拯救微积分学的基本定理

2013年6月18日，无穷小放飞互联网之后，微积分学的许多重要定理被菲氏微积分的徒子徒孙们在网络上公开玷污（强奸）了、原意扭曲了，令人心痛、可气。怎么办呢？努力拯救之！

比如说，微积分学基本定理，这原本是一个基本定理，被这帮家伙说成是一个计算公式，叫什么“牛顿-莱布尼兹”公式，完全掩盖了它的理论内容。

袁萌   10月6日   

附：微积分学基本定理的原文，供参考。

第4.2节 微积分学的基本定理

4.2 FUNDAMENTAL THEOREM OF CALCULUS

In this section we shall state five basic theorems about the integral, culminating in the Fundamental Theorem of Calculus. Right now we can only approximate a definite integral by the laborious computation of a finite Riemann sum. At the end of this section we will be in a position easily to compute exact values for many definite integrals. The key to the method is the Fundamental Theorem. Our first theorem shows that we are free to choose any positive infinitesimal we wish for dx in the definite integral.

THEOREM 1

Given a continuous function f on [a, b] and two positive infinitesimals dx and du, the definite integrals with respect to dx and du are the same.

From now on when we write a definite integral _____f(x)dx, it is understood that dx is a positive infinitesimal. By theorem 1, it doesn’t matter which infinitesimal.

The proof of Theorem 1 is based on the following intuitive idea. Figure 4.2.1 shows the two Riemann sums _____f(x)dx and ___ f(u) du. We see from the figure that the difference ______f(x) dx -_____ f(u) du is a sum of rectangles of infinitesimal height. These difference rectangles all lie between the horizontal lines y= -ε and y=ε, where ε is the largest height. Thus -ε(b-a) ____ _____ f(x) dx -____f(u) du≤ ε (b -a). Taking standard parts,

Figure 4.2.1

Theorem 1 shows that whenever Δx is positive infinitesimal, the Riemann sum is infinitely close to the definite integral,

This fact can also be expressed in terms of limits. It shows that the Riemann sum approaches the definite integral as Δx approaches 0 from above, in symbols

Given a continuous function f on an interval I, Theorem I shows that the definite integral is a real function of two variables a and b,

We now formally define the area as the definite integral shown in Figure 4.2.2.

Figure 4.2.2

DEFINITION

If f is continuous and f(x) ≥0 on[a,b], the area of the region below the curve y=f(x) from a to b is defined as the definite integral:

The next two theorems give basic properties of the integral.

THEOREM 2 (The Rectangle Property)

Suppose f is continuous and has minimum value m and maximum value M on a closed interval [a, b]. Then

That is, the area of the region under the curve is between the area of the rectangle whose height is the minimum value of and the area of the rectangle whose height is the maximum value of in the interval [a,b].

The Extreme Value Theorem is needed to show that the minimum value m and maximum value M exist. The rectangle of height m is called the inscribed rectangle of the region, and the rectangle of height M is called the circumscribed rectangle. From Figure 4.2.3, we see that the inscribed rectangle is a subset of the region under the curve, which is in turn a subset of the circumscribed rectangle. The Rectangle Property says that the area of the region is between the areas of the inscribed and circumscribed rectangles.

Figure 4.2.3 the Rectangle Property

PROOF By Theorem 1, any positive infinitesimal may be chosen for dx. Let us choose a positive infinite hyperinteger H and let dx=(b-a) /H. Then dx evenly divides b-a; that is, the interval [a, b] is divided into H subintervals of exactly the same length dx. Then

For each x, we have m≤ f(x)≤M. Adding up and taking standard parts, we obtain the required formula.

One useful consequence of the Rectangle Property is that the integral of a positive function is positive and the integral of a negative function is negative:

The definite integral of a negative function f(x)= -g(x) from a to b is just the negative of the area of the region above the curve and below the x axis.

This is because

f(x) dx = -g(x)dx,

(See Figure 4.2.4.)

Figure 4.2.4

THEOREM 3 (The Addition Property)

Suppose f is continuous on an interval I. Then for all a, b, c in I,

This property is illustrated in Figure 4.2.5 for the case a