Discover the beauty of mathematics - the origin and use of calculus (understand calculus in one article)

Mathematics, the cornerstone of changing the world. Calculus is one of the three natural discoveries of the 19th century. Descartes established analytic geometry, combining numbers and graphs. The discovery and creation of calculus is a new milestone in mathematics, solving problems that cannot be solved by conventional methods. It is a great revolution. Descartes' analytic geometry and Newton's calculus are both creations, interconnected and indispensable.

Preface

Mathematics, the cornerstone of changing the world. Great mathematicians include Newton, Archimedes, Leibniz, Euler, Lagrange, Laplace, Gauss, and John Bernoulli,Poincaré, Riemann, Euclid, Klein, Descartes, Dirichlet, Fermat, Pythagoras, Lópida, etc.

Archimedes, Newton, and Gauss are the figures standing at the top of the pyramid and are geniuses in the history of mathematics. Together with Euler, these four people are recognized as the four greatest mathematicians in the history of mathematics.

Mathematics is actually quite beautiful and not boring. The reason why you feel bored is because you are accustomed to memorizing formulas due to the knowledge you have been instilled, but you have not understood the principles and put them to use. Take calculus as an example and look at its origins and powerful uses.

Calculus is one of the three natural discoveries of the 19th century. Descartes established analytic geometry, combining numbers and graphs. The discovery and creation of calculus is a new milestone in mathematics, solving problems that cannot be solved by conventional methods. It is a great revolution. Descartes' analytic geometry and Newton's calculus are both creations, interconnected and indispensable.

Calculus is an important branch of mathematics, involving many concepts such as functions, limits, derivatives, integrals, differential equations, etc. It is widely used in engineering, science, economics and other fields. Calculus is divided into five main chapters: derivatives and differentials, integrals and definite integrals, differential equations and applications, multivariate functions and partial derivatives, vector functions and curve integrals.

If you truly understand the "fundamental theorem of calculus", you will feel that this thing is not complicated. However, the great thing about this theorem is that it has a wide range of applications. Although it looks ordinary, it is very practical. Unfortunately, university courses often only talk about formulas and not principles, leaving people who listen in confusion.

Let’s take a real application to analyze what calculus is. Calculus is not an independent word, it includes differential calculus and integral calculus.

Start with points

Let’s first understand integrals.Why start with integrals?

Archimedes studied integrals for calculating area in the 3rd century BC, and Newton and Leibniz figured out differential calculus in the 17th century. There is a difference of more than 1800 years between the two.

Almost all college mathematics textbooks first explain differential calculus and then introduce its inverse operation - indefinite integral. Furthermore, the definite integral used to calculate area is defined as the difference of indefinite integrals. While it certainly makes sense to teach mathematics logically in the order outlined above, historically the order of development has been exactly the opposite. Archimedes studied integrals for calculating area in the 3rd century BC, and Newton and Leibniz figured out differential calculus in the 17th century. There is a difference of more than 1800 years between the two.

Historically, points were discovered first, and there are certain reasons for this. Integral is directly related to the calculation of specific quantities such as area and volume. In addition, before studying differential calculus, you must first accurately understand concepts such as infinitesimals and limits. For example, the speed of an object's movement needs to be defined through differential definition. However, because there was no concept of determining limits in the ancient Greek period, Zeno's "flying arrow does not move" paradox appeared.

Therefore, before learning complex differentials, it is best to correctly grasp the integrals that are relatively easy to understand intuitively, and then think about their inverse differentials. Therefore, let’s explain the integral first. Whether you are confused about calculus in college or you are planning to start learning calculus, you can try to "start with integrals first."

Integration starts by calculating the area of the figure. The units of area include square meters, square kilometers, etc., which all contain the word "square". The area of a square with side length 1 meter is equal to 1 square meter. In other words, the area is based on squares, and the area of the calculated figure is equivalent to several squares. If it is a rectangle, how to calculate the area? In primary school, we have learned that the area of a rectangle is the product of its length and width, but now we pretend that we don’t know how to calculate the area of a rectangle. We only know the area calculation of a square and have not learned the calculation formula of a rectangle.

Assume that the rectangle is 1 meter wide and 2 meters long. If you divide the rectangle into two parts vertically from the middle, you will get two squares with side lengths of 1 meter, so the area of the rectangle is equal to 2 square meters. That is, the product of length and width equals the area of the rectangle.

Next, assume that n and m are both natural numbers. It is known that the rectangle is n meters wide and m meters long. As long as the width is equally divided into n parts and the length is equally divided into m parts, we can get n × m sides with a length of 1 meter. square (Figure 7-2). The area of the rectangle is exactly n × m times the area of the square, or n × m square meters. The result is still equal to the product of length and width.

What do points count for?

Using Archimedes' pinching theorem, the area of more complex curved figures can be calculated. Expressed in Cartesian coordinates, as shown in Figure 78, the straight line can be expressed as y = ax + b, and the parabola can be expressed as

Suppose we know a certain function f(x), then let's think about the curve y = f(x). As shown in Figure 7-9, assuming that the value of f(x) is always greater than 0 in the interval a ≤ x ≤ b, then let’s study the curve y = f(x) and y = 0, x = a, x = b The figure A enclosed by these three straight lines (the shaded part in the figure). If you know how to calculate the area of figure A, you can calculate the area of any figure enclosed by a curve through the grouping method.

The curve y = f(x) rises or falls along the y-axis. For the convenience of calculation, assume that y = f(x) is always increasing in the interval a ≤ x ≤ b. In other cases, the interval a ≤ b is divided into two parts, namely the interval where f(x) is increasing and the interval where f(x) is decreasing. Just substitute the following methods into the above two intervals respectively.

In order to use Archimedes' pinching theorem to calculate the area of the figure A, first divide the interval a ≤ x ≤ b into n parts, such as the figures Bn and Cn in Figure 710. Graph A contains graph Bn and is also contained in graph Cn. The graphics Bn and Cn are both rectangular sets, so the area can be calculated.

As shown in Figure 711, the difference between area (Cn) and area (Bn) is equal to

That is, the area of the rectangle with base ε = (b − a)/n and height = (f(b) − f(a)). The larger the value of n, the smaller the value of ε, so the areas of the graph Bn and the graph Cn are closer. When the value of ε reaches the limit, which is equal to 0, the areas of the two figures are equal. The value when reaching the limit is the area of graph A.

The area of graph A calculated according to the above method is called "the integral of function f(x) on the interval a ≤ x ≤ b", which is expressed as

Leibniz, who founded calculus at the same time as Newton, invented the symbol

It is the elongation of the initial letter "S" of "sum". Moreover, the d in "dx" refers to the first letter of "difference". When approximating a figure as a set of rectangles, the base length of a rectangle is equal to x + ε and

The "difference" of x. The area of a rectangle with height f(x) and base e is equal to f(x)e, so the symbol dx can be used instead of e, that is, f(x)dx. That is to say,

It contains Leibniz's idea, that is, "integration means arranging rectangles with a height equal to f(x) and a base dx on the interval from x = a to b, and finding the sum of their areas."

The integral explained above follows the definition of the 19th century German mathematician Bornhard Riemann, so it is called the "Riemann integral". In fact, integrals include many categories, such as the "Lebesgue integral" proposed by the French mathematician Henri Lebesgue, the "Ito integral" proposed by the Japanese mathematician Ito Kiyoshi, etc. The Riemann integral is sufficient to handle the function problems we learned in high school, but when we need to deal with random fluctuations in values such as stock prices, we need to use the Ito integral. Ito points are also used to determine the price of options, so Ito Kiyoshi is considered "the most famous Japanese on Wall Street."

Indefinite and definite integrals

Definite integral and indefinite integral are two important concepts in calculus. The definite integral can be understood as the area of a function within a certain interval, while the indefinite integral is the antiderivative (original function) of the function. Integral is another important branch of calculus theory, which can be used to calculate the area, arc length, volume, center of mass, etc. under the curve.

Integral is divided into indefinite integral and definite integral. Definite integral is a number, while indefinite integral is an expression. When calculating the definite integral, we need to first find the original function of the integrand, and then calculate the definite integral. Through the Newton-Leibniz formula, we can convert the calculation of definite integrals into the calculation of indefinite integrals. The existence of indefinite integrals and definite integrals is also different. There must be an indefinite integral for a continuous function, and a definite integral only exists when the function has only a finite number of discontinuous points in a finite interval and the function is bounded.

In short, indefinite integral is an important concept in calculus. It is the process of finding the original function or antiderivative of a function.

Newton-Leibniz formula

The Newton-Leibniz formula, often called the Fundamental Theorem of Calculus, reveals thedefinite integralThe connection with the primitive function of the integrand orindefinite integral. The content of the formula is the definite integral of a continuous function on the interval [a, b]< a i=8> is equal to the increment of any one of its original functions on the interval [a, b].

The discovery of the Newton-Leibniz formula enabled people to find general methods for solving the problems of the length of a curve, the area enclosed by a curve, and the volume enclosed by a curved surface. It simplifies the calculation of definite integrals. As long as the original function of the integrand is known, the exact value of the definite integral or the approximate value with a certain precision can always be obtained.

The Newton-Leibniz formula is a bridge between differential calculus and integral calculus. It is one of the most basic formulas in calculus. It proved thatdifferentiation and integral are inverse operations of each other. It also marked the formation of a complete system of calculus in theory. From then on, calculus became a real discipline. Subject.

The Newton-Leibniz formula simplifies the calculation of definite integrals. This formula can be used to calculate the arc length of a curve, the area enclosed by a plane curve, and the three-dimensional volume enclosed by a space surface. This formula is widely used in practical problems, such as Calculate the filling volume of the dam body.

The Newton-Leibniz formula is also widely used in physics to calculate the distance of a moving object, the work done by a variable force along a straight line, and the universal gravitation between objects.

The Newton-Leibniz formula promoted the development of other branches of mathematics. The formula is reflected in differential equations, Fourier transforms, probability theory, functions of complex variables and other branches of mathematics.

The Newton-Leibniz formula provides a simple basic method for calculating definite integrals, that is,To find the value of the definite integral, you only need to find the integrand f(x) An original function F(x), and then calculate the increment F(B)-F(a) of the original function on the interval [a,b]. This formula reduces the calculation of definite integrals to the problem of finding the original function, revealing the intrinsic relationship between definite integrals and indefinite integrals.

For example, how to find the shaded area of the following red area? S = S moment-S1-2S2, similar to the problem of solving the shadow area. Did you find it scary when you were a student? I thought I might as well find the shadow area in my heart first.

How to find the green shaded area of S2? You can use integrals, from point a to point b,

can be written as:

So what’s next? The expression is expressed, how to calculate the area of the green shade from a to b? How to calculate the definite integral?

It is generally very difficult to calculate definite integrals using definitions. The following Newton-Leibniz formula is not only a definite integral The calculation provides an effective method, and theoretically links the definite integral and the indefinite integral.

F(x) is called the original function of f(x), and f(x) is called the derivative of F(x).

According to the Newton-Leibniz formula, finding the integral becomes simple and becomes one> finding the original function.

Calculate the area A of the curved trapezoid enclosed by the parabola y=x^2 and the x-axis (assuming it goes from 0 to 1):

F(b) - F(a), is equal to 1/3 - 0 = 1/3. Therefore, the area of the enclosed part of the curve is 1/3.

Therefore, the green shaded area of s2 in the above figure can be solved using calculus. s2 = (1/3)*b^3 - (1/3)*a^3

The above f(x) = x^2 is a parabola. It is easier to see if it is replaced by a straight line. If f(x) = x, it can be seen as a uniform motion starting from the 0 coordinate:

Assume a=0,b=2, then the area of the triangle on the picture = base*height/2, which is equal to 2*2/2 = 2. If you calculate it using points:

S(b) = x^2/2 = 2, S(a) = 0, S(b) - S(a) = 2. You will find that using the triangle area formula is consistent with using integrals.

Calculation example

Below are some examples of definite integrals for easy understanding.

The inverse function of the function y=sinx is called the inverse sine function, recorded as x=arcsiny. It is customary to use x to represent the independent variable, with y represents function, so the inverse sine function is written as y=arcsinx, the domain is [-1, 1], and the value range is y∈ [-π/ 2 , π/2] ; If x = sin t, then t = arcsin x. where arcsin x is the arcsine function, which represents the corresponding sin t value for a given x value. The value of arcsin 1 is π/2. In trigonometric functions, arcsin x represents the corresponding sin t value for a given value of x. In this example, the value of x is 1, and the corresponding value of sin t is sin(π/2) = 1.

ln is the logarithm operator, and e is the exponential operator. Their relationship is the same as that of addition, subtraction, multiplication and division, indicating the relative relationship. Two operations of inverse. If y=lnx, then x=e^y (e raised to the yth power). e^x and ln(x) are the natural exponential function and the natural logarithm function respectively, which are a pair of functions and inverse functions. e is a natural constant, approximately equal to 2.718182

The conversion formula between e and In is as follows: e^ln(x)=x, ln(e^x)=x, lne=1, lnx=y, x=e^y

Derivative of a logarithmic function:

Derivatives and Differentials

Since the integral has been expressed above using the Newton-Leibniz formula, why do we need to mention derivatives and differentials? Since you simply used the formula above, how do you solve the integral? Where did the formula come from? This has something to do with derivatives and differentials.

The derivative describes the rate of change of the function at a certain point, while the differential describes the change of the function over the entire domain. Derivative (finding tangent lines) is an important part of calculus. On this basis, mathematicians have completed the solution of area by studying its inverse operation (integral).

By studying the derivatives and differentials of a function, we can obtain more information about the properties of the function, allowing us to better understand and analyze the behavior of the function. Furthermore, derivatives and differentials also play an important role in numerical calculations. By calculating the derivative of a function, we can determine information such as the extreme points, inflection points, and tangent equations of the function. Differential equations are an important tool for studying many phenomena in nature.

Derivatives and differentials are the basis of calculus theory. They can be used to describe the rate of change of curves, tangent slopes, and local growth and decrease trends. The derivative is the rate of change of the function at a certain point, expressed in the form of a limit; the differential is the approximation of the tangent line of the function at a certain point, expressed in the form of a differential operator. Derivatives and differentials are closely related. While solving the derivatives, you can also find the differential value of the function at a certain point.

Are derivatives and differentials the same thing?

Differentials and derivatives are not the same thing, and the two concepts are easily confused and confusing.

The two have different definitions, different essences, and different geometric meanings. The derivative is the quotient of the differential (the name is not rigorous, it is just called that. Modern mathematics strictly redefines the derivative with limits). The geometric meaning of the derivative can be seen as the slope of the function at a certain point, and the differential is at The increment of the dependent variable of the function in the tangential direction.

1. Definition of differential and derivative

Differential definition: From the function B=f(A), we get two number sets A and B. In A, when dx is close to itself, the limit of the function at dx is called the differential of the function at dx. The central idea of the differential is Infinite division.

Derivative definition: when the increment of the independent variable approaches zero, the limit of the quotient of the increment of the dependent variable and the increment of the independent variable.

The difference between derivative and differential is ratio and increment.

The derivative is the slope of the function graph at a certain point, that is, the ratio of the ordinate increment (Δy) and the abscissa increment (Δx) when Δx-->0.

Differential refers to the increment obtained by the vertical coordinate after the tangent line of the function image at a certain point obtains the increment Δx on the abscissa, generally expressed as dy.

2. The relationship between differential and derivative

For the function f(x), the derivative f'(x)=df(x)/dx, the differential is df(x), and the relationship between the differential and the derivative is df(x)=f'(x)dx .

Taking the function f(x) = x^2 as an example, the calculation formula of its derivative is:

In calculus, due to the existence of limits, we can gradually reduce the value of Δx to obtain f(x ) at a certain point. The geometric meaning of differential is, let Δx be the increment of point M on the curve y = f(x) on the abscissa, Δy be the increment of the curve on the ordinate corresponding to Δx at point M, and dy be the increment of the curve on the point M The tangent line of M corresponds to the increment of Δx on the ordinate. When |Δx| is very small, |Δy-dy| is much smaller than |Δx| (high-order infinitesimal), so near point M, a tangent segment can be used to approximately replace the curve segment.

The derivative describes how fast a function changes, and the differential describes the degree of change of a function. The derivative is the local property of a function. The derivative of a function at a certain point describes the rate of change of the function near this point. The differential is a functional expression used to calculate the approximate value of the dependent variable when the independent variable changes slightly.

The geometric meaning of the derivative is the slope of the tangent line, and the geometric meaning of the differential is the increment of the ordinate of the tangent line. Therefore differentiation can be used for approximate operations and error estimation. In the simplest unary case, the derivative is a definite value, its geometric meaning is the slope of the tangent line, and its physical meaning is the instantaneous velocity.

Differentials can be calculated using basic differential formulas.

The basic differential formula is one of the most basic formulas in calculus, which can be used to calculate the differential of various functions. Basic differential formulas include constant differential formulas, power function differential formulas, exponential function differential formulas, logarithmic function differential formulas, trigonometric function differential formulas, etc.

The constant differential formula means that for any constant c, its differential is 0. The power function differential formula means that for any power function y=x^n, its differential is y=nx^(n-1). The differential formula of exponential function means that for any exponential function y=a^x, its differential is y=a^xlna. The differential formula of logarithmic function means that for any logarithmic function y=loga(x), its differential is y=1/(xlna). The differential formula of trigonometric functions means that for any trigonometric function y=sin(x), y=cos(x), y=tan(x), the differentials are y'=cos(x), y'= -sin(x),y'=sec^2(x).

The basic differential formula has a wide range of applications. It can be used to solve the derivatives of various functions, thereby helping us better understand the changing rules of the function. In practical applications, differential calculus can also be used to solve problems such as tangents, extreme values, and concavity and convexity of curves. Therefore, mastering basic differential formulas is very important for learning calculus and solving practical problems.

History of Calculus

This is first due to the analytic geometry founded by Descartes, which is the product of the combination of algebra and geometry, making it possible to quantitatively express the relationship between variables, thus setting the stage for the creation of calculus.

Newton (1643-1727), great physicist, mathematician, astronomer, natural philosopher and alchemist. In 2005, he beat Einstein and was named "the most influential person in the history of science." Newton was born prematurely into a peasant family in England and barely survived after birth. In his childhood, Newton was not a child prodigy and his grades were not outstanding, but he loved reading and making toys. At the age of 17, Newton, who was still in middle school, was called back to the farm by his mother to work in farming. However, under the persuasion of Newton's uncle and the principal of the middle school, Newton was allowed to return to school after nine months of farming. The principal's persuasive speech to Newton's mother included a sentence that can be said to be the luckiest prophecy in the history of science. He said to Newton's mother: "It would be a shame for the world to bury such a genius in the complicated farming work. What a huge loss!”

In 1661, Newton was admitted to Trinity College, Cambridge, where he was taught by Barrow. While in college, he studied the works of Galileo, Kepler, Descartes, and Wallis. In terms of the formation of mathematical ideas, Descartes's "Geometry" and Wallis's "Infinite Arithmetic" had the most profound influence on him. It was these two works that led Newton to the path of creating calculus.

Newton was a genius, there is no denying this.

In 1665, Newton was 22 years old. He had just received his degree. The university was closed due to the Great Plague in London, and Newton had to go home. During the eighteen months at home, he proposed the binomial theorem, which later developed into a new mathematical theory, which is calculus in advanced mathematics. During this period he also studied optics and the laws of gravity.

Newton invented differential calculus (derivative) and integral calculus between 1665 and 1966. He called them "forward calculus" and "reverse calculus" and compiled it into "A Brief Theory of Calculus" for circulation only to colleagues. From the naming A kinematic background to Newton's calculus can be seen. Regarding the naming of flow numbers, Newton explained: "I regard time as a continuous flow or growth, and other quantities as continuous growth with time. Starting from the fluidity of time, I call the growth rate of all other quantities It’s a flow.”

Newton expressed the basic problem of calculus as "the relationship between the known flow rates, find the relationship between the flow numbers" and "the equation of the relationship between the flow numbers of the known variables, find the relationship between the flow rates".

Newton used "derivatives" to solve the problem of finding velocity with known displacement, and used "integral" to solve the problem of finding displacement with known velocity. He also pointed out how to calculate area through this inverse operation, thus establishing the basic theorem of calculus.

Although the reciprocal relationship between area calculation and tangent finding has been vaguely pointed out by a few people on special occasions in the past (such as Barrow's "Lectures on Geometry"), Newton was able to realize this reciprocity with enough sensitivity and ability. The relationship is clearly revealed as a general law and used as the basis for establishing a universal algorithm of calculus. Just as Newton himself said in "A Brief Theory of Fluids": Once the integral problem is solvable, many problems will be easily solved. Newton applied the unified algorithm he established to 16 types of problems such as finding the tangent of a curve, curvature, inflection point, finding the length of a curve, finding area and volume, finding gravity and the center of gravity, etc., demonstrating the great universality and system of his algorithm. sex.

In this way, Newton unified the various special techniques for solving infinitesimal problems since ancient Greece into two general types of algorithms - forward and reverse numerology, that is, differential (derivation) and integral, and proved the reciprocity of the two. Relationship, and further unifying these two types of operations into a whole, this is his achievement that surpasses his predecessors. It is in this sense that we say that Newton invented calculus.

At that time, there was a famous family of mathematicians in Switzerland, the Bernoulli family. John Bernoulli raised the steepest descent line problem to his brother Jacob Bernoulli. After John Bernoulli solved it, he challenged European mathematicians.

Faced with the challenge, Newton quickly solved the problem. It is worth mentioning that there were many talented mathematicians in Europe at that time, and Leibniz, Lobida, Jacob Bernoulli and others also solved this problem.

To talk about how powerful Newton is, in the "List of 100 People Who Impacted the Course of Human History" written by Mike Hart, Muhammad ranked first and Newton ranked second, before Jesus.

What's even more amazing is that Newton lived to be eighty years old, but he spent the next forty years studying theology. In other words, Newton's achievements were all developed in the first half of his life, and around the age of 22 is an important period.

Newton studied calculus mainly to serve physical calculations. Let’s take a look at how Newton derived calculus:

Newton's Calculus

The overall idea of Newton's inductive calculus is:

It is proved that derivation is the inverse operation of indefinite integral, that is, the first fundamental theorem of calculus ("Advanced Mathematics" Tongji Edition is to find the derivative of the upper limit function of the integral).
This led to the Newton-Leibniz formula, the second fundamental theorem of calculus.

1.1 The First Fundamental Theorem of Calculus

Newton tried to prove the following conclusion:

Under the known function curve, the area of the interval is:

If the upper limit a is replaced by x, then the area under the curve is a function, which we call the integral upper limit function:

Below I will introduce these two steps respectively. From these two steps we can see respectively:

Step 1: Derive the first fundamental theorem of calculus
Step 2: Show how Newton solves derivatives

1.1.1 Step 1

At that time, the word derivative did not exist, but there was an equivalent word, which was rate of change, so Newton started by finding F(x)

starting from the rate of change. To find the rate of change, Newton thought like this:

Make a rectangle with B and b as the base, the part enclosed by the dotted line in the picture.

It can be seen that the smaller o is, the closer the area enclosed by the dotted line on the figure is to the product of o and f(x).

Newton asserted that when o is small enough

The increment of F(x) with respect to o is o*f(x). According to the definition of rate of change,

Note that there is a small dot on the head for the flow number of F(x) (the flow number here refers to the rate of change), which is the current derivative.

So far:

In fact, the first fundamental theorem of calculus has been reached here:

It can be seen from here that the area function F(x) is actually a primitive function of f(x).

1.1.2 Step 2

The next step is to calculate what f(x) is equal to. Fermat and Cavalieri calculated:

Just replace it and you'll get:

Based on this conclusion, Newton continued to push forward (binomial refers to n being a natural number, while generalized binomial refers to n being a rational number. The generalized binomial formula is a mathematical inference that Newton was very proud of):

Proof completed

1.2 Newton-Leibniz formula

Starting from the first fundamental theorem of integral calculus, it is easy to derive the famous Newton-Leibniz formula (the second fundamental theorem of calculus).