Leibniz studied the problems and methods of research that were different from Newton's, but in essence the same, using limit calculations. Leibniz first defined the function, using the term function in one of his 1673 manuscripts to denote the ordinate of any quantity that changes as the points on the curve change, and then he studied the tangents of the curve. The tangents of the curve are related to the derivative, are more geometrically intuitive than the velocity, and are closely related to optics as well as planetary motion. For a given curve
y=f(x) and point x0, we want to get tangents of curves passing through points A(x0, f(x0)). As shown in the following figure:
Figure 1
Point A(x0, f(x0)) is on the curve y=f(x), where f(x0) is the coordinate of the y-axis corresponding to x=x0. By definition , a tangent is a straight line that passes through point A and has only one intersection point with the curve in its attachment to point A. According to the straight-line equation, we only need to find the slope of the tangent again to get the tangent equation. But how do you calculate the slope? Similar to Newton's thinking, an increment h is given at x0 of the x-axis, so that a corresponding increment m=f(x0)-f(x0) can be obtained at f(x0)-f(x0) on the y-axis. As shown in Figure 1, the ratio m/h is the slope of the serant AB, where the coordinates of B are (x0+h, f(x0+h)). Obviously, when the increment h tends to 0, the increment m also tends to 0. It is conceivable that the secant AB and the curve will have only one intersection, so Leibniz defines the ratio at this time as the slope of the tangent, and is represented by the symbol dy/dx. This symbol has been used to this day, and we call dy/dx the derivative of the function y to x. After about twelve years of effort, Leibniz published his first paper on calculus in the Faculty Journal in 1684, which was also the first to systematically expound calculus. Comparing the "instantaneous velocity = [f(t0+△t)-f(t0)-f(t0)]]/△t" in the previous article "Analysis of The Thought of Calculus--- Newton" shows that Leibniz's method is essentially the same as Newton's method, and like Newton, Leibniz cannot explain the limit operation ground rules very well. But Leibniz was a great philosopher, and in the face of criticism from all sides "excessively harsh", he gave a philosophical and still valuable answer in his 1695 article in the Teacher's Journal: "Excessively prudent steps should cause us to abandon the fruits of creation." At the same time, Leibniz further considered the order of infinitesimals, arguing that when h is an infinitesimal, any power of h such as h2, h3 would be a smaller amount to be ignored. In a letter to a friend in 1699 he wrote:
"It would be useful to consider such an infinitesimal quantity, when calculating their ratios, not to treat them as zeros, but to discard them whenever they appear with incomparable quantities. For example, if we have x+dx, we discard dx. ”
As you can see, Leibniz has already spoken of the idea of the higher order infinitesimals that we often use in analytics today. If the curve equation is y=ax2, it is similar to the previous article "Calculus Thought Analysis--- Newton"
m/h=(39.2h+4.9h2)/h
=39.2+4.9h
the calculation of dy/dx=2ax, if you let a=4.9 and x=4, then dy/dx=39.2, which is consistent with the results of the above calculation.
Differentiation is far less intuitive than derivatives, but is closely related to derivatives. When the derivative dy/dx=2ax, the corresponding differential form is dy=(2ax)dx. We know that when the derivative, the differentiation is an approximate representation of the function increment, and when x gets an increment dx, y gets an increment dy, which is a linear function of dx with an intercept of 0 and a slope of the derivative. Of course, this increment must be very small, otherwise it will cause a large error.
The original purpose of the integral was to calculate the area of the area enclosed by the curve. This is a very old question that goes all the way back to Theodox and Archimedes of ancient Greece. By the 17th century, with the help of a Cartesian coordinate system, such a problem could be explained more clearly.
Figure 2
As shown in Figure 2, to calculate the area of curve y= x2, a≦ x ≦b. Because we calculate the area of the rectangle, we think of a solution to the problem from the rectangle. Divide the interval [a,b] into n equal parts and the points x1,... xn-1, xn, where xn = b, so that n small rectangles with a width of (b-a)/n and a height of yi =xi2 can be obtained, and the sum of the areas of these small rectangles is
(b-a)•(x12+...+xn2)/n （1）
The sum of this area is obviously greater than the area under the curve, but as n gradually increases, the difference between the areas will gradually decrease. As with the idea of instantaneous velocity, if n tends to infinity (equivalent to 1/n toward 0), the sum of the above areas is equal to the area under the curve.
Let's calculate the equation (1), which is known by definition that for i=1,...,n, there is xi=a+i(b-a)/n, so equation (1) can be written as
[(b-a)/n]•[a2+(2a/n)(b-a)∑i+[(b-a)2/n2]∑i2]
where ∑i means sum of i from 1 to n, and we know that this sum is equal to (1/2)n(n+1); ∑i2 means sum of i2 from 1 to n, and this sum is equal to (1/6)n(n+1)(2n+1). By calculation we can get the above equation as:
(b-a) [a2+(1+1/n)a(b-a)+(b-a)2(1/3+1/2n+1/6n2)]
According to Leibniz, terms of the higher order infinitesimals 1/n and 1/n2 can be ignored, so we get the area under curve y = x2 on the interval [a,b] = (1/3) (b3-a3)
What a wonderful calculation method, what a wonderful result!
The above calculation method can be generalized to the general, if we want to calculate the area of curve y = f(x), a≦ x ≦b, corresponding to the "instantaneous velocity = [f(t0 +△t)-f(t0)]]/△t" formula can get the sum of the areas of the small rectangle
(b-a)∑(1/n)f(xi)
The sum is then calculated, ignoring the higher-order infinitesimals. Leibniz was a master of symbol-making, and he replaced this series of processes with an elongated ∑ symbol, and replaced (b-a)/n with the differential symbol dx that he had invented, so that the area under the curve y=f(x) on the interval [a,b] ∫ baf(x)dx
Thus, the points are established. From analytic geometry, a function can always correspond to a curve, so the integral has a good intuitive explanation: the integral of a function is the area under the corresponding curve.
But from the above operations, we can see that summation is not a simple thing, is there a simpler way to calculate the integrals of common functions? Again, to analyze the function y=f(x)=x2, we already know the integral of this function, and if F(x)=x3/3, then the result of the integral can be written as F(b)-F(a). It is easy to verify that the derivative of F(x) is exactly f(x), so a bridge is built between the derivative (differentiation) and the integral: if the derivative of F(x) is f(x), then
∫baf(x)dx=F(b)-F(a)
In honor of the contributions of Newton and Leibniz, the formula is called the Newton-Leibniz formula.
It is easy to see that the essence of the integral is also the use of limit operations, but for limit operations with such power, people still cannot clearly express the rules of this operation, so they cannot give a reasonable explanation. This problem will be explained in the follow-up "Establishment of Limit Theory".