When I talked about differential in the book I read, I just mentioned that to find differential is to find derivative, plus DX, dy, What is the meaning of differential?

Tsinghua calculus (1) textbook for differential is explained as follows: differential and derivative are to discuss the local properties of function, differential is to discuss the possibility of continuous function approximated by linear function locally, derivative is to discuss whether the change rate of continuous function value near a certain point tends to a fixed value, in univariate function

Several meanings of differential are

Geometric meaning
Let Δ X be the increment of point m on the curve y = f (x) on the abscissa, Δ y be the increment of point m corresponding to Δ x on the ordinate, and Dy be the increment of point m tangent corresponding to Δ x on the ordinate. When | Δ x | is very small, | Δ y-dy | is much smaller than | Δ y | (infinitesimal of higher order), so near point m, we can use tangent segment to approximate curve segment
Algebraic meaning
An element X of function f of one variable in the domain of definition_ The geometric meaning of the derivative at 1 is the graph (f) corresponding to the function f (which is a curve) at point (x)_ 1,f(x_ 1) The differential is an infinitesimal, not a number, It has no geometric meaning unless it is extended to "external differential forms of a function"; the extended external differential is a fiberwise linear homomorphic mapping from the tangent vector bundle of a domain to the tangent vector bundle of a real number set, Each coordinate of a sequence is a directed line segment. For example, DX can be understood as a group of directed line segments sent to the right at point x (fix this x first). The starting point of each directed line segment is point x, and the ending point is changed, but the change is "directional", that is to say, the "module length" of the second coordinate of the directed line segment is half less than the "module length" of the first coordinate, The module length of the third is half smaller than that of the second, and so on. The limit of this infinite sequence is a zero vector (that is, the vector that x points to itself). It should be noted that these directed line segments can be regarded as vectors, but they are not "free vectors", Because their starting points are fixed, which is X

In differential

Equal to, means not equal to

Differential? What is the definition?

Let the function y = f (x) be defined in a neighborhood of point x, and give the increment Δ x (x + Δ x is still in the neighborhood) to the independent variable. The corresponding function has increment Δ y = f (x + Δ x) - f (x), if Δ y can be expressed in the following form Δ y = a * Δ x + O (Δ x), where a is independent of Δ x, and O (Δ x) is infinitesimal of higher order than Δ X

Why do we define the concepts of higher order infinitesimal in mathematics? What does it mean that a is a higher order infinitesimal of B?

This is different from low order infinitesimal

Understanding of the concept of infinitesimal
"If limf (x) = 0, then f (x) is called infinitesimal under the change trend of independent variable x
I don't understand why infinitesimal is a variable whose limit is 0. 0 is not the smallest. What about negative numbers?
Why absolute value?

Here, the negative infinity is not infinitesimal, but is called negative infinity. The sign is a minus sign before an inverted 8,
The infinity of the same positive number is called positive infinity,
So here in mathematics, only when it tends to zero is infinitesimal

Is infinitesimal and infinitesimal a concept? What's the difference?

No, infinitesimal is a microcomponent and can be used for integration, such as DX, while infinitesimal corresponds to infinity and is a number

Definition of equivalent infinitesimal, infinitesimal of the same order and equivalent infinitesimal?

In the same change process of independent variable, f (x) - > 0, G (x) - > 0, and limf (x) / g (x) = K. if k = 0, then f (x) is said to be infinitesimal higher than g (x); if K does not = O, then f (x) is said to be infinitesimal of the same order than g (x); especially, when k = 1, f (x) and G (x) are said to be equivalent infinitesimal, denoted as f (x) ~ g (x)

The definition and expression of finding infinitesimal of order k

If & nbsp; Lim β / α ^ k = C ≠ 0, κ & gt; 0, then β is the infinitesimal of order κ about α. For example: Lim1 cosx ^ 2 / x ^ 2 = [2Sin (x / 2) ^ 2] / x ^ 2 = sin (x / 2) ^ 2 / 2 (x / 2) ^ 2 = 1 / 2C ≠ 0, (x → 0), so when (x → 0), 1-cosx ^ 2 is the second-order infinitesimal about X

Who can give the definition of equivalent infinitesimal?

Popular point, is the division of two infinitesimals, the result is an integer (1,2,3 ）So these two numbers are equivalent infinitesimal! OK, ha ha

When I talked about differential in the book I read, I just mentioned that to find differential is to find derivative, plus DX, dy, What is the meaning of differential?