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Knowledge related to mathematics:

Set function limit, derivative, differential, partial derivative vector sine cosine theorem least square matrix, orthogonal matrix

Set: refers to the totality of things with certain properties, and the things that make up the set are called elements.

  usually represents collections in uppercase and elements in lowercase; enumeration, description

  enumerated method: a = {A1 ∈ a2 ∈ a3 ·a}, A1 recital A

  description: B = {x | x ^ 2-1x 0}, {x | the properties of x}. The solution of the equation is the element of the B set.

Set properties:

  ≠ B if all the elements of An are elements of the B set, it is said that A (BMague An is contained in B, if A contains B, then the set AB is equal; if A ≠ B, then An is a proper subset of B, A ∈ / ≠ B.

To hand in and supplement:

  A ∩ B, A ∪ B, A ^ c complement

Function

Parity function: F (- x) =-f (x), f (x) = f (- x)

Elementary function:

Power function: y = X ^ u u ∈ constant exponential function: y = a ^ x; (a > 0 and a ≠ 0) logarithmic function: y=logaX (when a > 0 and a ≠ 0 is y=ln x) trigonometric function: y=sin x Magi y = cos x Magi ylematan x inverse trigonometric function: y=arcsin x Magnec arccos x Magi yarctan x

Properties of continuous functions with closed interval

Boundedness and maximum and minimum Theorems

There are defined f (x), x 0 ∈ I on interval I, such that for any x ∈ I, there are f (x) ≤ f (x 0), f (x 0) ≥ f (x), that is, f (x 0) is the maximum and minimum value of f (x) on interval I.

Zero point theorem

If x0 makes f (x0) = 0, then x0 is called the zero of f (x).

Let f (x) be continuous on the closed interval [a], and the sign of f (a) is different from that of f (b), that is, when f (a) * f (b) N, inequality | Xn-a | limit of ∞ function

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Slope of MN line: tan θ = (y-y0) / (x-x0) = [f (x)-f (x0)] / (x-x0)

The line is tangent to the curve when there is x-> x _ 0.

Derivative definition:

The above slope can be reduced to the limit: lim [f (x)-f (x0)] / (x-x0), x-> x0.

Definition: let the function yfolf (x) be defined in some field of point x0. when the independent variable x obtains the increment △ x at x0 (x0 + △ x in the domain), then △ yfolf (x0 + △ x)-f (x0), when △ x-> 0 (that is, x-> x0), the function is said to be derivable at x0, and the derivative of the function is called f `(x0).

  f `(x0) = lim (△ y / △ x) = lim [f (x0 + △ x)-f (x0)] / △ x

  can also be recorded as: Y* | x=x0, dy/dx | x=x0

Geometric meaning of derivative: slope of tangent

Derivatives of commonly used elementary functions

1. (C)'= 0 2. (x ^ u)'= ux ^ (x*lna 1) 3. (sinx)'= cosx4. (cosx)'=-sinx5. (tanx)'= sec^ 2 x 6. (cotx)'=-CSC ^ 2 x 7. (a ^ x)'= a ^ x * lna8. (e ^ x)'= e ^ x9. (logaX)'= 1 / (x*lna) 10. (lnx)'= 1/x11. (1max)'= 1 / (x ^ 2)

The law of derivation: the derivation of compound function

[U (x) ±v (x)]'= u'(x) ±v'(x)

[U (x) v (x)]'= u'(x) v (x) + u (x) v'(x)

[U (x) / v (x)]'= [u' (x) v (x)-u (x) v'(x)] / v ^ 2 (x)

Dy/dx= (dy/du) * (du/dx)

The relationship between differential and derivative of function

Dy = f'(x) * dx

Differential definition

Let the edge length of the sheet be x 0 and the area A, because the area of the sheet changes when the temperature changes, the corresponding length increases △ x, and the area increases △ A correspondingly.

△ A = (x0 + △ x) ^ 2-x0 ^ 2 = 2x0 △ x + (△ x) ^ 2

= = > General: △ yearly A △ x + 0 (△ x)-> replace (△ x) ^ 2 that is (△ x) is very small

When △ x higher order infinite hour A ≠ 0, △ yearly A △ x

The function is expressed as: △ yinvf (x0 + △ x)-f (x0) = A △ x + 0 (△ x). It is said that the function yquadf (x) is differentiable at point x0, while A △ x is called the differential of the function at point x0 corresponding to the independent variable △ x, denoted as dy, dy=A △ x.

When △ xmuri-> 0; △ y / △ xmura + o (△ x) / △ x = > A=lim (△ y / △ x) = f'(x0) thus it can be seen that the necessary and sufficient condition that the function f (x) is differentiable at x0 is that the function is derivable at point x0: dy=f'(x0) △ xmure-> dy=f'(x) dx.

In the sense of differential geometry, tangents can be used instead of curve segments, linear instead of nonlinearity, approximate calculation, error estimation differential theorem.

Fermat Lemma, that is, the definition of the maximum and minimum value of the continuous interval property mentioned above makes f (x) ≤ f (x 0), f (x 0) ≥ f (x), then f'(x) = 0, which can be proved by the derivative condition and limit of f (x) at x 0. The point where the derivative is equal to 0 is usually called the stationary point or critical point of the function.

If the function f (x) satisfies continuity on the interval [a]; it is derivable in the open interval, and the function f (a) = f (b) at the end point, then there is at least a little e in the interval such that the monotonicity and extremum judgment of the function f'(e) = 0 (judging minima and minima by finding the stationary point of the function from the above theorem)

Let the function yquof (x) be continuous on [a. B] and derivable on (a. B):

If f'(x) > 0, then   f (x) increases monotonously on the interval.

  if f'(x) 0, then the graph of y'(x) is concave on the interval = > minimum

  if f'(x) Maxima

If the second derivative is 0. The size is judged directly by monotonicity. If f'(x) ≠ 0, the size can be judged by the second derivative, as above.

Note: the maximum value problem: F (x) is derivable except for a finite number of points in the open interval, and there are at most a finite number of stationary points and non-differentiable points. The extreme value may be a stationary point or a non-differentiable point.

Partial derivative

When   studies the unary function, we introduce the concept of derivative from studying the rate of change of the function, and we also study the rate of change of the multivariate function, but there is more than one independent variable of the multivariate function, and the dependent variable and the independent variable are more complex than the univariate function. In this case, the independent variables are considered one by one, and the other independent variables are considered as constants. The derivative in this case is called the partial derivative. Similar to the definition of a unary function.

  corresponds to the differential of one variable, and the multivariate introduces total differential: dz= (∂ z / ∂ x) △ x + (∂ z / ∂ y) △ y: △ Xmure-> dx

The extreme value problem of   bivariate function can generally be solved by partial derivative, which is similar to unary.

Theorem 1: suppose that the function zpropagf (x0) has a partial derivative at the point (x0) and an extreme value at the point f (x0), then fx (x0) = 0 (x0) = 0 (0)

The solution of   isomorphic first order partial derivative = 0 is called stationary point, which is not necessarily the extreme value. Theorem 2: study whether the stationary point is the extreme value.

Let the function z _ quof (x _ 0) be continuous and have a first-order second-order continuous partial derivative in a certain field of a point (x _ 0), and fx (x _ 0 ~ 0) = 0refy (x _ 0 ~ y _ 0) = 0, such that the second order partial derivative: fxx (x _ 0 ~ Y0) = A ~ (x _ 0 ~ ~) = B ~ fyy (x _ 0 ~ y0) = C, then the function obtains the extreme value at the point (x _ 0 ~ (0) ~ (0) ~ (0)).

  1.AC-B ^ 2 > 0 has extreme value, A0 has minimum value.

  2.AC-B ^ 20, y > 0)

The value of   Ax=2 (ymur2 / (x ^ 2)) = 0 Magi Ayko2 (xMurray 2 / (y ^ 2)) = 0 colors = > x y

The above extreme value is limited to the definition domain, and there are no other conditions. Lagrangian multiplication adds conditional extreme value.

The formula: l (x _ ()) = f (x _ ()) + r φ (x _ ()), and the condition φ (x _ ()) = 0, which can be extended to multivariate application.

Simultaneous solution equation:

  fx (xpene y) + r φ x (xpene y) = 0

  fy (xpene y) + r φ y (xpene y) = 0

  r φ (XBI y) = 0

For example, when the conditional surface area is a ^ 2, and the volume is the largest?

φ (x ~ (th) y ~ z) = 2xx ~ 2yz ~ 2xz-a ^ 2 = 0, v=xyz joint solution can be obtained. Least square method, linear regression prediction: a method commonly used in practice for algorithms with the above extremes

The method of selecting constants for unary linear equations based on the condition that the sum of squares of deviations is minimum is called the least square method.

  example: in order to measure the wear speed of the tool, an experiment is carried out: after a certain period of time (such as every hour), the thickness of the tool is measured, and the data is obtained: sequence number i01234567 time ti/h01234567 tool thickness yi/mm27.026.826.526.326.125.725.324.8

In order to determine the relationship between time and tool thickness, the tracing point method is used to observe the data in the Cartesian coordinate system.

The point in the graph is roughly close to a straight line and has a negative linear correlation. We can set f (t) = at+b,a,b constant.

Because these points are not in a straight line,   can only require that the value of the function at each point of the experiment is as small as possible from the experimental results, that is, the error of each point should be minimized: ▲ = yi-f (ti).

Can each deviation be minimized by the sum of deviations: ∑ [yi-f (ti)]. It can be seen from the figure that the data points are distributed on both sides of the straight line, and if the summation method is used, the deviations will be positive or negative and will cancel each other out. The offset deviation can be avoided by taking an absolute value: ∑ | yi-f (ti)] | (iMag0, 1, 2, 7), but it is not easy to analyze and discuss. The square of any real number is positive or zero: M = ∑ [yi-f (ti)] ^ 2.) this method is the least square method.

At this time,   finds out when M takes the minimum value and what is the value of yi,ti b: because the yi,ti is known, the function is reduced to solving it, and the independent variable is regarded as aMaginb: the extreme value of the partial derivative mentioned above is discussed:

  Ma (aformab) = 0

  Mb (aformab) = 0

At this point,   calculates the relevant terms of aformab and can find out: yawning attainable breadth *

The prediction of the univariate linear regression model uses the least square estimation of the parameters, and the above general formula is the solution of the regression line, which is also one of the important features.

Vector: both size and direction (vector)

The magnitude of the vector is called the module of the vector; note here bold represents the vector, I, j, k space Cartesian coordinate system unit vector

Vector linear operation: starting point-> end point

A+b=AB+BC=c

B+a=AD+DC=AC

AB=AO+OB=OB-OA=b-a

Let a = (ax,ay,az) b = (bx,by.bz) = = > a=axi+ayj+azk

Aaccounb equals to the addition of corresponding coordinates

Module-Pythagorean Theorem of Vectors

Let r = (x _ ray _ y _ 2 _ z) = OM, OP=xi, OQ=yi, OR=zi

OM=OP+OQ+OR

| | r | = | OM | = √ [| OP | ^ 2 + | OQ | ^ 2 + | OR | ^ 2] |

| | r | = √ x ^ 2 + y ^ 2 + z ^ 2 |

Quantity product

In the problem of physical work, the result of doing such an operation on individual vectors an and b is a number, which is equal to the cosine product of the cosine product of | a |, | b | and their included angle θ, which is called the quantitative product of these two vectors and recorded as a.

  a ·b = | a | × | b | cos θ

Coordinate representation: a ·b = axbx+ayby+azbz-- the corresponding coordinates are multiplied and added

Note: the vector product is a vector: Centra × b, which can be calculated by using the third-order determinant, point multiplication and × multiplication distinguish sine and cosine theorem: cos θ cosine similarity judgment attribute similarity

Positive: any triangle whose sides are equal to the sine of its opposite angle and equal to the diameter of the circumscribed circle.

The square of either side is equal to the sum of the squares of the other sides minus twice the cosine of the angle between the two sides.

You can also have the above picture: c=AB=b-a to prove that both sides are squared and get the communication knowledge of cosine theorem according to the definition of vector product.

The signal is the carrier of the message

Information and its measurement

The degree of uncertainty of   events can be described by their occurrence probability. The amount of information contained in the message is closely related to the probability of occurrence of the message. The smaller the probability of occurrence of a message, the greater the amount of information contained in the message. Suppose p (x) represents the probability of the occurrence of the message, and I represents the amount of information in the message. According to the relationship described: I = I [p (x)]

The smaller the   p (x) is, the larger the I is, and vice versa, and when p (x) = 1, the ∞ p (x) = 0my.

  I=loga [1max p (x)] =-loga [p (x)]

The unit of information content of   is related to the base number of a, which is bit bit at axiom 2 and Harlett Hartley when it is Knight nat;a=10.

  for non-equal probability discrete data sets; average information quantity representation is also known as entropy of information sources

  H (x) = p (x1) [- log2 p (x1)] + p (x2) [- log2 p (x2)] +. + p (xm) [- log2 p (xm)] =-∑ p (xi) * log2 p (xi)

Logarithmic operation

Nature:

Multiplication property of logarithm: division property of log (ab) = loga+logb logarithm: division property of log (a) loga-logb logarithm: log (b ^ n) = (nagoc) logb, m is the base of logarithm: log (b) = log (b) / log (a) commonly used: log (b) = log (b) / log (a) (based on 10) log (b) = ln (b) / ln (a) (based on e)

In linux, use:

Log (x) returns the natural logarithm of x e such as finding the natural logarithm of 10: awk 'BEGIN {fl=log (10); print fl}' if you seek log (2 and 10), with 2 as the base, the logarithm of 10: awk 'BEGIN {fl= (log (10) / log (2)); print fl}' # awk 'BEGIN {a = (log (4) / log (2)); printf "% d\ n"

Matrix.

Matrix elementary (row, column) transformation

Swap two lines

Multiply the non-zero parameter k by all elements of a row, or add it to a row

Determinant operation

Algebraic codeterminant of n-order determinant between principal diagonal and secondary diagonal

High-order conversion to low-order 3muri-"2"

In the third-order determinant, after the I row and j-th column of the element aij are crossed out, the remaining elements form a second-order determinant in meta-order, which is called the algebraic cofactor of aij, which is marked as Mij. The sign (- 1) of the remainder is preceded by the power of (iSj).

Then the value of the third-order determinant is equal to the sum of all the meta-elements of any row or column of the determinant and their algebraic remainder.

Kramer's Law for solving Linear equations

Gaussian elimination, how to solve n unknown variables and equations

Note: the Kramer method is only applicable to the system of linear equations in which the number of unknown numbers is equal to the number of equations, and it is not applicable if it is not equal.

Inference:

If the system of homogeneous linear equations has a non-zero solution, then the coefficient determinant D must be equal to 0.

Eigenvalues and Eigenvectors

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