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Video tags
Subtitles
Russian
00:00:09
what is there to consider today?00:00:11
Last time we are almost finished, but we still00:00:14
need to talk about this a little more,00:00:16
the chapter dedicated to the surface00:00:19
integral and two famous theorems, the00:00:23
Ostrograd Gauss theorem and the00:00:27
Stokes formula or Stokes theorem,00:00:29
well, there will be a short afterword about this, so to00:00:32
speak,00:00:33
and then the last one is enough a00:00:36
short chapter on geometric00:00:38
applications of differential calculus00:00:40
and so I will remind you to formulate00:00:45
the theorem about the Stokes formula and the formula itself00:00:48
looks like this means int is a00:00:51
curvilinear integral of the second kind along a00:00:54
closed contour00:00:55
hey this is an integral of the general form pdx00:01:01
+ q d y +00:01:05
r&d z pqr functions x y z just for00:01:12
We don’t write the arguments for brevity, so this00:01:15
integral is equal to the00:01:18
surface integrals over the surface00:01:23
fi means fi00:01:27
the requirement of the surface was such that00:01:30
this surface is either smooth or piecewise00:01:33
smooth00:01:34
limited by this very piecewise00:01:37
smooth contour and,00:01:39
what is also very important, this00:01:42
surface is one-to-one00:01:44
projected onto any the coordinate00:01:48
plane is what we say, so we call00:01:51
it the surface x y z projected, so here is the00:01:54
integral over this surface and the00:01:58
integrand is like this here is a00:02:00
rather long expression deco pdx00:02:05
minus dppg y close the bracket to cosine00:02:12
gamma plus means alpha beta gamma these00:02:16
will be angles as usual, this is the angle gamma00:02:20
angles have arisen in which the00:02:22
normal vector to the selected side of00:02:25
the surface is formed by the coordinate axes00:02:28
due to gamma, this is the Sochi angle z plus, well, I’ll00:02:33
tell you honestly, I00:02:36
never remember this formula from00:02:39
beginning to end, and I already told you that00:02:42
it’s enough to remember that the first00:02:44
expression is in parentheses exactly the same as in00:02:47
Green's formula, there, actually, under the sign of the00:02:51
double integral, it is precisely this00:02:53
difference of duku that the star dad falls to y00:02:55
and then we write the next two terms by00:02:59
performing a cyclic permutation,00:03:02
which means pqr p goes boredom goes into00:03:07
p-r goes into p00:03:08
means q goes into r it means there will be a dryh00:03:15
goes into y means dr. pads y minus p00:03:19
goes into the deck00:03:22
y to the transition z yes cube a to z by cosine00:03:26
alpha and plus 3 more similar terms00:03:35
d.p.00:03:37
go z minus dr codes x close the bracket00:03:49
on cosine beta00:03:52
make sure I wrote everything down correctly00:03:55
based on this rule based on the00:03:58
cyclic permutation p goes into p00:04:00
q goes into r y and z z vx d.s.00:04:05
and so this is the famous00:04:10
Stokes formula and what else needs to be specified size with00:04:14
it that we choose a certain side of00:04:17
the surface and depending on this00:04:20
alpha beta and gamma are the angles of the vector on the00:04:23
normal directed in one00:04:25
direction or the other and the direction of traversing the00:04:29
closed loop and is consistent with the00:04:32
orientation of the surface, that is, if we00:04:36
have chosen,00:04:37
let's draw a picture, let's say some of00:04:39
our surface fi here00:04:42
we select, well, let's say, let's say, the00:04:48
top side of the surface00:04:50
so that the upward normal vector is directed00:04:52
to then00:04:54
for such a choice of the side of the surface,00:04:57
the positive direction of traversing the contour00:04:59
will be like this as indicated, we00:05:04
stipulated00:05:05
how one chooses to put or the direction of00:05:07
traversing the contour, so this00:05:11
Stokes formula can be written in a very short,00:05:14
easy-to-remember form,00:05:17
namely, let's enter this we have already00:05:20
done, introduce a vector function and with00:05:25
coordinates p q r, let me remind you once again pqr00:05:31
function x y z so this is a vector00:05:33
function also depends on the point from x y z00:05:38
from the coordinates of the point and another vector00:05:41
function00:05:43
which is called a rotor rotor and00:05:50
is denoted like this by the first three letters of00:05:53
the word rotor, the Latin letters rotor00:05:56
say so the rotor of a vector field, but00:06:01
we have introduced vector fields with components00:06:03
pqr and we will introduce another vector function00:06:06
curl of a vector field and it is written00:06:10
as this is the determinant of the third order, the00:06:13
first line is the coordinate vectors of Izhik, the00:06:21
second line is unusual,00:06:24
this is a line consisting of00:06:29
partial differential operators dpd x dpd y dpdz and the00:06:36
third line is the coordinates of the vector and00:06:40
that is, p q r equals00:06:45
let's write this out00:06:48
by expanding the determinant into elements00:06:52
first line00:06:53
what will be the coefficient at and this is00:06:56
this second-order determinant,00:06:59
cross out the first column and the first00:07:01
row how this determinant should be interpreted00:07:04
and here is how the product along the main00:07:07
diagonal is the operator dpd y acting00:07:10
on the function p this will be d rpd y minus the00:07:14
products along the side to diagonal00:07:17
is yes cube a to z means it is equal to dr00:07:21
under and y minus deco will run this is the00:07:28
coefficient at and this is the x avaya component of00:07:32
this vector plus what will be the00:07:36
coefficient of pressure since the00:07:38
element is the first row of the second00:07:41
column the sum of 1 plus 2 is odd then it will be Well,00:07:46
we cross out the first row and the second00:07:47
column and take the00:07:51
second-order determinant with a minus sign, which means00:07:53
first along the side diagonal it will be00:07:55
dpp z minus dr pdx dpp z minus dr pdx00:08:05
is the coefficient of yarn00:08:08
and plus, finally, the coefficient for k00:08:12
we find it the same way this will00:08:15
add up and x minus dppg y to the vector k, but00:08:27
if we now look at well and00:08:31
remember, let’s also write next to it00:08:33
that if m the normal vector00:08:38
has unit length, its length is equal to unit of the00:08:41
normal vector, then its coordinates, that00:08:44
is, the direction cosines,00:08:46
well, let’s do this let's write cosine alpha on00:08:50
i plus cosine beta00:08:54
knives plus cosine gamma00:09:00
on ko and now if we look at the00:09:04
integrand on the right side of the00:09:09
Stokes formula, what we see is that let's00:09:14
follow the terms in the wrong order, but00:09:16
in what order are these average terms00:09:19
drp y minus additional purchase z Dr. gifted additional00:09:23
purchase and z is their coordinates of the00:09:25
rotor vector and is multiplied by the cosine00:09:29
alpha further dppg z minus Dr. pdx by the00:09:35
cosine beta00:09:36
dpp z minus Dr. pdx is the game you and the00:09:39
coordinates of the tarot vector multiplied by the00:09:41
bone beta00:09:42
and finally 1 there is a term that is up to the navel00:09:45
up to x minus y by posting it, thus we00:09:49
see that what is in square00:09:51
brackets in the Stokes formula00:09:53
is what is this the scalar product of the00:09:56
vectors rotor a and the normal vector00:09:59
and the left side is when we are with you we were00:10:04
working on curvilinear integrals, we00:10:08
wrote this way and that, the Stokes formula00:10:11
can now be written as a00:10:15
closed loop integral00:10:16
el large and the left side can be written00:10:20
as the scalar product of vectors a and00:10:24
b and vector a is a vector with coordinates00:10:29
pqr since we introduced it as one vector with00:10:33
coordinates dxd y and z are like an00:10:36
elementary vector directed along the00:10:39
curve, and so the left side will be this00:10:43
kind of curvilinear integral, and the right00:10:46
side will be the surface integral over the00:10:48
surface phi from the scalar00:10:50
product rotor a by the unit00:10:55
normal vector00:10:56
VAT in this form,00:10:59
this formula is easy to remember and it it00:11:04
reads like this, we already said such a word as00:11:08
circulation00:11:10
and the flow is what is on the left is the00:11:13
circulation of the vector field and along00:11:16
the contour hey and what is on the right is the00:11:19
flow of the vector field rotor and through the00:11:23
surface fe00:11:24
well, let's write this formula00:11:27
it reads like this circulation of the vector field00:11:30
a circulation of the vector field a along00:11:37
the contour and large is equal to the flow of00:11:46
the vector field a rotor00:11:48
a through the selected side of the surface fi00:11:56
limited by the contour and00:11:58
so the circulation of the vector field along00:12:03
the contour hey is equal to the flow of the00:12:07
vector field rotor a through the selected00:12:10
side of the surface ufe00:12:12
limited by the contour and we are still with you We00:12:17
will return to this formula at the beginning of the00:12:18
next semester, I have already talked about this00:12:21
when we study scalar and00:12:24
vector fields in general, this formula is very00:12:27
physical00:12:28
and has numerous applications in physics,00:12:32
you00:12:33
will also meet it many times in physics itself, and the last thing00:12:37
in this paragraph is that we are with you Let's00:12:38
consider here we had in the chapter on00:12:41
curvilinear integrals a00:12:43
theorem about the conditions for the independence of a00:12:46
curvilinear integral of the second kind from the00:12:49
path of integration on the plane,00:12:53
exactly the same theorem holds00:12:57
for curvilinear integrals of the second00:13:00
kind in space,00:13:02
why should we consider it here00:13:05
they have that theorem00:13:07
in In the proof of that theorem,00:13:09
I used Green's formula,00:13:11
and here, in the proof, I need the00:13:14
Stokes formula, and the proof itself is absolutely the00:13:17
same as there, and so Theorem 5, we have00:13:20
another Theorem 5,00:13:22
and let's write in parentheses this00:13:27
long name about the conditions00:13:30
of independence about the conditions for the independence of a00:13:37
curvilinear integral of the second kind00:13:43
in conditions of independence of fastening played of the00:13:46
second kind from the path of integration in00:13:48
space from the path of integration in00:13:51
space the formulation of the theorem is very00:14:03
very similar, almost the same as00:14:07
similar theorems for a curvilinear00:14:10
integral of the 2nd kind on the plane in the tower00:14:15
two statements statement 1 we, as00:14:19
there, denote 1 Roman00:14:22
means let let functions p well let's00:14:28
point out once that they are functions x y z p00:14:32
from x y z kuat x y z00:14:37
r from x y z 3 functions let functions pqr00:14:45
continuous in the domain is large00:14:50
continuous in the domain is large then the00:15:00
following three conditions are00:15:02
equivalent absolutely also00:15:05
the theorem was formulated on the plane00:15:08
only there was two functions in ecu00:15:10
that depended on x and y then the00:15:12
following three conditions are equivalent to the00:15:15
first condition for any closed00:15:23
piecewise smooth00:15:27
contour el large belonging to the00:15:36
same area the curvilinear integral along the00:15:42
contour ey pdx00:15:46
plus where y + rd z is equal to zero the second00:15:57
condition for any two points a and b00:16:05
belonging to the region, the00:16:11
curvilinear integral along the curve a bpd x00:16:17
plus where y + rd z does not depend does not depend00:16:28
on the choice of the curve connecting points a and b00:16:33
well, of course, we won’t even write this, the00:16:36
entire curve must already lie in the region,00:16:39
it cannot be somewhere you you then enter again,00:16:42
so for any points a and b adjacent to00:16:45
the region, such a curvilinear00:16:48
integral from along the curve both does not depend on the00:16:53
Vyborg00:16:55
connecting points a and b and finally the third00:16:58
condition00:17:00
exists a function of x y z such that00:17:08
such that its differential dau is00:17:13
exactly equal this integrand00:17:16
of the expression pdx plus cube y + r&d z, in00:17:25
other words, dry such a function00:17:29
whose partial derivatives00:17:32
with respect to x and with respect to y and with respect to z are equal, partial derivatives with respect to00:17:35
x, etc.00:17:36
according to the game and according to z ttr well, and this is00:17:41
true throughout the entire region, that is, at00:17:45
any point in the region, but this is00:17:51
not all condition 3, while the integral along the00:17:56
curve a bpd x plus where y + rd z is equal to00:18:07
y at point b minus y at the point and here is this00:18:12
very function y and00:18:14
at the same time it is true that such ravens00:18:16
was absolutely the same and00:18:19
for integrals on the plane only there00:18:21
were only two terms pdx bus where y00:18:24
well they also say so here is the expression00:18:29
this sum pdx bus where the games are simply rg z00:18:32
is by a complete differential, what is the00:18:34
meaning of these words,00:18:37
exactly this one, that there is a function00:18:39
whose differential is equal to this00:18:41
expression, this is the first statement 200:18:47
Roman second Roman if, in addition, well,00:18:54
besides what, actually we00:18:57
had one condition that the functions pqr are continuous and00:19:00
we will now add another something if in addition the00:19:03
functions pqr the functions pqr00:19:07
have in the domain they already have00:19:12
continuous partial derivatives of the first00:19:15
order in the domain they already have continuous00:19:20
partial derivatives of the first order in the domain and the00:19:26
domain and here are some additional00:19:29
conditions on the image it would be00:19:33
superficially simply connected what is it00:19:37
we will say a little later and closer is00:19:39
superficially unrelated,00:19:46
then each of the conditions 1-3 that we00:19:53
listed above,00:19:54
then each of the conditions 13 is equivalent to00:19:57
condition 4 equivalent to condition 4 and00:20:05
condition 4 is this, look at the00:20:08
left board at and the Stokes formula is here00:20:13
each parenthesis is equal to zero,00:20:15
that is, let's write it like this: pdx gakupo00:20:21
dx equals dppg y then what do we have dr00:20:30
under y equals to buy in addition and z dr under up to y00:20:37
equals for lips and even well, the last00:20:42
bracket dppg z equals d -р pdx dppg z00:20:50
is equal to dr pdx well, and this holds throughout the00:20:55
entire region,00:20:59
well, now, as if in parentheses,00:21:01
an addition to this theorem, the region is already00:21:05
called superficially 100:21:09
connected; the region is called00:21:21
superficially 1 connected if if for00:21:27
any closed contour hey00:21:30
for any closed contour el00:21:37
large belonging to the area,00:21:43
there is a surface fi00:21:46
surface and large with the boundary and with the00:21:52
boundary or entirely belonging to the00:21:56
area already00:21:57
means there is a surface fig with the00:22:01
boundary el entirely belonging to00:22:03
the area well,00:22:09
what example can be given of a superficially00:22:12
1 connected area and vice versa which is00:22:14
not is, according to version 1, connected, let's say00:22:17
about the lepidus cube, this is superficially connecting with00:22:23
an area, it is completely clear if inside the00:22:25
cube we take any closed contour,00:22:28
not necessarily flat, on it, as00:22:30
they say, you can stretch a surface00:22:32
whose boundary will be this contour and00:22:35
this entire surface lies in the cube,00:22:37
but here is an example of a superficial one one00:22:39
connected area and to top the00:22:41
steering wheel, let's take a closed contour00:22:45
passing like this through this entire00:22:47
steering wheel,00:22:48
whatever surface we take as the00:22:51
boundary that will be this contour, it will00:22:54
definitely be somewhere inside, in this00:22:57
hole from the donut, they will jump out of the00:22:59
area beyond limits of the torus, the00:23:03
proof of this theorem is carried out in00:23:05
exactly the same way for the plane, we00:23:07
proved it and so I advise you, and even better,00:23:11
without looking there, try00:23:14
to prove it yourself, well, something will not be00:23:16
forgotten, look at how the00:23:18
proof of this theorem is carried out00:23:20
absolutely according to the same scheme as00:23:22
similar ones theorems for the plane means the00:23:26
first statement is proven that from 100:23:29
follows 2 from 2 follows 1 and 2 follows 300:23:34
from 3 follows 1 and the second statement00:23:41
is proven like this is-3 follows 4 from 400:23:47
follows one now or from 4 I already forgot,00:23:53
look here I will put question, that00:23:56
is, four follows 3 answers from 34,00:24:00
and from 41, that’s exactly it, and the00:24:04
proof itself is absolutely the same as00:24:08
for the plane, except for this last00:24:10
moment, to prove that from 400:24:13
one follows, we used Green’s formula there,00:24:16
but here we will need to use the00:24:20
Stokes formula Well, and just the last00:24:24
remark,00:24:25
this is condition 4, if you introduce a vector00:24:30
with these coordinates, the rotor and then00:24:32
condition 4 can be written very00:24:35
briefly conditions 4, the rotor a of the vector00:24:40
field a is equal to zero,00:24:43
well, the rocker is a vector, so on the right there is00:24:46
not a number zero, but a 0 vector00:24:49
here you need to be able to prove this theorem,00:24:52
it’s not difficult, as I00:24:54
said, absolutely the same, so those00:24:57
applying for the highest score, of course,00:25:00
must what theorem, after all,00:25:03
we have finished the chapter on surface00:25:05
integrals and are moving on to the last chapter, which00:25:10
means in our numbering this is chapter 14,00:25:15
let’s call geometric00:25:18
applications that way differential calculus00:25:21
would be carried out to express some00:25:23
geometric applications well,00:25:25
some words will not become geometric00:25:27
applications of differential calculus00:25:34
those applications will be discussed00:25:36
what in general we can do with the help of00:25:41
differential calculus in relation to the00:25:44
study of the behavior of curves we can00:25:48
for graphs of functions y equals f of x00:25:51
well as well as for of parametrically defined00:25:53
curves, find extremum points,00:25:56
monotonicity, inflection points, direction of00:25:59
convexity, asymptote, this is all we00:26:03
learned back in the first semester, and here we will00:26:06
consider a new series of questions, namely00:26:08
the question of the so-called tangency of curves, the00:26:12
so-called envelope of a family of00:26:16
curves, and the curvature of a curve, here are three00:26:19
paragraphs we will have paragraph 100:26:22
paragraph 1 tangency of plane curves00:26:30
tangency of plane curves00:26:34
about what else we will give a definition of what it00:26:39
means that two curves touch each00:26:41
other at some point means if if00:26:49
curves and we will denote them like this curves00:26:52
el big first goal big second00:26:55
if two curves or 102 have a common00:27:01
point, let us denote it as00:27:03
m0, have a common point and me, and at this00:27:08
point a common tangent00:27:10
and at this point a common tangent, then they00:27:17
say that they say that the curves or 10200:27:22
touch at the point m 0, well, they clearly touch00:27:26
each other, then they say that the curves or 10200:27:31
touch at a point with me, sometimes instead of00:27:33
the word touch they say touch and00:27:36
both terms are used00:27:37
let's draw a visual picture, let's00:27:41
say we have a curve or one on00:27:46
it.00:27:47
and here00:27:51
a tangent passes through me, so I’ll just denote it with00:27:54
letters and without indices, and in addition there is 200:27:57
curves that also pass through the point00:28:00
m 0, here’s the curve l200:28:02
and tangent and they have a common one, then00:28:06
they say that the curves or 102 touch at the00:28:09
point with me further we will consider00:28:15
not just some curves, but curves00:28:18
that are graphs of functions y00:28:20
equals f from x, so let let00:28:23
the curve or one be the graph of00:28:27
the function y equals00:28:29
f1 from x and curve r2 is the graph of00:28:36
the function y equals f2 from x00:28:42
and let00:28:45
these curves touch and let these curves00:28:49
touch at a point by me with coordinates x000:28:54
f1 from x0,00:28:58
well, of course, since they touch at point 0, then00:29:02
this is common. This means that instead of f1 from x0, we00:29:06
could write in 230 this is the00:29:08
same.00:29:09
let's also draw a picture here,00:29:14
now we already have a00:29:16
coordinate system on the x y plane, and here there is00:29:23
one curve, let's say this is a curve or one00:29:27
here is 2 curve00:29:29
l2 this point m 0 has an abscissa x000:29:42
and at this point the common tangent is00:29:48
again designated with the letter l,00:29:50
now we will take some point00:29:55
other than x0, so let's mark it here00:30:00
.00:30:01
x is different from x0 and we will be00:30:04
interested in drawing a vertical00:30:06
line through this point and selecting00:30:11
this segment, the00:30:12
length of which is the distance00:30:16
from one curve to another in this00:30:18
place, it is clear what the length of this00:30:23
segment is equal to, this is the difference f1 minus 200:30:28
but modulo because that my f100:30:31
is greater than two, but it could be the other way around, the00:30:33
length of this segment is the modulus of f1 from x00:30:38
minus r two from x00:30:43
now further00:30:45
let m-n small00:30:49
natural number some natural00:30:52
number that is 1 2 3 some00:30:56
consider the following limit let's consider the00:31:00
limit as x tends to x0 such a00:31:06
fraction in the numerator will be the00:31:09
length of this segment about which we00:31:11
just talked about modulus f1 from x minus00:31:16
f 2 x00:31:18
and in the denominator modulus x minus x but00:31:25
now I’ll write it will be in some00:31:27
degree, what is modulus x minus x 0 is00:31:29
this for us from this horizontal00:31:31
segment modulus x minus x 0 to the power00:31:36
n plus one, let's consider the indicated00:31:40
limit 1, so let's consider such a limit and00:31:46
now we will formulate such a definition of00:31:49
definitions if the limit is 1 and by the way, it00:31:56
represents what type of00:31:57
uncertainty what00:31:59
the numerator tends to what the denominator tends to00:32:01
bows zero when x tends to x0 at the00:32:05
very point x0 we have f1 and f2 equal00:32:08
also the numerator to zero and when x00:32:10
tends to x0 it’s clear the denominator of00:32:12
destroyers just put before itself00:32:13
tends to zero00:32:14
so the technique can exist can00:32:16
exist, so if limit 100:32:18
exists and is different from zero, if limit00:32:24
1 exists and is different from zero,00:32:29
then they say that they say that the order of00:32:34
tangency of00:32:36
curves or 1 and r 2 at a point is introduced by me, that is, the00:32:41
concept of the order of tangency of curves is introduced, then00:32:43
they say that the order of tangency of curves00:32:45
or 12 at point 0 is equal to n is equal to and00:32:55
if the limit 1 is equal to zero if the limit 1 is00:33:00
equal to zero then they say that the order00:33:03
of tangency is higher than n00:33:09
and finally if the limit if the order of00:33:14
tangency is higher than any n this can be00:33:17
now we will give an example of a simple not and00:33:19
finally if the order of tangency is higher of any00:33:21
ent they say that passap the order of tangency is00:33:24
infinite and so it is with the concept of the00:33:28
order of tangency of curves, but if limit 100:33:30
exists and is not equal to zero then the order of00:33:33
tangency is equal to n if it is equal to zero then00:33:36
the order of tangency is higher and if higher than any00:33:38
ent they say that the order of the braid is not00:33:40
infinite let's consider a pair00:33:43
simple examples,00:33:44
the first example is let the curve or 1 be00:33:49
the graph of the function y is equal to the sine of x,00:33:52
which means f1 is our f1 of x is the sine of x00:33:57
and the curve n 200:34:01
is the graph of the function y is equal to x, that is, f200:34:06
of x is x00:34:08
and as . and let’s take the origin of00:34:13
coordinates as a point with coordinates 0 0, well,00:34:17
it’s clear that the sine of x and x is equal to zero, that00:34:22
is, the graph passes through this00:34:24
point, you can even draw a picture,00:34:30
which means that the x and y axes are well. and now we have a lot of00:34:34
origins,00:34:36
here’s the sine graph, we won’t draw the whole graph,00:34:42
here’s a piece of it, and here’s00:34:45
the graph y equals x, which means we have this00:34:51
or two, but this or one, the question is00:34:56
what is the order of tangency of these curves, well,00:35:00
one of them is straight, but nevertheless, it is00:35:02
glue, the same applies to the words curve curve00:35:04
at the same time level y equals x00:35:06
what is the order of tangency of these curves in . and00:35:08
I will consider the limit as x00:35:13
tends to zero in this case00:35:16
x 0 we have equal to zero in the numerator the modulus of00:35:21
sine x minus x in the denominator is00:35:26
simply modulus x since, by the way,00:35:29
we rushed to the power n + 1 equals curly00:35:34
bracket this limit will depend on n00:35:37
let's Let's remember from McLaren's formula00:35:40
sine x is equal to what00:35:42
x minus x cube by 3 and then00:35:47
higher powers of x minus x00:35:50
will cancel, which means the numerator behaves00:35:53
like a sine like x cube by 3 means if n is less than 2 n00:36:01
less than two by isin plus one is less than00:36:03
three a here x cube00:36:06
and here they are to a lesser degree then the limit00:36:08
will be equal to zero this is if n is less than 200:36:13
if it is equal to two here it will be x cube and00:36:16
then what will the limit be equal to 1 300:36:20
since here modulo everything is taken00:36:22
one third if n is equal to two but not to00:36:27
is natural more than two then in the numerator00:36:31
x to a greater extent than vice versa to a00:36:35
lesser extent in the numerator x cube00:36:36
and in return, to a power greater than three we00:36:40
will get infinity that is, does not00:36:42
exist00:36:43
if n is greater than 2 well, from here, in accordance00:36:47
with the definition, we can say that00:36:50
the order of tangency is00:36:53
order touching these two curves at a00:36:56
point and me with coordinates 0 0 is equal to 200:36:59
is equal to long00:37:01
and here is an example, put the number 1 there and00:37:05
this is 2 2 examples, let's look at two such00:37:08
functions, the first function y is equal to zero,00:37:12
here we have f1 from x00:37:14
i.e. . the graph is el1 and y graph is the00:37:20
x axis an-200:37:24
this is the curve given like this y is equal to the curly00:37:28
bracket00:37:29
e to the power of minus00:37:32
one divided by x square this is if x is00:37:36
not equal to zero and 0 if x is equal to zero00:37:44
check for yourself that the order of tangency of these00:37:48
curves is in point again. m0 is the same00:37:51
with coordinates 00, this other function00:37:55
at x is equal to zero is equal to zero so it does not00:37:57
pass through this point, you will see for yourself00:38:00
that the order of tangency is higher than any, and00:38:03
that is, the order of tangency of these curves at00:38:06
point 0 is equal to infinity, well, here00:38:11
you need to calculate simple limits such00:38:13
as you calculated in the first semester,00:38:17
well, usually00:38:19
further, well, I told you the theorem,00:38:24
there is a theorem that allows you00:38:27
to determine the order of sandy based not on00:38:30
the definition,00:38:31
but on the basis of this: if we have functions00:38:33
n plus one time differentiable f1 from00:38:37
xf-2 tx, that is, they have a derivative din00:38:39
coward of the first order00:38:40
if the functions themselves and all their derivatives of00:38:44
long order at the point x0 coincide and the00:38:48
derivative plus the first order are not00:38:51
equal at the point x0 then the order of tangency00:38:53
is n and vice versa if before the tangency is00:38:55
equal to in the butt domain derivatives00:38:58
they all coincide,00:38:59
well, this is a rhema in itself is of some00:39:03
interest but00:39:05
will not be used anywhere further so let’s go and00:39:06
won’t even consider such a00:39:09
short first paragraph00:39:11
now paragraph 2 envelope00:39:20
1 parametric will be written down in one00:39:25
word one parametric family of00:39:27
curves00:39:29
envelope 1 parametric family of00:39:32
curves00:39:37
this is the concept for us is more important than the00:39:41
theorem mentioned above because the00:39:44
concept of an envelope00:39:46
will be used exactly one year later00:39:51
in the spring of your second year you will take a00:39:53
course on differential equations and00:39:56
there the concept of an envelope will arise in connection00:40:00
with the so-called special solutions of a00:40:02
differential equation in general everything00:40:06
that we are doing now is mathematical00:40:08
analysis We guess what are the properties of functions of00:40:10
different types, integrals, different formulas, all00:40:14
this is a certain mathematical apparatus that00:40:17
will be used in courses but already00:40:19
directly adjacent to physics in the00:40:22
course of differential equations in the course of00:40:24
methods of mathematical physics in the course of00:40:26
mathematical modeling what00:40:28
will happen next so the envelope for a00:40:31
parametric family of curves00:40:33
before to talk about the envelope itself, we00:40:35
will talk about the so-called special points of00:40:37
curves, which means there will be 2 points in our00:40:41
paragraphs, the first point is special points00:40:44
of curves, special points of curves,00:40:51
well, first of all, let's indicate what are00:40:58
the ways to define a curve on a plane,00:41:01
everywhere we will talk about plane curves, we wo00:41:05
n’t talk about them every time to say that we00:41:07
are on a plane means a plane00:41:11
curve a plane curve can be defined in00:41:15
three ways a plane curve that is a00:41:20
curve on a plane well everything is just a00:41:24
plane curve on a plane about x and y that is00:41:27
we will consider a curve in a00:41:31
given rectangular coordinate system a00:41:33
plane curve00:41:34
on the x plane The game can be00:41:36
specified in three ways: the first method00:41:42
and the simplest is an explicit assignment; an00:41:47
explicit assignment means00:41:54
that the curve is specified by the equation y equals f00:42:00
from x or vice versa x equals f from y, that is,00:42:07
one of the coordinates of a point on the curve00:42:09
is specified as a function of the other coordinator,00:42:12
this is the most a simple and well-00:42:15
studied way of specifying for such00:42:17
curves, we can plot graphs there, find the00:42:19
extremum points of inflection, find and00:42:22
so on, the second implicit task, the implicit00:42:28
task is when the curve is given by00:42:34
an equation, write it like this and in the larger00:42:37
of x and y is equal to zero,00:42:39
from which y can be you can’t express it through x,00:42:43
and vice versa, x through y,00:42:48
well, for example,00:42:51
x square plus y square minus one is00:42:56
equal to zero,00:42:57
here’s our function f big from x y00:43:01
is x square plus y square minus is00:43:05
from here, of course,00:43:11
y can be expressed through x, but this is an expression00:43:14
ambiguous what a00:43:17
curve is geometrically at the same time00:43:19
this equation is well known that the00:43:21
circle has a radius of one and it has an00:43:23
upper semicircle and a lower one that there is00:43:26
no unambiguous expression here and00:43:28
finally the third way is the00:43:32
parametric definition of the curve00:43:38
parametric00:43:40
is when x and y are given as functions of00:43:45
some parameter t x is equal to fiat and y00:43:49
is equal to psy at&t00:43:51
well, you belong to someone00:43:54
I will designate that large to some00:43:57
interval that large and so the00:44:01
simplest and most convenient is an explicit task, but00:44:05
clearly the curve is given implicitly or00:44:09
parametric then it can have so-00:44:12
called singular points in the case of an explicit00:44:14
assignment, there are no special points, all00:44:16
points are equal and they are called00:44:18
ordinary and let’s say that means in the00:44:22
case of an implicit assignment of an implicit ip or a00:44:26
parametric assignment of a curve, in the case of an00:44:30
implicit or parametric yes, a00:44:35
curve may have special points, what00:44:41
are they now we will say but getting ahead of ourselves00:44:44
I'll go ahead and say that at all points we would00:44:47
subdivide in this case into two classes:00:44:49
special00:44:50
and ordinary, here's an ordinary one.00:44:54
now there is no need to write this yet, we will now00:44:57
write it in the neighborhood of which, if00:45:00
the curve is at the same time implicit or parametric,00:45:02
then in the neighborhood of an ordinary point the00:45:05
curve can be specified by an explicit equation,00:45:09
but if it is special. then it turns out that there00:45:12
may not be such a task, and so let us00:45:16
consider these two cases, let the00:45:19
curve be given implicitly by the equation f from x00:45:22
and y is equal to zero, let us denote the curve and let the00:45:28
Elsa curve be given implicitly by the equation f from00:45:31
x y is equal to zero and let . and by me with the00:45:40
coordinates x0 y0 belongs to the curve and00:45:44
that is, what does this mean, that is, the00:45:47
coordinates of this point satisfy00:45:49
the equation, that is, f of x0 y0 is equal to zero00:45:55
and now let’s define what00:46:01
point we will be, yes, well, I didn’t say that,00:46:04
let’s really say that I say in the00:46:06
future, no matter what task interruptions00:46:09
we consider explicitly and implicitly or00:46:11
parametrically, we will assume that00:46:13
these functions that define the curve will not stipulate00:46:16
this every time, have00:46:18
continuous derivatives, do they have the right and00:46:20
derivatives, so now we will give a00:46:23
definition definition00:46:28
. and by me with coordinates x0 y000:46:34
belonging to the curve l is called special00:46:44
in brackets00:46:45
ordinary ordinary if if00:46:56
such a condition is met, we will00:47:01
call this point special if the00:47:06
partial derivative with respect to x00:47:08
we denote it simply and in large with respect to00:47:10
x with an index their environment without a prime00:47:12
because there we have degrees00:47:16
ef with respect to x squared from x0 y0 + f by y00:47:22
squared from x0 y0 equals zero in parentheses00:47:30
not equal to zero,00:47:31
that is, a point is called singular a point on a00:47:35
curve if at this point both partial00:47:39
derivatives with respect to x and y are equal zero00:47:42
because when the sum of squares is equal to00:47:43
zero00:47:44
when both terms are equal to zero, but00:47:47
if at least one is different from zero and00:47:49
thus the sum of squares is not equal to zero then.00:47:52
It's called ordinary, well now00:47:54
we'll take a break00:47:55
and after the break we'll show what if.00:47:57
ordinary00:47:58
then in the vicinity of this point from00:48:01
such an implicit task you can go to an00:48:05
explicit simpler one. The00:48:07
first thing we will now show is what00:48:09
if. ordinary00:48:11
then in its neighborhood a curve can be specified by an00:48:15
explicit equation, which means initially we have a00:48:18
curve defined by an implicit equation00:48:21
with the equation f of x y equal to zero and so00:48:24
let m 0 let m0 with coordinates x0 y000:48:34
adjacent to the curve be an ordinary spruce.00:48:38
ordinary which means then the sum of the00:48:43
squares of the derivatives at this point is not00:48:45
equal to zero, which means at least one of the partial00:48:47
derivatives is not equal to zero, even if, for example,00:48:50
in y at the point x0 y0 is not equal to zero, and00:48:57
then look for yourself. or rather,00:49:02
the coordinates satisfy the equation00:49:06
since the curve or point lies on the00:49:08
curve f of x0 y0 is equal to zero and the00:49:11
derivative with respect to y is not equal to zero but this is00:49:15
how we know let's remember this is the00:49:17
condition of the implicit function theorem00:49:19
then then by the implicit function theorem00:49:25
then by the theorem they are explicit functions in00:49:31
some neighborhoods of the point and by me then,00:49:35
by the theorem, implicit functions in some00:49:39
neighborhoods of the point and by me00:49:43
the equation f of x and y is equal to zero00:49:51
has a unique solution relative to00:49:54
y has a unique solution00:50:00
relative to y let us denote it as we00:50:04
denoted earlier y is equal to iv small of x00:50:07
and so under these conditions00:50:13
that f from x0 y0 is equal to 0 in y is not equal to00:50:16
zero this condition is the implicit function theorem00:50:19
according to the neural function theorem inside around the00:50:21
point 0 this is the equation to have a00:50:24
solution relative y and00:50:26
thereby the curve el in this neighborhoods and00:50:30
thus the curves in this kristen can be00:50:33
specified by an explicit equation y equals f of x00:50:37
can be specified by an explicit equation y equals iv00:50:41
small of x00:50:47
well, let’s remember something else; the function00:50:51
f small of x is differentiable and its00:50:56
derivative f prime of x is expressed by i Let me00:51:00
remind you with this formula minus the fraction in00:51:04
the numerator00:51:05
f large and in x from x and y in the denominator00:51:10
and in the large in y from x and y, provided00:51:16
that instead of y you need to substitute rf small00:51:19
tanks00:51:20
formula-1 so if . and I’m00:51:25
ordinary, then in the vicinity of this point00:51:30
here it is necessary 2 at two points to distinguish such a00:51:33
function exists,00:51:35
whether we can find it or not is a completely00:51:37
different question, sometimes it can be found in00:51:39
explicit form, there is no one, but such a function00:51:41
exists which gives an explicit00:51:44
equation for our curve and the derivative00:51:48
of this function is expressed by formula 100:51:51
if . im 0 special00:51:54
if . im 0 is special then in the neighborhood of00:52:02
this point the curve may not have an explicit00:52:06
equation. Now we will give a simple00:52:08
example if . im 0 special then in the vicinity of00:52:12
this point the curve may not have an explicit00:52:15
equation00:52:19
let's consider such a simple00:52:22
equation x square minus y square00:52:26
equals zero00:52:27
night function f large of x and y in our00:52:32
example this is x square minus y square00:52:35
as a point and I have this we have00:52:39
an equation of some curve hey,00:52:43
implicitly as point 0, let’s take point00:52:46
zero zero, it’s clear that it belongs to00:52:49
our curve and00:52:51
but now let’s look carefully00:52:53
at the equation again and what it actually00:52:55
defines y square equals x square00:52:59
here y equals either plus x or minus x, that is,00:53:03
the curve or represents two straight lines,00:53:06
let's draw them, so here are the very00:53:11
coordinates x y, here is the straight line, the game is equal to x00:53:15
and here is the straight line, y is equal to minus x,00:53:22
that is, our curve and well, we use the00:53:25
term curve, in fact, this curve00:53:28
consists of, there is a combination of two here These straight00:53:31
lines now look at what memory the00:53:34
neighborhood of the origin was not taken00:53:37
from us now. im 0 is the origin of coordinates00:53:39
here each x corresponds to 200:53:42
players and each player 2x and that is, there is no00:53:49
such equation y equals f from x00:53:52
f small from x which would describe both00:53:56
lines, that is, it is obvious that in the00:54:00
vicinity of point 0 with coordinates 0000:54:03
our curve n consisting here the zealous ones do not00:54:07
have an explicit equation,00:54:13
well now consider the case when the curve is00:54:19
at the same time parametrically00:54:21
night let now the curve l is given00:54:26
parametrically let the curve or now and00:54:31
we have formulated what is special. for00:54:33
this curve or not, he already lost00:54:38
memory at the end of the first hour or we haven’t00:54:41
gotten to that, so let’s do it again,00:54:44
the body of the curve at the same time parametrically x00:54:46
equals fiat and y equals psy from and let .00:54:56
m0 with coordinates feat t 000:55:01
psy from other00:55:06
ordinary .00:55:07
that is, that is, the sum of the squares of00:55:14
the derivatives fish 3 x squared of t 000:55:17
plus psyche 3 x squared of 0 is not equal to00:55:22
zero00:55:24
special. this is when the sum of squares is equal to00:55:28
zero and thus both derivatives and fish00:55:32
tereh ipsy stroke at point t0 are equal to zero and00:55:34
let go. ordinary we will now see00:55:37
very simply and that then in00:55:40
the vicinity of this point m 0 the curve00:55:43
can again be specified by an explicit equation means00:55:46
the sum of squares is not equal to zero here at00:55:48
least which of the terms is not equal to00:55:50
zero let for example let for example fi the00:55:53
stroke at point t0 is00:55:55
not equal to zero00:56:01
then then the00:56:04
fi prime from t is not equal to zero and retains00:56:10
the sign and preserves the sign well in some00:56:16
neighborhood of the point t0, well, we told00:56:22
you that we assume that the functions00:56:24
that define our curves are incorrectly00:56:27
differentiable, that is, they have00:56:28
continuous derivatives, which means if the fi00:56:30
prime is at the point then but not is equal to zero,00:56:33
then we remember this property: the stability of the00:56:36
sign of a continuous function means that the prime00:56:38
from t will be different from zero and will00:56:42
retain the same sign.00:56:58
monotonically00:57:01
in this neighborhood of the point t0 in this00:57:05
neighborhood of the point t0 and that means it has an00:57:11
inverse function and that means it has and00:57:15
let out then there is an inverse00:57:18
function00:57:21
you are equal to fi to minus 1 of x and so in this00:57:29
case the function x is equal to fiat strictly00:57:32
monotonically and that means there is an inverse00:57:35
function such theorem we were in the first00:57:36
semester substituting substituting this00:57:40
expression for t into the equality00:57:43
y equals psy from and here we have the curve for00:57:50
one equation mixer again from y equals 500:57:52
substituting this expression for t into the00:57:54
equality to good at&t we get y00:57:58
equals psy pot fi in minus 1 from x well, and this00:58:04
is let's denote this function00:58:07
by iv small from x00:58:11
and thus and thus in the00:58:16
nectar neighborhood of the point by me and00:58:20
thus in some pieces m0 curve n00:58:23
has an explicit assignment thus not in the00:58:27
habit of me curve n has explicit00:58:32
tasks if . and me with coordinates00:58:40
feat you zero all pull special special00:58:48
that is, both derivative fish 3 at point00:58:54
t0 ipsy stroke at point t0 is equal to zero00:59:01
then in the vicinity of point m 0 then in the00:59:06
vicinity of point m 0 the curve may not00:59:10
have an explicit definition, it may or may00:59:13
not have but claim that it has we00:59:15
cannot cross point m 0 the curve may00:59:19
not have an explicit definition, well, we could00:59:22
give a lot of all sorts of examples here,00:59:24
let’s save time, we won’t give an example,00:59:26
so now we know what00:59:28
special ordinary points are00:59:29
if the curve is given implicitly or00:59:33
parametric then the entire set of points on00:59:36
the curve is divided into two classes:00:59:38
ordinary and special special. There may00:59:41
not be, but there may be a special one.00:59:44
differs from an ordinary one in that in the00:59:46
neighborhood of an ordinary point there is an00:59:50
explicit assignment for this very curve,00:59:52
but in the neighborhood of a special one there may not be,00:59:56
but now let’s move on to the main goal of00:59:58
this paragraph, paragraph 22, envelope of a01:00:02
family of curves,01:00:05
envelopes of a family of curves, well, in the title01:00:09
of the paragraph we said envelopes 101:00:11
parametric family of curves, so01:00:14
sometimes we will omit the words 1 parametric,01:00:16
although we will consider exactly01:00:18
1 parametric family of curves, what is this01:00:21
now, we will clarify with you,01:00:25
yes, I was in a hurry, when we01:00:32
said that in the vicinity of an ordinary01:00:35
point of a curve given parametrically01:00:38
there is an explicit task and we got01:00:42
this the very formula01:00:43
x is equal to psy01:00:47
from phi minus 1 from x which denoted01:00:52
f from x additions01:00:55
after that f prime from x01:00:59
how the derivative of a function01:01:02
given parametrically is calculated we will need01:01:04
this formula I will remind you01:01:06
this fraction numerator psy prime from t01:01:10
denominator fi prime from t01:01:13
where instead of t we need to substitute this01:01:16
very inverse function where t is equal to fi01:01:22
minus 1 of x01:01:24
let's number this puddle form 201:01:27
we will also need it and now let's01:01:33
move on to the envelope let's look at01:01:36
the equation and in large01:01:40
from three variables x and y and01:01:44
denote the third variable the letter a is equal to zero,01:01:47
equation 301:01:50
where a variable changes over a certain01:01:54
interval, it could be the entire number01:01:58
line, it could be some interval,01:02:03
but this is not three intervals, let let01:02:07
for any value a, equation 301:02:14
clearly gives me some curve, if we01:02:18
fix it, then this is already an implicit equation01:02:21
with two variables x y,01:02:23
so let for any equation 301:02:27
implicitly define a certain curve,01:02:34
the set of these means by changing a we will01:02:37
get different curves, the set of these01:02:39
curves is called 1 parametric01:02:42
family of curves, the set of all these01:02:46
curves is called 1 parametric01:02:50
family of curves, variable a is01:02:54
called parameter a, equation 301:03:00
equations of a family of curves, a variable01:03:05
is called a parameter01:03:06
and equation 3 equations of a family of01:03:10
curves let's look at a simple01:03:13
example 1 of a parametric family of01:03:15
curves consider such an equation example01:03:21
y minus in brackets01:03:24
x minus a squared equals zero our01:03:29
function f large of x y01:03:32
and in this our example it's this one01:03:36
function y minus x minus a squared but01:03:39
this equation can be like this y equals x01:03:41
minus a squared01:03:43
what kind of curve is this parabola which,01:03:49
depending on a, crawls as if along the01:03:52
x axis here are the curves of this family01:03:56
let's draw the coordinate axes x y here01:04:00
if a is equal zero then this y is equal to x01:04:04
square it’s just such a well-01:04:06
known school time it was a equal to01:04:11
zero if a is more than 0 well, it was time to01:04:15
move somewhere to the right and more than 0 if a is01:04:20
less than zero then to the left well and I’m all that01:04:27
family of all such a parabola changes01:04:30
from minus to plus infinity, it01:04:32
fills the entire upper half-plane,01:04:34
well, let’s pay attention to the fact that all01:04:38
these parabolas have a common tangent axis01:04:42
x, and so let’s note that01:04:45
all these parabolas touch the x axis, at01:04:52
their vertices, there’s such a line,01:04:56
such a curve, but in this case it01:04:58
we will now call a straight line an envelope01:05:02
definition definition01:05:07
means a curve a curve that at each of01:05:14
its points01:05:20
touches and, moreover, only one curve of the01:05:24
family01:05:25
and at different points touches various01:05:32
curves of the family us a curve which at01:05:37
each of its points touches and, moreover,01:05:39
only one curve family and at01:05:41
different points touches different01:05:43
curves of the family is called the envelope of01:05:46
this family of curves01:05:49
is called the envelope of this family of01:05:51
curves01:05:56
well, based on this definition, we01:05:58
can say in the example considered the01:06:01
x axis is the straight line or which01:06:05
is given by the equation y is equal to zero this is the01:06:08
envelope 1 of the parametric family of01:06:11
parabolas here in this example, the x axis,01:06:15
that is, the curve is actually a straight line, the01:06:18
x axis is the envelope of 101:06:23
parametric family of a parabola,01:06:29
well, now our goal will be01:06:32
this, yes, but specifically and the equations,01:06:34
of course, in this simple example01:06:37
we can immediately see that each at each and this01:06:39
pair was here we drew them, we see01:06:42
that this family has an envelope, but01:06:45
if the level and more complex one01:06:47
is asked how to find the envelope,01:06:49
then this is our task: we will derive01:06:52
the necessary conditions for the envelope, we will derive the necessary01:06:57
conditions for the envelope of the family of01:07:00
curves given by equation 3, we will01:07:02
simply say families of curves 301:07:04
we will derive necessary condition for the envelope of a01:07:08
family of curves 3 now look if01:07:15
the necessary conditions for the envelope are what is01:07:19
given and what must be deduced once the necessary01:07:23
condition means there are envelopes, let one01:07:26
now we write like this let the family01:07:28
3 have envelopes from here something follows01:07:31
with necessity this will be the01:07:32
necessary condition and so let the01:07:35
family of curves 3 have an envelope01:07:38
let the family of curves 3 have an envelope let’s01:07:46
denote it with the letter r big well and so I01:07:55
’ll draw let this be the01:07:57
envelope of the family of curves some01:08:00
curve or01:08:02
consider an arbitrary point m with01:08:07
coordinates x y its envelope01:08:11
consider an arbitrary point m with01:08:14
coordinates with coordinates x y of its envelope, but what01:08:20
happens at this point is the spinning of the01:08:23
envelope at this point the envelope01:08:26
touches and, moreover, only one curve of the01:08:28
family, so let’s01:08:29
write at this point the envelope is afraid and,01:08:33
moreover, I’ll draw only one curve for the family,01:08:36
let this be the very curve of the01:08:38
family that touches the envelope at01:08:44
the point and as for, we know it01:08:48
was in the first paragraph, it means at this01:08:49
point they have a common tangent,01:08:51
so we took an arbitrary point on the01:08:56
envelope at this point the envelopes touch, I will01:08:59
come only for whom and the01:09:00
family, in turn, this curve of the01:09:03
family corresponds to neither that01:09:05
value parameter a, in turn, this01:09:09
curve of the family corresponds or01:09:12
corresponds to either value of the parameter a,01:09:19
while different values of a01:09:24
correspond to different points of the envelope,01:09:28
while different values of the sail by01:09:31
definition of the envelope, while different01:09:33
values of a correspond to different01:09:36
points of the envelope01:09:37
and therefore the x and y coordinates of the point m01:09:44
are functions of the parameter a and01:09:47
therefore the x and y coordinates of the point m01:09:51
for each point have their own parameter value01:09:55
and vice versa each parameter value has01:09:56
its own point on the envelope and therefore the01:09:59
x and y coordinates of the point m01:10:01
are functions of the parameter let’s denote01:10:04
these functions as x equals feat a y equals01:10:10
psy from well, these are the parametric01:10:19
equations of the envelope, these are the parametric01:10:23
equations of the envelope, and01:10:30
well, now, as the necessary01:10:33
conditions that we mentioned, we will derive01:10:36
and write, we will derive the equations01:10:39
that these functions satisfy, we01:10:43
will derive the equation that01:10:46
these functions satisfy since since since01:10:49
. m let's start with x and y coordinates, we01:10:52
designated feat obse and then since . m with01:10:56
coordinates feat obse and that means01:11:03
since .01:11:05
m with coordinates fiat and everything from a01:11:12
also belongs to this point on the envelope01:11:16
and on the other side we see this is a point on the01:11:18
curve of the family on which curve the01:11:19
corresponding parameter y a since .01:11:22
they also belong to the curve of the family the01:11:30
curve of the family01:11:36
the idea coordinates satisfy01:11:40
equation 3 according to its coordinates01:11:43
satisfy equation 3 that is,01:11:46
this equality is true and in large01:11:48
from fiat01:11:50
a psy from a a is equal to zero01:11:56
equality 4 for which a is replenished01:12:02
equal to y-4 for any we took01:12:05
any point on the envelope it01:12:07
corresponds to some value a and01:12:09
vice versa, each value corresponds to its01:12:12
own a. on the envelope so that was born 401:12:16
is an identity with respect to the01:12:18
variable01:12:19
equality 4 is an identity with respect to01:12:22
the variable01:12:27
we differentiate this identity by a01:12:30
variable we differentiate and01:12:32
triumph variable well we must01:12:34
differentiate as a complex function and in01:12:37
large depends on x01:12:38
and x is fiat from y and y from obse and even01:12:42
from and therefore when we take01:12:44
derivatives it will turn out like this fpx from01:12:49
all these arguments, for the sake of brevity I01:12:51
will not write them in fig tree had a + f by01:12:57
y on psy barcode a + f by a all this is01:13:06
taken instead of x we need to substitute feat a01:13:09
and instead of y we need to substitute psy from a this is equal to01:13:14
zero01:13:16
rules 105 when differentiated by01:13:19
and now we have not yet used anywhere01:13:23
the fact that the01:13:26
envelopes and the curve of the family touch at01:13:29
this point now we will use this01:13:32
since since01:13:34
the envelopes and the curve families touch at01:13:37
point m with coordinates fiat and psy and then01:13:41
since the envelope and curve of the family01:13:45
touch at point m with coordinates feat01:13:49
obse from and then they have a common tangent at this point,01:13:58
simply by definition a tangency then they01:14:01
have a common tangent at this point01:14:07
researcher on equal angular01:14:10
coefficients of the tangents and therefore01:14:16
equal angular coefficients of the tangents01:14:18
and what is the angular coefficient of the01:14:23
tangent, if the curve is at the same time an explicit01:14:26
equation y is equal to f of x then the angular01:14:31
coefficient of the tangent is f the stroke of x is the01:14:33
tangent of the angle of inclination of the tangent, it is01:14:35
not by chance that we wrote out the formulas01:14:38
for the derivatives of f stroke from x the case01:14:43
when the curve is at the same time parametric and01:14:45
this formula 2 and when the curve for for the01:14:47
implicit equation equating the01:14:50
angular coefficients of the tangents for the01:14:53
envelope and01:14:55
and for the family curve, equating the01:14:58
angular coefficients of the tangents01:15:01
for the envelope and for the family curve at01:15:05
point m with the coordinates feat all and then01:15:08
we get equality, we get the equality, well,01:15:13
I’ll write first for the envelope, the envelope is01:15:17
given to us, but with these equations x01:15:20
equals fiat y equals psy, which means psy01:15:23
stroke from and on the filter move and this is01:15:29
apply formula 2 equals apply01:15:34
formula 1 to agree on the coefficient of the01:15:36
family curve f minus / fpx a01:15:45
x y a knife by y from x y01:15:53
and this should be taken with x equal feat a y01:15:59
equal to psy from and this is we equated the01:16:06
angular coefficients of the tangents01:16:08
tangent to the 1st floor, so the angular01:16:11
coefficients are calculated differently using01:16:14
this formula on the left for the envelope on01:16:17
the right for the family curve01:16:19
from here follows from here it follows01:16:22
from there to alternate to and such an equality01:16:25
fpx01:16:26
on fig tree had a + f by y on psy01:16:32
barcode a is equal to zero, well, let's01:16:38
write in more detail with x equal feat01:16:41
a y equal everything from a is equal to zero, which means01:16:48
we have some kind the number there was 4501:16:50
number 601:16:51
and now let's look at equality 5 here are the01:16:57
first two terms here fpx on01:17:00
the filter had a + f in y for the entire stroke then01:17:03
with x equal fiat y is equal to all then this is01:17:06
exactly what is on the left of equality 601:17:09
means due to 601:17:11
filu 6 equality 5 takes the form filu 601:17:16
equality 5 takes the form01:17:24
our mind and the first two terms are equal to01:17:27
zero,01:17:28
then this remaining third is equal to01:17:31
zero01:17:32
equality 6 takes the form f from fiat01:17:37
a psy from a01:17:40
a is equal to zero rules 107 well, now01:17:49
let's summarize what we got in01:17:51
this way in this way if01:17:55
the family of curves 3 here we had01:17:58
one equation 3 thus if the01:18:01
family of curves 3 has an envelope has an01:18:09
envelope01:18:10
then the function x is equal to fiat y is equal to psy from and01:18:14
then the function and which fiat y is equal to psy and01:18:18
then the ones that01:18:19
define this envelope satisfy the01:18:22
equality 4 and 7,01:18:24
so if the family of curves 301:18:30
has an envelope, then the functions x equals fiat01:18:33
y equals the entire one that defines this envelope is01:18:35
removed by the equality 47, that is, these01:18:41
functions are the solution to the system01:18:45
of equations, that is, these functions are the01:18:49
solution to the system of equations01:18:55
f the largest of x y a is equal to zero and vpo and the01:19:03
partial derivatives with respect to x and y a are equal to01:19:08
zero01:19:09
system of equations 8 these are the01:19:14
necessary conditions for the envelope01:19:17
this is the necessary condition for the envelope01:19:21
means that we have deduced that if there01:19:24
is an envelope then the functions x equals fiat y01:19:28
equals xy otherwise they are a solution to a system of01:19:30
equations of 8 functions describing the01:19:32
envelope, what is the conclusion01:19:34
from this? I with system 8 does not01:19:38
have a solution regarding x y what01:19:41
could it be if system 8 does not have a solution for color01:19:44
x y01:19:45
then family 3 does not have an envelope then01:19:52
family 3 does not have an envelope because I’m01:19:56
afraid of dogs I was the same system01:20:01
80 necessary would have solutions and if it01:20:04
has a solution and still fiat and Herka,01:20:06
actually we cannot say that this01:20:08
will necessarily be an envelope,01:20:10
so we will write like this if system 801:20:13
has a solution I serve system 8 has a01:20:17
solution x equals feat a y is equal to psy and01:20:27
then this solution may not be an01:20:30
envelope, it may or may not be then01:20:33
this solution may not be an envelope.01:20:35
Now we will explain why,01:20:45
well, in fact, in fact, how we01:20:49
got this last second01:20:51
equation and exactly we01:20:53
used zero with the equality 6 a is equal to01:20:56
dry we wrote based on the equality of the01:20:58
angular coefficients in fact equal to01:21:01
106 is also satisfied in that case01:21:04
equal to dry is also satisfied in the01:21:07
case when either and in by y under feat a01:21:16
psy from a a is equal to fpx01:21:23
from feat a seat01:21:27
a a is equal to zero when both parts come01:21:32
actually look at before 601:21:34
if fpx with x equal to 5 grams of satay by01:21:38
y are equal to zero the crowd of terms is equal to zero01:21:41
either or fig tree at a is equal to psy stroke01:21:51
from t is equal to zero in this case also01:21:57
each term is equal to zero, the sum is equal to01:22:00
zero and what does this mean in the first case01:22:04
in the first case.01:22:07
m with coordinates feat a seat and in the first01:22:12
case.01:22:15
m with coordinates fed up is the singular01:22:17
point 7 of the singular point of the family of01:22:20
singular curves. family of curves, each01:22:25
curve is at the same time an implicit equation,01:22:28
well, not even a family of curves, a singular01:22:30
point of the curve of the family along and the curve01:22:33
that corresponds to the parameter, which means in the01:22:35
first case. and being a point of the01:22:38
family curve in the second case special01:22:41
. gebo is a singular point of the curve01:22:46
given by the equations x is equal to fiat obse01:22:49
from but we assumed that01:22:52
there is a solution but it may not be an envelope, we01:22:56
explain why because01:22:58
106 is satisfied not only in the case01:23:02
when the envelopes of the curve of the01:23:04
family touch, but also in the case when either01:23:06
this or that In the first case . it01:23:10
is a special point of the curve of the family01:23:12
in the second case, a special point of the curve that01:23:15
is described by these equations01:23:18
anyway fiat y is equal to psy and01:23:21
therefore, considering system 7 system 801:23:26
we and having received what theory of solution we01:23:30
cannot state01:23:33
a priori as they say that this will01:23:35
necessarily be an envelope therefore here01:23:37
are special names for the solution of the01:23:40
solution x is equal to fiat y is equal to psy and then01:23:43
the system 8 is called the discriminant of the01:23:47
curve of the family of curves 301:23:49
the solution x is equal to the fiat game all and then01:23:55
the system 8 is called the discriminant of the01:23:59
curve of the entire family of curves 301:24:04
thus the discriminant on the i curve01:24:07
thus the discriminant on i the curve01:24:11
can be either an envelope01:24:21
or a set of singular points01:24:25
so the discriminant on i the curve can01:24:27
be that is, the solution to system 8 can01:24:30
be either an envelope or a set of01:24:34
singular points01:24:35
or partly one partly another or partly one01:24:39
partly another now we will01:24:42
look at such an example very such a01:24:44
vivid illustrative example, and so01:24:48
what is the general conclusion, it no longer needs to01:24:50
be written down that if we have01:24:52
families of curves that are given by01:24:53
the equation f of x and y a is equal to zero for01:24:56
each a, some curve of01:24:58
the family is obtained, then in order to find the envelope we need to01:25:01
consider a system of equations 8 if01:25:05
it has no solutions regarding x y, then there is01:25:07
no envelope, and if there is a01:25:09
solution, then we still need to explore it further;01:25:12
it may be an envelope, it may be a01:25:14
set of singular points, or perhaps there is01:25:16
more than one solution, some gives an envelope of01:25:19
some set of singular points this is what01:25:21
we will now see in an example example let's01:25:25
look at the following equation x minus a01:25:30
cubed minus y minus a squared is01:25:35
equal to zero,01:25:37
which means our function f is large from x01:25:40
y01:25:41
and this is such a function for each and we01:25:47
get some kind of curve on the x01:25:49
y plane look, if we give, for example, let's put it01:25:53
equal to zero, x cube01:25:55
minus y square is equal to zero, which means that01:25:57
with a equal to zero, we get y square01:26:02
equals x cube01:26:04
and that means y is equal to plus or minus x to the01:26:09
power of three, the second01:26:12
two curves y plus x minus, well, here I am01:26:18
here it costs me three second and a square then01:26:21
it’s a parabola and sometimes say01:26:24
quadratic parabola if a cube they say01:26:26
cubic parabola01:26:27
and from three second how would you call this01:26:29
curve semi cubic parabola01:26:32
so they say this is the family of semi01:26:34
cubic according to the slaves of the family of semi01:26:37
cubic parabolas now a little bit01:26:39
after a couple of minutes of my drawing, let's01:26:42
add cancer to this, we'll look for the01:26:44
envelope of the family, we'll add to this01:26:46
equation, here in system 8 there is also01:26:50
the equation ef by, we calculate the derivative of ef by, which01:26:53
means ef by is equal to the derivative by, here01:26:59
from the first term will be minus 3x01:27:03
minus a squared and I hope01:27:08
We all know how to calculate derivatives, so I don’t01:27:09
explain why there is a minus about this01:27:11
minus plus 2y minus a equals zero,01:27:18
well, system 8 is such an equation and01:27:22
it can be solved very simply, from here01:27:26
y minus and this will be the second three x minus and01:27:28
in the square we will substitute the system here easy01:27:32
to solve means from this system I’ll01:27:36
write right away we get two solutions 1 x01:27:40
equals y equals a let’s denote this solution01:27:43
el1 and the second solution01:27:48
x equals a + 4 ninths y equals a +01:27:56
820 sevenths let’s denote this solution01:28:00
l2 check it yourself it’s very easy to solve and01:28:06
you you can easily verify that this is what the01:28:09
first solution is,01:28:11
but it can be written in the form or one01:28:15
is a direct game equals x01:28:19
a or 201:28:25
is a straight line if we exclude the parameter from01:28:28
these two equations01:28:29
we get the game equals x minus01:28:33
check it yourself -4 20 sevenths x -fi01:28:38
protection 27 and so the discriminant on i the curve01:28:42
of this family of curves01:28:45
represents two straight lines and parallel01:28:49
straight lines01:28:51
but at the points of the curve or 1, or better to01:29:00
say on the curve or one on or one01:29:05
what we have look x is equal to y is equal to a01:29:09
and the derivative f with respect to x what is equal to 3x minus a01:29:14
squared so that it is equal to zero for01:29:17
x equal to a01:29:18
and the derivative of f with respect to y and what is equal to minus01:29:21
2y minus and it is equal to zero for y equal to and01:29:25
so on the curve or 1f with respect to x equals f with respect to01:29:33
y equal zero01:29:36
that is, a curve or one or a straight line in01:29:40
fact or one is the set of singular01:29:43
points of the curves of the family this is the set of01:29:46
singular points of the curves of the family and the curve01:29:49
r2 is an envelope now we will draw01:29:52
a picture it will be clearly visible and the01:29:54
straight line or 2 is an envelope but this is01:29:58
what the picture means we draw the coordinate axes01:30:07
here is a straight line or one game equals x here is a01:30:15
straight line l 2 parallel to it01:30:18
y equals x minus 4 20 sevenths it is01:30:21
lowered if you look along the y axis it is01:30:25
lowered by 420 sevenths in relation to the01:30:28
straight line or 101:30:30
and if we take let’s say a equals01:30:33
zero here this curve y is equal to plus and to the01:30:41
power of three second y is equal to minus it01:30:45
will be like this curve that comes out01:30:48
of the origin of coordinates01:30:51
touches somewhere on a curve or two and goes01:30:56
up like x to the power of three second a and y01:31:00
is equal to minus x to the power of three second that’s01:31:03
symmetrical goes down if we take01:31:07
any other point, then we need this very same01:31:12
semi-cubic parabola with01:31:15
such a beak, you can01:31:17
move it along, so to speak, or 1 means it will01:31:20
behave like this, here it will touch somewhere here, so01:31:24
I will highlight these points of01:31:28
tangency of the01:31:29
straight line curve m2 well, here it will go down01:31:32
and so on, so you can move this semi-01:31:36
cubic parabola, the positive branch01:31:38
and deny love, move it01:31:41
parallel to itself so that01:31:43
the tip moves along or one,01:31:45
then we will depict the whole family of these01:31:48
semi-cubic parabolas and now we see01:31:50
what is really and here01:31:55
the points of tangency that I have highlighted, that is,01:31:57
they barely touch all these curves and or one is the01:32:01
geometric locus of singular points, well,01:32:06
everything is bending, the only thing is like what I01:32:10
already said at the beginning, this concept will be01:32:12
used in the theory of differential01:32:13
equations01:32:15
when you study ordinary equations in the second year01:32:18
differential01:32:19
equations then it turns out that let's say if the01:32:22
simplest differential equation of the01:32:24
first order01:32:25
is an equation of this type yes and flu dx01:32:28
yes and flu dx is equal to f of x and y01:32:33
this equation is an infinite set of01:32:36
solutions for things from an arbitrary01:32:38
constant that is, solutions are a01:32:40
general solution as they say there will be01:32:43
some function nu fi01:32:46
of x and of parameter a, usually it is01:32:51
denoted in the theory of the species of creatures by the letter01:32:53
c, but I will write a because here we have01:32:55
the designation a and this is nothing more than01:32:59
equation 1 of a parametric family of01:33:01
curves for their thing from the parameter and01:33:04
this family may have an envelope and01:33:06
this envelope is the so-called01:33:09
special solution of this differential01:33:11
equation, well, in detail about this in the course of01:33:13
differential equations, we have01:33:16
one short paragraph left, I think in about01:33:19
20 minutes we must meet the curvature of01:33:23
the curve, let's agree or01:33:25
read it yourself about the curvature of the curve, or01:33:29
let's take 20 minutes, what will we spend, we01:33:32
can take a break now for a couple of01:33:33
minutes, we can not do what we'll do,01:33:36
tell us,01:33:39
the nods are basically like this, okay,01:33:42
shall we take a break or not,01:33:45
okay then paragraph 3, the01:33:47
curvature of a flat curve, paragraph 3, the01:33:50
curvature of a flat curve well, everyone has some sort of01:33:55
initial intuitive idea of01:33:58
curvature, well, there are straight01:34:01
lines, there are curved lines, let’s01:34:04
draw a certain curved line, look at the01:34:07
curved line, that’s how I01:34:11
’ll draw it01:34:12
and highlight two sections of the same length on it,01:34:15
label them 1 and 2 so on the curve01:34:24
we denote the curve el large,01:34:27
two sections of the same length 1 and 2 are highlighted,01:34:33
which we can say01:34:35
based on visual representations, but what is01:34:39
intuitively clear is that the01:34:41
curvature of section two is greater than01:34:44
section one, so it is intuitively clear01:34:47
that the curvature of section two is greater01:34:50
than section 1,01:34:56
well here is our task and how to quantitatively01:34:58
describe curvature to quantitatively01:35:01
describe the fact that the now curved01:35:03
co2 is more than section one, this01:35:05
quantitative measure is curved steam and01:35:07
a little later we’ll call it curvature pretty01:35:09
quickly we’ll call it curvature let’s look at the01:35:12
curve and look at the curve hey well that’s01:35:16
how I and let us01:35:19
depict a curve, at each point of01:35:23
which there is a tangent and at each01:35:27
point of which there is a tangent,01:35:34
we will consider at each point a01:35:36
directed tangent, we will01:35:43
consider at each point a01:35:44
directed tangent, a conscience01:35:46
directed as the term was already01:35:49
used, the direction of the tangent, we will choose the direction of the tangent01:35:52
corresponding to the direction of movement along the01:35:55
curve, we will choose the direction of the tangent01:35:58
corresponding to the direction of movement along the01:36:01
curve, we mark two points on the curve m 0 and01:36:07
m1, here we mark two points on the curve m 001:36:12
m 1, well, we will assume that we are moving from01:36:15
m0 com 1,01:36:17
here the directed tangent at the point in01:36:19
my direction corresponds to the movement01:36:24
from m0 com 101:36:26
and here is the directed tangent at point m01:36:29
1 but let's continue down so01:36:33
here a certain angle has formed between the01:36:36
angle at which the directed01:36:40
tangent turned at point 0 when we move01:36:42
only madden turns and now it01:36:45
turns at this angle let's denote the01:36:48
length of the01:36:50
sections of the curve m 0 m 1 so that this01:36:56
was perceived as a curve, so I’ll01:36:58
put the arc icon here, I’ll put the length of the section01:37:01
curve and 01 through the deltas, and01:37:04
this is the angle between the directed01:37:07
tangents01:37:08
from delta fe and it doesn’t matter which01:37:13
way it turned, we’ll assume that01:37:15
delta fit is greater than or equal to zero,01:37:20
sometimes delta fees are introduced, but we’re familiar it01:37:24
will be quite enough to consider it doesn’t matter01:37:25
in which direction to retell the delphi01:37:27
the larger one is equal to zero, now look01:37:29
from visual considerations it is clear that the01:37:34
more curvature the01:37:35
greater the angle of the curve, the greater the angle the01:37:40
tangent will turn and vice versa, the greater the01:37:43
angle the tangent will turn, the more it is01:37:46
bent if we say this the straight line01:37:48
then the tangent coincides with the straight line itself;01:37:50
it will not rotate through any angle at all;01:37:56
we will use these visual considerations as the basis for the definition of curvature;01:37:59
definition; definition of the01:38:06
average curvature; average curvature of the section of the curve m 0 m 1;01:38:11
average curvature of the section of the curve m 0 m 1; let's call01:38:17
the relationship delta fi to delta hey,01:38:26
and we will denote this average01:38:29
curvature with the letter k01:38:31
and below indicate which section of the curve k01:38:34
and below we write m 0 m 1 so the average01:38:39
curvature of the section curve many one01:38:41
we call this ratio of the delta fig01:38:43
del del we01:38:44
denote it to to many 1a by the curvature of the01:38:48
curve at a point and I, curvature is a01:38:54
point concept, and the curvature of blood at a01:38:58
point mole is the limit of this01:39:03
average curvature to m 0 m 1, provided01:39:09
that .01:39:10
m1 tends to point 0 and below01:39:15
we will also write m1 and belongs to01:39:17
l01:39:18
that is. and me. m1 tends to point01:39:21
m moving along the curve l01:39:23
and the curvature of the curve at point m 0 we will call01:39:26
this limit well and we will denote it01:39:29
like this from m0 to from and by me this is similar to01:39:38
how we introduced the concept of derivative,01:39:40
treating the derivative as a speed we01:39:42
first take the average relations delta y01:39:45
delta x01:39:46
if x is time and y is the coordinate01:39:50
we talked about delta y delta x is the average01:39:52
speed on the interval delta x01:39:55
and the limit the average speed is the instantaneous01:39:58
speed there and here there is the average01:40:00
curvature and the limit of the average curvature this01:40:02
curvature at a given point, well, let's consider01:40:05
simple examples simple examples 101:40:10
example01:40:16
first example let's01:40:19
take a straight line as a curve, let's say01:40:23
hey this is a straight line, here we take any01:40:27
point m 0 here we take the directional tangent at01:40:31
this point we take an arbitrary point m 101:40:34
here we take the directional tangent at this point it is01:40:37
absolutely clear that delta fi is equal to zero01:40:41
tangentially does not rotate in any way,01:40:44
which means the average curvature for any01:40:48
section of a straight line is zero and the curvature at01:40:53
any point is zero, but sometimes they say that a01:40:58
straight line has no curvature, well, in01:41:02
fact, it’s not entirely accurate curvature is a number, it’s01:41:04
just01:41:05
curvature as a mathematical concept,01:41:08
but it is equal to zero and so on it’s as if01:41:11
a commentator is counting a hockey01:41:13
match, they say the score is not open there, but01:41:17
if we draw an analogy when they say01:41:19
that a straight line has no curvature, well, it’s01:41:22
similar to the counting, well, why not, the count is 00,01:41:25
the count is 000, also its straight line has curvature,01:41:29
it is equal to zero01:41:30
second example, let's look at a01:41:34
circle of radius r large, here is01:41:38
a circle of radius r, take two points on it01:41:42
m 0 and01:41:45
m1, here is a directed tangent, we move01:41:51
from m0 com1 here is a directed tangent at point m01:41:53
0, here is a directed tangent01:41:57
at point m, one here is this angle delta fi01:42:01
and here is this angle it is not difficult to figure out that the01:42:05
angles corresponding to the perpendicular01:42:07
sides are also delphi01:42:09
and the length is m 0 m 1 g t l and that is, r on01:42:18
delta fi01:42:19
is true of delta fe, but you and I know this well01:42:23
from the planimetry course;01:42:26
this is the average curvature k m 0 m 1 equal to01:42:32
delta fi in delta r is equal to one on r,01:42:38
that is, the average curvature of any section of01:42:42
the circle is a constant value01:42:44
inversely proportional to the radius, and the01:42:47
curvature at any point, the limit of this01:42:51
constant is,01:42:52
of course, equal to this constant itself,01:42:55
that is, k at any point m 0 is equal to01:42:58
one on r-well,01:43:01
this is at some point then it’s clear to the best of our ability, we’ve taken01:43:04
a circle and now let’s increase the01:43:05
radius of the circle for more and more01:43:07
more and it’s absolutely clear that the larger01:43:11
the circle when we take01:43:13
any piece on it, the closer it is to a01:43:15
straight line01:43:16
with h tending to infinity,01:43:18
of course the curvature tends to zero01:43:21
third example 3 examples let's01:43:25
now take not a circle01:43:27
but an ellipse given by the canonical01:43:30
equation,01:43:36
so the third example curve n is an ellipse01:43:39
given by the equation x square by square01:43:43
plus y square by b square is equal to01:43:46
one let us have a greater than b01:43:50
semi-axis and greater than semi-axis b let's01:43:54
draw this ellipse that means the x y axes, well,01:44:01
I’ll take an ellipse whose a is01:44:05
two times larger than b,01:44:07
let’s mark the two vertices of this ellipse,01:44:10
here at point m 1 m1 and m201:44:17
the vertices of the ellipse lie on the01:44:21
coordinate axes. Because of intuitive considerations,01:44:24
it’s clear that curvature k at a point means01:44:29
this here we have a semi-axis and this is a semi-axis b01:44:33
that is attached to at point m 1, what sign01:44:38
is connected with the curvature k at point m 25 from01:44:42
visual considerations where the curvature is01:44:44
greater at which point and at point m 101:44:47
of course, that is, I will jump off, this01:44:50
inequality is asking, well how01:44:52
to prove this and you need to learn how to calculate01:44:55
curvature, now we will get a formula01:44:57
for curvature and so to prove this we01:45:00
strictly need a formula for curvature,01:45:03
we will consider two cases when the curve is01:45:06
at the same time an explicit equation and when the curve is01:45:09
given parametrically let the curve el is01:45:14
given by the equation y equals f of x01:45:18
let the curve el is given by the equation y01:45:23
equals f of x01:45:27
let's draw a picture of the x and y axes,01:45:33
well, let's even count here we'll01:45:37
need the second derivative01:45:38
it turns out the formula for curvature will include01:45:41
the second derivative let the function f of x01:45:43
be twice differentiable let the01:45:45
function f of x 2 rda be differentiable Well, here’s01:45:48
some kind of curve and01:45:50
and y is equal to f from x, let’s take here the point01:45:54
m 0 m 0 with coordinates x001:46:00
f from x 0 for the direction of the tangent,01:46:06
let’s take the direction when moving, well, in01:46:10
my drawing up, here’s the directional01:46:12
tangent at point m 0, so we’ve moved to01:46:15
point m 1. m 1 is a point with coordinates01:46:21
x f from x that is . and the mole is fixed,01:46:25
which means this is what we have x 0 and this is01:46:30
x1 and here is the directed tangent01:46:36
at point m 1 and this corner we have is01:46:43
what we designated delta fi the angle at01:46:45
which the directional01:46:47
tangents turned when we moved from point 0 to01:46:49
point they are one now further the angle between the01:46:55
directed tangent at point m 1 with01:46:59
coordinates arbitrary x f from x let01:47:02
's denote by alpha alpha we01:47:07
change from minus p in half to plus and in01:47:11
half for the point and I01:47:18
naturally denote this corner by01:47:20
alpha zero01:47:23
perez let's look at the geometric01:47:26
picture angle alpha is the external angle of a01:47:30
triangle whose two angles are not01:47:34
adjacent to alpha equal to alpha zero and01:47:38
this delta fi01:47:40
means from here we get that delta fe01:47:43
well in my picture alpha is greater than alpha01:47:46
zero but if I took the convexity of the other01:47:49
side it would be the other way around so delphi01:47:51
this is the module delta alpha and by delta01:47:56
alpha we mean alpha minus alpha zero01:48:00
divide japheta module delta alpha01:48:06
so the average curvature is the average01:48:10
curvature of the section of the curve k m 0 m 1 this is01:48:18
delta fig delta el since we01:48:21
defined the average curvature in our01:48:23
case this is the module01:48:25
delta alpha Well, beit-el is always01:48:28
positive, but for simplicity I’ll write it01:48:31
like this: the modulus of deltas alpha gt01:48:33
and the curvature at a point by me is the limit as01:48:40
delta h tends to 0.01:48:43
Well, the modulus delta alpha delta and01:48:48
but this limit, by the very definition,01:48:51
is nothing more than module of the derivative01:48:54
d alpha podi hey01:48:58
taken at point 0 then removed for x equal to01:49:03
x0 and let’s write this module derivative of01:49:06
dolf pdl for x equal to x0 now next01:49:11
we have tangent alpha this is the tangent of the01:49:14
tangent angle this is f prime from x01:49:19
we know this well said many times01:49:21
in the Peru semester,01:49:23
it means alpha is the arctangent f prime from x01:49:32
d alpha let01:49:36
's find the differential of the arctangent f prime01:49:39
from x d alpha this will be taken first the01:49:44
derivative of the arctangent this is a complex01:49:46
function throughout the argument01:49:47
in the denominator will be one plus f prime01:49:50
squared from x which is in the numerator01:49:56
Well, the derivative of f is a prime, that is, f is two01:49:59
primes from xf-2 a prime from x and what else to01:50:02
multiply by dx01:50:04
is the differential alpha now what is01:50:08
d.l.01:50:09
we will count the length of the arc el from the01:50:12
point m 0 naturally and then let's01:50:16
remember that the length of a thing is the length of the01:50:26
section of the curve m 0 m 101:50:31
this is the integral from x0 to x dx01:50:39
x0 this is the abscissa of the point 0x abscissa. im01:50:42
one square root of unit + f prime01:50:46
squared let's integrate the variable01:50:48
since x is occupied in the01:50:51
upper limit let's denote the letters with f301:50:54
squared from vat and then d.l.01:51:04
this is the derivative on dx the derivative01:51:08
of this function is the derivative of the01:51:10
integral with a variable upper limit01:51:13
we know well that it is equal to the01:51:14
integrand taken at the01:51:17
upper limit01:51:18
this is the root of one plus f the prime01:51:22
squared of x on dx and so we calculated01:51:28
to alpha as a function of x dx01:51:30
dr as a function of x dx well, now dividing01:51:34
one by the other we get01:51:41
means the curvature at the point by me linking the01:51:48
point m 0 to from m 0 equals again I01:51:54
will remind the module d alpha under hey with x01:51:59
equal to x0 if we divide by before alpha01:52:03
drill text will be reduced and put x01:52:06
equal to x0 then we get a fraction in the01:52:10
numerator module01:52:11
f two strokes at the point x 0 in the denominator01:52:17
one plus f squared stroke at the point01:52:21
x0 and all this is to the power of three second01:52:26
understandable root vdm included in d alpha01:52:32
included units plus four squared and01:52:34
even when we divide by duel it01:52:37
will turn out like this, the curvature at a point01:52:43
I’ll remind you once again that01:52:46
m0 is a point with coordinates x001:52:49
f from x 0 the curve given to Aries by01:52:53
the equation, this is how the curvature is subtracted,01:52:55
what we see is that the curvature is greater01:52:58
than is greater than the second derivative, but01:53:03
if the second derivative f two strokes at the01:53:09
point x0 is01:53:11
not equal to zero, then the curvature at the point m 0 is not01:53:19
equal to zero, and in this case, in this case,01:53:24
let's denote by p a01:53:30
large01:53:34
value the inverse of the curvature, we introduce such a01:53:38
value r large equals one unit .01:53:44
and many, and now let’s introduce the concept of01:53:52
a circle, a circle of radius r large, a01:53:59
circle of radius r large, touching the01:54:04
curve and at the point m 0 at this very01:54:09
point, but the circle of this radius can be01:54:16
attached on one side of the curve and on the01:54:18
other, now we will continue the phrase, I will01:54:20
first draw it so that it is It’s clear from01:54:23
which side this circle is taken, here’s the01:54:26
x y axis, here’s01:54:28
our curve, hey, given by the equation, y01:54:32
is equal to f of x01:54:34
here. I have with coordinates x0 x001:54:40
circles of radius r touching the curve01:54:43
el large at point 0 and having in01:54:46
the neighborhood of point 0 the same direction of01:54:49
convexity as the curve and having at . and01:54:54
by me the same direction of the convexity as the01:54:58
curve,01:54:59
but this means to surround this one, we must01:55:02
attach it from here so that there is the01:55:06
same direction of the convexity,01:55:09
somewhere its center01:55:10
is its radius r and so the circle of01:55:14
radius r is large touching the credit at the01:55:17
point by me and having in the vicinity of point01:55:19
0, for example, the same release as the curve01:55:22
is called, oddly enough, the01:55:24
circle of curvature of the elf curve. they are 0 well, a01:55:28
circle is called a circle a little bit01:55:30
like that, well, as if the terminology is a circle01:55:35
from not a circle, of course, nevertheless, this is how it is01:55:36
customary to say this circle01:55:38
is called the circle of curvature of the curve or at the01:55:41
point m 0 and the center of this circle and the radius01:55:44
is called the center of the ir01:55:46
at the curvature of the curve itself at point m 0 and the01:55:50
center and radius of this circle01:55:52
is called the center and radius of curvature01:55:55
these are circles at point m feet01:55:58
such a small problem for you, consider the01:56:02
curve el01:56:03
given by the equation y equals x square01:56:08
parabola as a point and I take the01:56:13
origin of coordinates 0001:56:21
well, you can draw a picture of the x axis and01:56:27
y and here is our parabola is a circle cos01:56:34
and here is . m0 of the origin of coordinates is a01:56:37
circle something like this,01:56:38
the task will be like this for you to write01:56:43
the equation of this circle, but for this you01:56:46
need to find it simply for the sake of finding the curvature01:56:48
using this formula01:56:49
f two strokes and the second it is easy01:56:52
to find for the function x the square of them must be01:56:56
set equal to zero you will find curvature at01:56:58
point m 0 p is units for curvature, it is clear due to01:57:02
symmetry that the center is circling and01:57:04
lies on the y axis,01:57:06
and then easily write the equation this01:57:10
curvature of the curve in the case when the curve is01:57:13
given by an explicit equation, well, without any01:57:19
unnecessary explanation, let now the01:57:22
Elsa curve be given parametrically x equals feat t01:57:27
y equals psy and then you d.l.01:57:35
this is let's remember the expression for the01:57:38
arc length of the curve back parametrically01:57:41
integrand this is d and01:57:44
this is the root of fish 3 x squared of t +01:57:49
psy prime squared at&t01:57:53
on dt tangent alpha f prime from x a f01:58:00
prime from x for a curve given01:58:02
parametrically this is psy stroke from t01:58:04
divided by fish 3 t alpha this will be the01:58:10
arctangent of this fraction arctangent psy in01:58:14
3 t on the filter although they mean for alpha01:58:20
it will be well, if you differentiate01:58:23
and this arctangent then you will get this is01:58:26
what I will write immediately in the denominator will be01:58:29
the sum of squares and in 3 x squared from t01:58:33
plus historic square plus psy prime01:58:36
squared and t a in the numerator will be psy 201:58:39
prime from t on fish 3 t minus01:58:44
phi2 prime from t on psy prime from t and on01:58:50
dt and on dt01:58:53
well and therefore the curvature if we take01:58:57
a point and me if we take y at the point m 001:59:05
on the curve01:59:08
with coordinates feat t 0 seat t0 to a01:59:15
specific value t equal to t001:59:17
then the curvature of the curve at the point m 0 is the01:59:26
modulus d alpha by d.l.01:59:29
with t equal to t0 if we divide d alpha01:59:36
by drill and set t equal to t0 then01:59:38
we get a fraction in the numerator module psy01:59:43
two primes feb 301:59:45
minus phi2 prime obse in 3 well of course01:59:49
this is taken at t equal to t0 and in the01:59:52
denominator01:59:53
fish 3x squared plus psy stroke01:59:57
squared to the power of three second and all this02:00:02
must be taken with t equal to 0, here is the02:00:06
curvature of the curve at the same time and parametrically,02:00:08
well, using this formula and the02:00:13
parametric equations of that same02:00:15
ellipse, now if we return to the02:00:16
example with the ellipse where we set the02:00:19
task of how to prove that the curvature in point02:00:21
m 1 is greater than at point m 2,02:00:24
which means the parametric equation of the ellipse02:00:26
x is equal to and cosine t02:00:31
y is equal to sine t. m102:00:39
this one. what value of m02:00:42
corresponds to you equal to zero yes m 1 is02:00:47
3 equal to zero. m202:00:50
this top or all those lying on the02:00:53
y axis corresponds to some pi in half,02:01:01
calculate using this formula using the02:01:05
phi function we have a cosine 3 function psy02:01:07
pool 100 calculate the curvature using this formula02:01:10
and thereby prove the thing in02:01:14
. m1 is greater than k02:01:16
at point m 2 well, dear friends, we02:01:25
have completed the second part of our 3 7 2 course of analysisDescription:
1. Формула Стокса 2. Независимость криволинейного интеграла II рода от пути интегрирования в пространстве 3. Геометрические приложения дифференциального исчисления. 3.1. Касание кривых 4. Огибающая однопараметрического семейства кривых. 4.1. Особые точки кривых 5. Кривизна плоской кривой
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How can I download "Бутузов В. Ф. - Математический анализ - Геометрические приложения (Лекция 21)" video to my phone?
You can download a video to your smartphone using the website or the PWA application UDL Lite. It is also possible to send a download link via QR code using the UDL Helper extension.
How can I download an audio track (music) to MP3 "Бутузов В. Ф. - Математический анализ - Геометрические приложения (Лекция 21)"?
The most convenient way is to use the UDL Client program, which supports converting video to MP3 format. In some cases, MP3 can also be downloaded through the UDL Helper extension.
How can I save a frame from a video "Бутузов В. Ф. - Математический анализ - Геометрические приложения (Лекция 21)"?
This feature is available in the UDL Helper extension. Make sure that "Show the video snapshot button" is checked in the settings. A camera icon should appear in the lower right corner of the player to the left of the "Settings" icon. When you click on it, the current frame from the video will be saved to your computer in JPEG format.
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