estimated strengths the approximation
that
the triple in the one of the diversion of the gradient
is approximately the volume times diversions of the break-in at the center
in that the treatment of key terms the humidity is approximately the volume
times katie mcginty at the center
that approximations
these approximations improve towards equality
in in the limits
we city every step of course cancel
the coleman
volume factor that we have here
as well it is here
in into the mix
we get the fact that the divergence of the great of you
is equal to key
times
do you
t and so that is
how he transfer works and that is
uh... diffusion works
into if you want to philosophically
you would like that this is why it works that way it's
what to do revision gives us
finally we have to have origins off
correla measures vorticity crow as something that's been inouye
and divergence measure source friends will something that's been ste
typically shouldn't have a source it one of the reason
but something that's been this doesn't come from anywhere because of course
from ten rounds
so
uh... physically speaking here were important to have to be speaking should
make sense if the governor divergence of a crawl is zero
well that actually turns out to be the case
because the countries we compute the divergence of a corrupt well that is
now operator dot
mab operator across your vector feel one out of a little bit across your vector
field we keep this from the same yes
d_r_ gyi minus d q jeezy
right here
it is d_r_ d_x_ minus d_p_ d_c_
with an extra negative signs that you can't easy months the or d_x_ into de ke
*** excellence t pty right here
now i a plaza dot product well that's easy that's
the second part of our with respected x and y minus the second part of q with
respect to extent z
plus the second part of p with respect to why in c_-minus is second part of our
with respect to why x
class but the second part of a few with respect to see index minus the second
part of p with respect to see and why and now we have to collect in terms of
intolerant bc
here we've got the second derivative p with respect to what i a m z and here we
have its negative so back was awake
here we have the second derivative keo with respect to z index and here we have
its negative so that part goes away
in here we have the second derivative of are with respect to x and y here we have
it's made it so that was a way and that means
the divergence
the coral
is equal to zero
that's
all over the issue it's after his own all presentations on that divergence the
arraignment stokes theorem i think we have
earned a quick weather
note what we have gotten q because off the
intuition that we have to bail out for all of the interpretation off the
divergence in the gradient we were evil to talk about very competent to you
about a mathematical situation that also his physical enter implications in that
we know the criterion but if you does it make you feel
we were to develop an integration week with derivations act that really is is
as watertight as i i would care to deliver i think
uh... four
uh...
four d he question okay
abstractly speaking i would still have to
for heat equation
uh... justify putting the right hypotheses as to why i've been in a
changing depression differentiation relief after the mass folks among hume
intense then the quick competition at the end of the coverage of the crew as
he put his ear a little bit of an idea what it means these is the thing that
you want to continue to develop the computation of the city of course is
something that you developed by brute force and you will have some homework
assignments where it simply is more off the same in terms of computing things
may be something working with maxwell's equations and routine with durai
deriving certain kinds of phoniness
that are important investor calculus but along the same lines we also have to use
situation now it's very good situations that we have
physical understanding what these operators mean so that in many ways
predict what the result ought to be in as a result isn't quite what we
predicted to be well there we just simply have to think about it some more
as we have to do
for perot and great able to relax kia there are singularities that can mess
things up
all right
have fun with the homework we've got one more to go which is um... green's
theorem at least in the line-up that i envisioned for the course
i'll see you in that