Differential of two changes at a point. Function differential. Why have a sense of private pokhіdnyh
Appointment: Relative differential of the function how many variables are called the sum of all її private differentials:
Example 1: 
.
Solution:
Oskіlki okremі pokhіdnі tsієї funktsії rivnі:
Then one can immediately write private differentials of these functions:
, 
,
So the last differential of the function looked like:
.
butt 2 Find the latest differential of a function
Solution:
This function is foldable, that is. can you tell me how
We know private holidays:
Analytical change of the total differential of the field lies in the fact that the total differential of the function of decals of the change is the main part of the total increase in the function, that may be equal is near: ∆z≈dz.
It is necessary, remember, that the approximations of equality are only valid for small differentials dx and dy of the arguments of the function z=f(x,y).
The calculation of the total differential in the approximation calculations is based on the alternative formula ∆z≈dz.
Obviously, as in the given formula, the increase in ∆z of the tax function in the view, and the last differential in the view 
, then we take:
≈ 
,
Otriman's formula can be tweaked for the approximate meaning of the "new" value of the function of two variables, as if it were not taken for a small increase in both arguments.
butt. Know the approximate value of a function , With the upcoming values of її arguments: 1.01, .
Solution.
Substituting private similar functions, previously found in the formula, we take into account:
When setting the value x=1, ∆x=0.01, y=2, ∆y=0.02, we take:
Scalar field.
If the function U(p)=U(x,y,z) is given to the skin point of the space D, then we can say that the scalar field is given in the area D.
For example, U(x, y, z) denotes the temperature at the point M(x, y, z), then it seems that a scalar temperature field is given. If the area D is filled with a native gas and U(x,y,z) means a vice, then it is a scalar field of a vice. If in the space there is a distribution of charges or massive bodies, then we can talk about the potential field.
The scalar field is called stationary, so the function U(x,y,z) does not change over time: U(x,y,z) ≠ f(t).
Whether a stationary field is characterized by:
1) on the surface of the scalar field
2) quickly change the fields for the given one directly.
On top of the level A scalar field is called a geometrically spaced point, for which the function U(x,y,z) has a constant value, so U(x,y,z) = const. The sukupnіst tsikh dots make up the deak on the surface. If we take another constant, then we take another surface.
Butt: Let a scalar field be given. The butt of such a field is the field of the electric potential of a point electric charge (+q). Here the surfaces of the equal will be equipotential surfaces 
Tobto spheres, in the center of which there is a charge that creates a field.
Directly the largest growth of the scalar function is determined by the vector, which is called gradient it is denoted by a symbol (abo).
The gradient of the function is known through private linear functions and is always perpendicular to the surface of the surface of the scalar field at a given point:
, de
Alone vectors and similar to the axes OX, OY, OZ
Pohіdna type of function U(x, y, z) for any other direction (λ) is assigned to the formula:
, de
α, β, γ - tse kuti between the coordinate axes in the same way OX, OY, OZ and straight ahead.
Excluded data of the sample:
ABOUT DIFFERENTIAL OF ANOTHER ORDER
Lovkov Ivan Yuriyovich
student of Moscow state university information technologies, radio engineering and electronics, RF, m. Serpuhiv
E- mail: alkasardancer@ Rambler. en
Taperechkina Vira Oleksiivna
cand. fiz.-mat. Sciences, Associate Professor, Moscow State University of Information Technologies, Radio Engineering and Electronics, Russian Federation, Serpukhov
ABOUT SECOND-ORDER DIFFERENTIAL
Lovkov Ivan
student of Moscow State University of Information Technologies, Radio Engineering and Electronics, Russia, Serpukhov
Vera Taperechkina
candidate of Physical and Mathematical Sciences, Associated Professor of Moscow State University of Information Technologies, Radio Engineering and Electronics, Russia, Serpukhov
ABSTRACT
The robots have looked at ways of finding similar and differentials of the first and other orders for folding functions of two changes.
ABSTRACT
Calculation methods derivative and first and second differentials for composite functions of 2 variables.
Keywords: private holidays; differential.
keywords: partial derivatives; different.
1. Intro.
Formulyuєmo deyakі z teorії funktsіy rich change, yakі nadoblyaetsya far away.
Designated: the function z=f(u, v) is called differentiated at the point (u, v), so that the increase in Δz can be represented as:
The linear part of the increase is called the upper differential and is designated dz.
Theorem ( enough mind differentiation) div.
If in the real neighborhood of t.
(du = ∆u, dv = ∆v). (one)
Designation: The other differential of the function z=f(u, v) at this point (u, v) is the first differential of the first differential of the function f(u, v), that is. 
From the appointment of another differential z=f(u, v), de u and v are independent changes, vyplivaє
Thus, the formula is valid:
When the formula was introduced, Schwartz's theorem about the equalness of the mixed relatives was victorious. Tsya jealousy is fair for the mind that 
assigned in the vicinity of t.(u, v) and without interruption t.(u, v). Div.
The formula for the value of the 2nd differential can be written symbolically in the offensive form: - Formally squaring the arch with further formal multiplications on the right by f (x y) gives the formula . The formula for the 3rd differential is similarly valid:
I started:
Deformally, the transition to the n-th step is carried out according to the Newton binomial formula:
; 
It is significant that the first differential of the function of two different powers is the invariance of the form. That is, since u and v are independent changes, then for the function z=f(u, v), it is necessary to (1)
Now let u = u (x y), v = v (x y), then z = f (u (x y), v (x y)), x і y - independent changes, then
Vikoristovuyuchi v_domі formulas for pokhіdnoї folding function:
Todi z (3) and (4) is taken:
in such a manner,
(5)
de 
- The first differential of the function u, 
- The first differential of the function v.
Por_vnyuyuchi (1) і (5), Bachimo, that the formal notation of the formula for dz is taken, but if (1) du=Δu, dv=Δv - increase of independent changes, then (5) du and dv - differentials of functions u and v.
2. Another differential folding function of two shifts.
We show that another differential cannot have the power of form invariance.
Let z=z(u, v) for different independent variables u and v have another differential known by formula (2)
Now let u=u(x y), v=v(x y), z=z(u(x y), v(x y)), de independent change x and y. Todi
.
Father, we took away the rest:
Formulas (2) and (6) do not change form, therefore, another differential cannot be invariant in power.
Previously, the formulas of private similar 1st order formulas for the folding function z=f(u, v), de u=u(x y), v=v(x y), de x and y are independent changing divs.
We will show formulas for calculating private similar differentials of a different order for the function z = f (u, v), u = u (x y), v = v (x y), where x and y are independent changes.
For functions u(x y), v(x y) of independent variables x, y, we can formula:
We represent formulas (8) (6).
In this way, they took away the formula for the differential of a different order of folding functions of two variables.
Por_vnyuyuchi coefficients for private similar to another order collapsible functions of two changing (2) and (9), we take the formula:
Butt 1 cm
Let z = f(u, v), u = xy, v =. Find another differential.
Solution: calculate private trips:
, , , ,
, 
,
Like Bachimo, the value of the differential needs to be multiplied by dx. Tse allows you to write down a different table for differentials from the tables of formulas for similar ones.
The new differential for the function of two changes:
The final differential for the function of three changing sums of private differentials: f (x, y, z) = d x f (x, y, z) dx + d y f (x, y, z) dy + d z f (x, y, z) dz
Appointment. The function y \u003d f (x) is called differentiated at the point x 0, so the increment at the points can be given at the sight ∆y=A∆x + α(∆x)∆x, where A is a constant, and α(∆x) – infinitely small as ∆x → 0.
In addition, the differentiation of the function at the point is equivalent to the basis of the similar one at the point, moreover, A=f'(x 0).
Let f(x) be differentiated at the point x 0 і f "(x 0)≠0 then ∆y=f'(x 0)∆x + α∆x, where α= α(∆x) →0 as ∆x → 0. The value of ∆y i of the skin of the appendages of the right part is infinitely small at ∆x → 0. 
, then α(∆x)∆x is infinitely small more high order, Chim f'(x 0)∆x.
, then ∆y~f'(x 0)∆x. Also, f'(x 0)∆x is the main and at the same time the linear part of the increase in ∆y (linear - means to avenge ∆x at the first step). The addendum is called the differential of the function y=f(x) at the point x 0 i designate dy(x 0) chi df(x 0). Again, for larger values of x
dy=f′(x)∆x. (one)
Consider dx=∆x, then
dy=f′(x)dx. (2)
Butt. Find similar and differentials of given functions.
a) y=4tg2x
Solution:
differential: 
b) 
Solution:
differential: 
c) y=arcsin 2 (lnx)
Solution: 
differential: 
G) 
Solution:
= 
differential: 
Butt. For the function y=x 3 to know the viraz for ∆y and dy with different values of x and ∆x.
Solution. ∆y = (x+∆x) 3 – x 3 = x 3 + 3x 2 ∆x +3x∆x 2 + ∆x 3 – x 3 = 3x 2 ∆x+3x∆x 2 +∆x 3 dy=3x 2 ∆x At to this particular typeα(∆x)∆x = 3x∆x 2 + ∆x 3 .
Enthusiasm...