Invariance of the first differential of a function of many variables. Folding function differential invariance of the form of the implicit function differential is equal to the plane and normal to the surface

We bachili that the differential function can be written in the form:
(1),

yakscho є independent change. Come on now є collapsible function , then.
,
and to that
. What other functions
і
prove that
like a foldable function. Differential
or. ale
and we can write
, then. otrimali viraz for
yak i (1).

Visnovok: formula (1) is correct if and when є is an independent change, so it’s in the mood, if є function as an independent change . In the first mood pіd
understand the differential of the independent change
, the other has a differential function (for which
, vzagali seeming). Tsya power saving form (1) and is called invariant form of differential.

The invariance of the form of the differential gives benefits when calculating the differentials of folding functions.

For example: please calculate
. Nezalezhnaya in addition, the fallow chi is independent of the mine we can write down. Yakscho - function, for example
, then we know
that, koristyuyuchis іnvariantnіstyu forms of the differential, may have the right to write.

§eighteen. Pokhіdnі naivishchih orderіv.

Let the function y \u003d  (x) be differentiated into the current interval X, (so it can be similar to y 1 \u003d  1 (x) at the skin point of that interval). Todі 1 (x) є y X itself is a function in x. Maybe, but at certain points chi override at all x 1 (x) itself can be lost, then. іsnuє pokhіdna vіd pokhіdnoї (y 1) 1 \u003d ( 1 (x) 1. For which type of її is called another pokhіdnoy or pokhіdnoy of a different order. pіdkreslity, which is pokhіdna in t.x 0, write

y 11 / x \u003d x 0 or 11 (x 0) or d 2 y / dx 2 / x \u003d x 0

pokhіdna in 1 is called the pokhіdnaya of the first order, or the first pokhіdnoy.

Otzhe, similar to a different order, is called a similar function in a similar first order.

As a whole, similarly, pokhidna (there, deva vona) in a different order is called pokhidna of the third order, or the third pokhidna.

Mean (y 11) 1 \u003d y 111 \u003d 111 (x) \u003d d 3 y / dx 3 \u003d d 3  (x) / dx 3

Functions y = (x) are named similar to the n-th order of the function. (as stinks are known, zvіsno).

signify

Read: n-a pokhіdna vіd y, vіd (x); d n at d x in n-th.

Fourth, fiveth, too thin. the order is not manually indicated by strokes, to write the number in the arches instead  v (x) write  (5) (x).

At the arms, so as not to stray the n-th order of the same and the n-th stage of the function.

Pokhіdnі order, vyschі for the first, call the pokhіdnim of higher orders.

From the most important point, it is obvious that for the recognition of the n-ї pokhіdnoї it is necessary to know successively all the forward ones from the 1st to (n-1)-ої.

Apply: 1) y \u003d x 5; y 1 \u003d 5x 4; y 11 = 20x3;

y 111 = 60x2; y (4) = 120x; y (5) = 120; y(6) = 0, ...

2) y = e x; y 1 \u003d e x; y 11 = e x; ...;

3) y \u003d sinx; y 1 = cosx; y 11 = -sinx; y 111 = -cosx; y (4) = sinх;

It is respectful that a friend may be a singing mechanical shifter.

Like the first one is like a path to the hour and the speed of a straight-line uneven movement

V \u003d ds / dt, de S \u003d f (t) - equal to the movement, then V 1 \u003d dV / dt \u003d d 2 S / dt 2 - the speed of change of the speed, tobto. quick rush:

a \u003d f 11 (t) \u003d dV / dt \u003d d 2 S / dt 2.

Otzhe, the friend's path is lost after an hour, є sooner the point of change - at whom the mechanical zmist another pokhіdnoy.

At a number of vipadkіv vdaєtsya write viraz pokhіdnoї no matter what order, passing through the middle.

Apply:

y=e x; (y) (n) = (e x) (n) = e x;

y = a x; y 1 \u003d a x lna; y 11 \u003d a x (lna) 2; y (n) = a x (lna) n;

y = x; y 1 = αx α-1; y 11 =
; y (p) \u003d α (α-1) ... (α-n + 1) x α-n, with =n maybe

y (n) = (x n) (n) = n! Pokhіdnі order vishchevsі equal to zero.

y = sinx; y 1 = cosx; y 11 = -sinx; y 111 = -cosx; y (4) = sinx; ... and so on.

y 1 \u003d sin (x + /2); y 11 \u003d sin (x + 2 /2); y 111 \u003d sin (x + 3 /2); etc., then (n) \u003d (sinx) (n) \u003d sin (x + n /2).

It is easy to insert the last derivations and global formulas:

1) (CU) (n) = C(U) (n); 2) (U ± V) (n) = U (n) ± V (n)

The formula for the n-th successive addition of two functions (U·V) (n) is more collaborative. Vaughn can name Leibnitz's formula.

Take away її

y = U V; y 1 \u003d U 1 V + UV 1; y 11 \u003d U 11 V + U 1 V 1 + U 1 V 1 + UV 11 \u003d U 11 V + 2U 1 V 1 + UV 11;

y 111 = U 111 V+ U 11 V 1 +2U 11 V 1 +2U 1 V 11 + U 1 V 11 + UV 111 = U 111 V+3U 11 V 1 +3 U 1 V 11 + UV 111;

Similarly, it is taken

y (4) \u003d U (4) V + 4 U 111 V 1 +6 U 11 V 11 +4 U 1 V 111 + UV (4) etc.

It is not important to remember that the right parts of all these formulas predict the layout of the steps of the binomial U+V, (U+V) 2 , (U+V) 3 and so on. Only the deputy steps of U and V should stand here in the lower order. Podіbnіst bude especially povnim, akshcho otrimanih formulas to write zamіst U і V, U (0) і V (0) , then. 0-th similarities of functions U and V (functions themselves).

Expanding this law in case of any n, we take away the general formula

y(n) = (UV)(n) = U(n)V + n/1! U(n-1) V 1 + n(n-1)/2! U(n-2) V(2) + n(n-1)(n-2)/3! U (n-3) V (3) +…+ n(n-1)…(n-k+1)/K! U(k) V(n-k) +…+ UV(n) – Leibniz formula.

Stock: know (e x x) (n)

(e x) (n) = e x, x 1 = 1, x 11 = 0 i x (n) = 0, so (e x x) (n) = (e x) (n) x + n/1 ! (e x) (n-1) x 1 \u003d e x x + ne x \u003d e x (x + n).

The formula for the differential of a function can be seen

de - Differential of the independent change.

Now let's have a collapsible (differentiated) function, de. Same for the formula of a similar folding function is known

so yak .

Otzhe, , then. the formula of the differential can be one and the same form for the independent change for the intermediate argument, which is a function, which differentiates itself.

Qiu power is accepted to be called power invariance of the formula or the form of the differential. Respectfully, scho is not good for the authorities.

Zvyazok mizh bezperevnіstyu and differentiation.

Theorem (Necessary mind differential function). Although the function is differentiated at a point, it is uninterrupted at this point.

Bringing. Come on function y=f(x) differentiated in points X 0 . Damo in tsіy point zbіlshennya argument  X. The function removes the increase  at. We know.

Otzhe, y=f(x) uninterrupted to the point X 0 .

Consequence. Yakscho X 0 is the point of development of the function, then the function is not differentiated.

The assertion, turned into a theorem, is not true. From bezperevnosti not viplivaet differentiation.

Differential. Geometric zmist. Zastosuvannya differential to the nearest calculation.

Appointment

Function differential is called a linear part of the function. Vaughn is signified as abo. In this manner:

Respect

The differential function becomes the main part of the її zbіlshennya.

Respect

Instructed to introduce the understanding of the differential to the argument from the understanding of the differential of the function. For appointment differential argumentє zbіlshennya argument:

Respect

The formula for the differential of a function can be written in the following way:

We must admit that

Otzhe, tse means, that it can be presented as a great difference - an extension of the differential function to the argument.

Geometric differential lock

The differential of the function at the point is equal to the increase in the ordinate of the dot, drawn to the graph of the function at the point, which confirms the increment of the argument.

Basic rules of differentiation. Pokhіdna postіyna, pokhіdna sumi.

Don't let the functions look like fun to the point. Todi

1. Constant you can blame for the badge of good luck.

5. Differential constant equals zero.

2. Pokhіdna sumi/retail.

Pokhіdna sumi/principles of two funktіnіy dоrіvnіuє sumіlіvіrіnі іnіhіd vіd derzhіnі їnії ї.

Basic rules of differentiation. Do some fun.

3. Pokhіdna robot.

Basic rules of differentiation. Pokhіdna folding and turning functions.

5. Folding function.

Pokhіdna foldable functions dorovnyuє pokhіdnіy tsієї funktsії behind the intermediate argument, multiplied by the pokhіdnu vіd intermediate argument behind the main argument.

I may be pokhіdnі vіdpovіdno to the point. Todi

Theorem

(About the dead function)

If the function is uninterrupted and strictly monotonic in the real neighborhood of the point, it is differentiated in the th point, then the reversible function can be lost in the point, moreover .

Differentiation formulas. Pokhіdna display functions.

How is the function of independent variables that differentiates, but the latest differential dz is more advanced? g, y) prove and be sure that the function r = f(x, y) is differentiated at the ith point. Geometric zmist new differential The normal to the surface Yak can be seen from formulas (2), w and w without interruption at the point ((, *?). Therefore, the function is differentiated at the point, it is acceptable to use the formula of the total differential for the function in the form of independent changes £ and m], perhaps replacing the right parts of equality (3) w and w їх virases of formulas (2), can be taken either as a mental function at a point ((,17) can be uninterruptedly private, then a stench at this point of differentiation and Z spіvvіdnosh (4) and (5) It is assumed that the Equation of formulas (1) and (6) shows that the new differential of the function z = / (i, y) is expressed by the formula of the same type as in the case, if the arguments x and y of the function / (r, y) є Invalid serpentine, so at the vipad, if the argument is in his worm with the functions of the senior snakes. With the rank of a differential, the function of the deeds is powerful, the Kintsevoye is not a melting. in deakіy area G on the plane хОу. th interval (ho - Lo, ho + ^o) is equal to one value of y, like at once from x satisfies equal (1), function y \u003d y (x) is determined, for which equalness is equal to x y to the specified interval. In some way it seems that the equation (1) signifies the value as an implicit function of x. In other words, the function assigned to equals, which is not allowed, is called an implicit function", it becomes explicit, as the occurrence of y species x is set without intermediary. Apply. the value of y is defined as a unique function x. points. Let's look at their graphs on the xOy plane (Fig. 11). parallel transfers vzdovzh osі Oh іrivoy r \u003d r sin y. It is geometrically obvious that with skin x the curves x = y and r = t + c $ 1pu can have a single point on the line, the order in the yakіy є function vіd x, as it is assigned equal to (2) implicitly. Through elementary functions, staleness is not expressed. 3. Rivnyannya for the next day's x does not change the function of the argument x. With such a sensor, one can talk about the implicit functions of a few changes. The theorem is coming yes enough mind unambiguous razvyaznostі equalization = 0 (1) schodo at the sevnіy outskirts of the given point (®o> Uo). Theorem 8 (the existence of an implicit function). Let's think like this: 1) the function is assigned and without interruption to a certain rectangle with the center at the point at the point the function y) turns into n\l; 3) the straight line D has no interruption private holidays positive number e there is a neighborhood tsієї vicinities іsnuє єdin ^ uninterrupted function(1) functions, vvazhayuchi іsnuvannya tsієї pokhіdnoї bring. Let y \u003d f (x) - an implicit function that differentiates, is equal to (1). Same in intervals) may be the same. Folding function differential. Invariant form of differential. Implicit functions. , y), what to lie on the curve, what to lie around the point (xo, yo) "maє coordinates, po'yazanі rivnyann Zvіdsi at y \u003d f (x) is taken, that i, otzhe, Butt. Know j * as a function y \u003d y (x), how it is equal to B to this particular type Zvіdsi z formulas (3) Respect. Theorem Given the conditions for the creation of a single implicit function, the graph of which is to pass through a given point (xo, yo). enough, but not needed. On the right, we can look at the equality Here we can uninterrupted private changes to zero at the point 0(0,0). prote, Dane equal there is only one solution, which is equal to zero at Zavdannya. Let the equalization be given - an unambiguous function that satisfies the equalization (P). 1) How many single-valued functions (2") are you satisfied with equal (!")? 2) Do the numbers of single-valued non-recursive functions satisfy the equality (!")? 4) How many single-valued, non-permanent functions, satisfying "equal (1"), so it's not enough to do it? The theorem is based, analogous to Theorem 8, but if the implicit function z - z(x, y) is two times the implicit function z - z(x, y), it is equal Privatni Pokhidni Todi for being sinking small e> o know the Ukolitza G2 Points (®o "UO) / In yaki іsnuu єdina, non -purified function z - / (f, y), cup of ore, Х = f, y, y, y, y. what satisfies the minds and beasts of equals (4) at the sameness: With this, the function in the division of Q can be without interruption in private holidays and GG We know the virazi for these wicked ones. Let's define z as an unambiguous differentiation function z = /(x, y) of independent changes. If the purpose of replacing z is to represent the function f(x, y), then we take the sameness Otzhe, more private similarities for w and y of the function y, z), de z = /(z, y), also due to but equal to zero. Differentiation, we know the names of the formulas to give in contrast to private similar implicit functions of two independent variables. butt. Know private proiaaodnіa vіd funktsії x(r,y) assigned to equals 4 Maєmo zvіdki §11. Dotichnaya plane and normal to the surface 11.1. Front view Let's go to the surface S assigned to equals Appointed*. The point M(x, y, z) of the surface (1) is called the primary point of the surface i, because at the point M all three are similar and without interruption, moreover, if only one of them is visible as zero. If at points Mu, z) on the surface (1) all three of them are equal to zero, or if one of them is not equal, then the point M is called a special point on the surface. butt. We can see a circular cone (Fig. 13). Here the cob of coordinates 0(0,0,0) is so singularly thin: the cob of coordinates 0(0,0,0) turns to zero at this point. Rice. 13 Let's look at the open space curve L, given by parametric equalities, Let the functions be able to move without interruption in the interval. Including from the view of the particular points of the curve, for some of them - the most important point of the curve L, as it depends on the values ​​of the to parameter. Todі - vector, scho stosuetsya curves in points. The visible area of ​​the surface Nehai surface 5 is given by equalities. Let's take a significant point P on the surface S and draw a curve L through it, which lies on the surface and is given by parametric equalities. , nowhere on (a) p), which do not turn back to zero at once. For the purpose, the curve L at point P is called tangent to the surface 5 at this point. L lie on the surface S, level (1) move at the sameness of t: Differentiation of the sameness with respect to t, according to the rule of differentiation of the folding function, we take Viraz at the left side (3) є scalar creation of two vectors: curve L at the th point (Fig. 14). If it is worth the vector p, then vin to lie only in the form of coordinates of the point and the type of function ^ "(x, y, z) and not to lie in the form of a curve to pass through the point Р. vector n vіdmіnna vіd zero, Te, which is a scalar addition means that the vector g, so that the curve L is standing at the point P, perpendicular to the vector p at the qiy point (Fig. 14). through the point P i lie on the surface S. Then, whether it is a straight line to the surface 5 at the point P, it is perpendicular to the vector p, i, then, all qi lines lie in the same plane, also perpendicular to the vector p. all straight lines extending to the surface 5, which pass through the given primary point P G 5, are called the dotic surface plane at the point P (Fig. 15). extremely P0 (®o, Uo" tsієї surfnі: As the surface 5 is set equal, then, having written down the tsієї і vglyadі otrimаєmo and ііnnja dіtіchі ї ploskoї і tochtsі, if it looks like this 11. 3. Geometrical change of the total differential If formula (7) is put, then it won’t look right \u003d / (x, y) of two independent changes x and y at point M0, which increas- , Uo)) WHEN passing from the point M0 (xo, Uo) to the point - 11.4. Normal to surface Designated. A straight line that passes through the point Po(ho, yo, tho) from the surface perpendicular to the dot- ic plane to the surface at the point Po is called the normal to the surface at the point Pq. Vector) L is a direct vector of the normal, and її the alignment may look Like the surface 5 is set to the alignment, then the alignment of the normal at the point) looks like this: at the point Here At the point (0,0) (0,0,0) swells the offensive look: (hoy area). Equation of normal

Virase of the total differential of the function of a few of the changeable ones itself looked independently, because of the fact that it is independent of the changeable functions of other independent changeful ones.

The proof rests on the formula of the total differential

What did it take to bring.

5.Overall similar functions- Pokhіdna funktsії after an hour vzdovzh traєktorії. Let the function be seen and її arguments lie in the hour: . Todi , de - parameters that set the trajectory. It is different functions (at the point) for such a different private time for an hour (at the point of departure) and can be calculated using the formula:

de - Private holidays. Significantly, what is known is sensible and cannot be extended to the bottom of differentials. In addition, the most important function to deposit is not only in the function itself, but also in the trajectory.

For example, there are similar functions:

Here, there are no shards on their own (“obviously”) not to fall into the water.

New differential

New differential

functions f (x, y, z,...)

at a vіpadku, if it’s blowing up in the face of total swell

f = f(x + x, y + y, z + z, ...) - f(x, y, z, ...)

by a value that is inexorably small in terms of

Dotichnaya flat to the surface

(X, Y, Z - current coordinates of the point on the dottic plane; - radius-vector of the point; x, y, z - coordinates of the dotik point (valid for normal); - dotistic vectors to the coordinate lines of the dotik v = const; u = const ; )

1.

2.

3.

normal to surface

3.

4.

The concept of differential. Geometric differential lock. Invariant form of the first differential.

Let's look at the function y = f(x), which is differentiable at this point x. Increasing Dy її can be imagined at a glance

Dy \u003d f "(x) Dx + a (Dx) Dx,

first of all, the addition is linearly Dx, and the other is at the point Dx = 0 an infinitely small function greater high order lower Dx. If f "(x) № 0, then the first addition is the head part of the increment Dy. This head part is the increment of the linear function of the argument Dx and is called the differential of the function y = f (x). If f "(x) = 0, then the differential of the function the value is equal to zero.

Designation 5 (differential). The differential of the function y = f(x) is the main linear distance Dx part of the increment Dy, which is similar to the increment of the independent change

It is important that the differential of the independent change is healthy for the increase in the price of change dx = Dx. Therefore, the formula for the differential is accepted to be written in such a way: dy = f "(x) dx. (4)

Z'yasuёmo some kind of geometric zmіst differential. Take on the graph of the function y = f(x) a sufficient point M(x, y) (Fig. 21.). Drawn to the curve y \u003d f (x) at the point M, so as to make f with a positive direct axis OX, then f "(x) \u003d tgf.

KN \u003d MNtgf \u003d D xtg f \u003d f "(x) D x,

then dy = KN.

Thus, the differential of the function is the increase in the ordinate drawn to the graph of the function y = f(x) at the qiy point, if x is taken to be the increase in Dx.

Significantly the main power of the differential, as analogous to the power of the poor.

2. d(c u(x)) = c d u(x);

3. d(u(x) ± v(x)) = d u(x) ± d v(x);

4. d(u(x) v(x)) = v(x)d u(x) + u(x)d v(x);

5. d(u(x) / v(x)) = (v(x) d u(x) - u(x) d v(x)) / v2(x).

Let's say one more power, which is a differential, but it can't be bad. Let's look at the function y = f(u), where u = f(x), so let's look at the collapsible function y = f(f(x)). Just as the skin functions f and f are differentiating, then the folding functions are similar to the theorem (3) better y" = f "(u) u".

dy \u003d f "(x) dx \u003d f "(u) u" dx \u003d f "(u) du,

because u "dx = du. So dy = f" (u) du. (5)

Remaining equality means that the formula of the differential does not change, so that the replacement of the function x looks at the function in the form of a change u. The power of the differential took away the name of the invariance of the form of the first differential.

Respect. It is significant that formula (4) has dx = Dx, while formula (5) du has only a linear part of the larger function u.

Integral calculation is a division of mathematicians, in which the powers of that method of calculation of the integration and their zastosuvannya are twisted. I. in. closely related to differential calculations and adding together one of the main parts of them