数学符号解疑 | Σ vs. ∫ / y(x) vs. f(x)-优快云博客

注：本文为 “数学符号解疑” 相关译文合辑。
英文引文，机翻未校。
如有内容异常，请看原文。

Integration vs. Summation

积分与求和

What’s the Difference?

二者的区别是什么？

Integration and summation are both mathematical operations that involve adding up values. However, they differ in terms of the type of values being added and the purpose of the operation. Summation is used to find the total sum of a series of discrete values, typically represented by the sigma notation. It is commonly used in arithmetic and algebra to calculate the sum of numbers or terms in a sequence. On the other hand, integration is used to find the area under a curve or the accumulation of a continuous function over a given interval. It involves adding up infinitely many infinitesimal values, represented by the integral symbol. Integration is widely used in calculus and physics to solve problems related to rates of change, areas, and volumes.
积分和求和均为涉及数值累加的数学运算，但二者在累加数值的类型及运算目的上存在差异。求和用于计算一系列离散值的总和，通常以希腊字母 Σ（西格玛）表示，常见于算术和代数中，用于求解数列中数字或项的总和。而积分用于求解曲线下的面积或连续函数在特定区间上的累积量，其核心是对无穷多个无穷小量进行累加，以积分符号 ∫ 表示。积分广泛应用于微积分和物理学领域，用于解决与变化率、面积及体积相关的问题。

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Comparison

对比表

Attribute 特性	Integration 积分	Summation 求和
Definition	The process of finding the integral of a function over a given interval. 求解函数在特定区间上积分的过程。	The process of adding a sequence of numbers together. 对一系列数字进行累加的过程。
Symbol	$\int$ (integral symbol) $\int$ （积分符号）	$\Sigma$ (sigma symbol) $\Sigma$ （西格玛符号）
Continuous	Applicable to continuous functions. 适用于连续函数。	Applicable to discrete values. 适用于离散值。
Result	A single value representing the area under the curve. 表示曲线下面积的单一数值。	A single value representing the sum of the sequence. 表示数列总和的单一数值。
Notation	$\, dx$	$a_n$
Geometric Interpretation	Area under the curve. 曲线下的面积。	Sum of values on a number line. 数轴上数值的累加和。
Applications	Calculating areas, volumes, and solving differential equations. 计算面积、体积及求解微分方程。	Calculating series, probabilities, and discrete sums. 计算级数、概率及离散总和。

Further Detail

详细说明

Introduction

引言

Integration and summation are two fundamental concepts in mathematics that involve the accumulation of values. While they may seem similar at first glance, they have distinct attributes and applications. In this article, we will explore the differences and similarities between integration and summation, highlighting their key features and providing examples to illustrate their usage.
积分和求和是数学中两个涉及数值累积的重要概念。尽管二者初看相似，但具有独特的属性和应用场景。本文将探讨积分与求和的异同点，重点阐述其核心特征，并通过示例说明其应用方式。

Definition and Notation

定义与符号

Integration is a mathematical operation that calculates the area under a curve. It is denoted by the symbol $\int$ and is often used to find the total accumulation of a quantity over a given interval. Summation, on the other hand, is a mathematical operation that adds together a sequence of numbers. It is denoted by the symbol $\Sigma$ and is used to find the total sum of a series of values.
积分是一种计算曲线下面积的数学运算，以符号 $\int$ 表示，常用于求解某一物理量在特定区间上的总累积量。求和则是对一系列数字进行累加的数学运算，以符号 $\Sigma$ 表示，用于计算一系列数值的总和。

Continuous vs. Discrete

连续性与离散性

One of the key distinctions between integration and summation lies in the nature of the values being accumulated. Integration deals with continuous functions, where the values change smoothly over an interval. It involves finding the area under the curve by dividing the interval into infinitesimally small segments and summing up the contributions of each segment. Summation, on the other hand, deals with discrete values, where the values are distinct and separate. It involves adding up the individual values in a sequence to obtain the total sum.
积分与求和的核心区别之一在于所累积数值的性质。积分处理连续函数，这类函数的数值在区间内平滑变化，其计算过程是将区间划分为无穷多个无穷小的子区间，再累加每个子区间的贡献以得到曲线下的面积。求和则处理离散值，这类数值彼此独立、互不连续，运算方式为直接累加数列中的各个数值以获得总和。

Geometric Interpretation

几何意义

Integration can be geometrically interpreted as finding the area enclosed by a curve and the $x$ -axis. By calculating the definite integral of a function over a given interval, we can determine the area under the curve within that interval. This concept is widely used in calculus to solve problems related to areas, volumes, and rates of change. Summation, on the other hand, can be geometrically interpreted as adding up the lengths of line segments or the areas of individual shapes. It is often used in discrete mathematics and combinatorics to count objects or calculate probabilities.
积分的几何意义是求解曲线与 $x$ 轴所围成的面积。通过计算函数在特定区间上的定积分，可确定该区间内曲线下的面积。这一概念在微积分中广泛应用于解决面积、体积及变化率相关问题。求和的几何意义则是累加线段的长度或单个图形的面积，常用于离散数学和组合数学中，用于计数物体数量或计算概率。

Applications

应用场景

Integration has numerous applications in various fields of science and engineering. It is used in physics to calculate the work done by a force, the displacement of an object, or the amount of energy transferred. In economics, integration is used to determine the total revenue or cost of a product over a given period. It is also employed in signal processing to analyze and manipulate continuous signals. Summation, on the other hand, finds applications in computer science, where it is used to calculate the average, maximum, or minimum values in a dataset. It is also used in finance to calculate the total returns on investments or the sum of a series of payments.
积分在科学与工程的多个领域具有广泛应用：在物理学中，用于计算力所做的功、物体的位移或能量传递量；在经济学中，用于确定某一时期内产品的总收益或总成本；在信号处理中，用于分析和处理连续信号。求和的应用场景包括：计算机科学中计算数据集的平均值、最大值或最小值；金融学中计算投资的总回报或一系列付款的总和。

Properties

性质

Integration and summation have several properties that make them powerful tools in mathematics. Integration has the property of linearity, which means that the integral of a sum of functions is equal to the sum of their integrals. It also has the fundamental theorem of calculus, which relates differentiation and integration. Summation, on the other hand, has the property of commutativity, which means that the order of the terms being summed does not affect the result. It also has the associative property, which means that the grouping of terms being summed does not affect the result.
积分和求和均具有若干重要性质，使其成为数学中的有力工具。积分具有线性性质，即函数和的积分等于各函数积分的和；此外，微积分基本定理建立了微分与积分之间的联系。求和具有交换律，即改变相加项的顺序不会影响结果；同时具有结合律，即改变相加项的分组方式不会影响结果。

Examples

示例

Let’s consider an example to illustrate the difference between integration and summation. Suppose we have a continuous function $f (x) = 2 x$ , and we want to find the area under the curve between $x = 0$ and $x = 3$ . To do this, we can calculate the definite integral of $f (x)$ over the interval $[0, 3]$ . The result will give us the total accumulation of the function over that interval, which in this case is 9 square units.
下面通过示例说明积分与求和的区别。假设有连续函数 $f (x) = 2 x$ ，我们需要求解该函数在 $x = 0$ 至 $x = 3$ 区间内曲线下的面积。通过计算 $f (x)$ 在区间 $[0, 3]$ 上的定积分，可得到该区间内函数的总累积量，此处结果为 9 平方单位。

Now, let’s consider a sequence of numbers ${1, 2, 3, 4, 5\}$ . If we want to find the sum of these numbers, we can use summation. By adding up each individual value, we get a total sum of 15.
再考虑数列 ${1, 2, 3, 4, 5\}$ 。若要求解该数列的总和，可使用求和运算，累加各项数值后得到总和为 15。

Conclusion

结论

In conclusion, integration and summation are both mathematical operations that involve accumulation, but they have distinct attributes and applications. Integration deals with continuous functions and calculates the area under a curve, while summation deals with discrete values and adds together a sequence of numbers. Integration has applications in physics, engineering, and economics, while summation finds applications in computer science and finance. Understanding the differences and similarities between integration and summation is essential for solving mathematical problems and analyzing real-world phenomena.
综上，积分和求和均为涉及累积的数学运算，但二者具有独特的属性和应用场景。积分处理连续函数并计算曲线下的面积，求和处理离散值并累加数列中的数字。积分应用于物理学、工程学和经济学领域，求和应用于计算机科学和金融学领域。理解积分与求和的异同点，对于解决数学问题和分析现实世界现象具有重要意义。

Is there a difference between $y (x)$ and $f (x)$

$y (x)$ 与 $f (x)$ 是否存在区别

Oftentimes functions described by $f (x) = 2 x + 4$ , and when this is mapped to the Cartesian plane, $f (x) = y$ . This surely implies that $y = 2 x + 4$ . Is there a difference between this and $y (x) = 2 x + 4$ ?
通常，函数可表示为 $f (x) = 2 x + 4$ ，当将其映射到笛卡尔坐标系时，有 $f (x) = y$ ，这显然意味着 $y = 2 x + 4$ 。那么，这与 $y (x) = 2 x + 4$ 之间是否存在区别？

asked Jun 30, 2016 at 19:51
Frank Vel

Answers

回答

Functions vs. coordinates

函数与坐标

Consider plots of two different functions: $f (x)$ and $g (x)$ on the same $x y$ -plane. One curve will be labeled $y = f (x)$ which means “this is a set of $(x, y)$ points that satisfy $y = f (x)$ condition”. The other will be labeled $y = g (x)$ .
考虑在同一 $x y$ 平面上绘制两个不同函数 $f (x)$ 和 $g (x)$ 的图像：一条曲线标记为 $y = f (x)$ ，表示“满足 $y = f (x)$ 条件的所有 $(x, y)$ 点的集合”；另一条曲线标记为 $y = g (x)$ 。

Which of these should define $y (x)$ ? Both? – certainly not, because $f$ and $g$ are different functions. I say: neither. The statement $y = f (x)$ is just a condition for some set of points (i.e. $(x, y)$ pairs) while $y = g (x)$ is another condition for another set of points.
那么，哪一个应该定义 $y (x)$ ？两者都可以吗？显然不行，因为 $f$ 和 $g$ 是不同的函数。我的观点是：两者都不能。表达式 $y = f (x)$ 仅表示某组点（即 $(x, y)$ 坐标对）需满足的条件，而 $y = g (x)$ 则是另一组点需满足的条件。

Explicit definition in a form $\dots$ does define a function (well, does or doesn’t, read the next paragraph). In this case $y$ is just an arbitrary name and may replace $f$ . The same symbol $y$ may be a coordinate on $x y$ -plane, which was $x f$ -plane before the name replacement. (It is only a custom to have $x y$ -plane.) This “union” of function name and coordinate name may cause a problem when there is another function $g (x)$ to plot.
以 $\dots$ 形式给出的显式表达式确实定义了一个函数（具体是否能定义，详见下一段）。此时， $y$ 仅为任意选定的函数名称，可替代 $f$ 。同一符号 $y$ 也可能是 $x y$ 平面上的坐标（名称替换前该平面可称为 $x f$ 平面——使用 $x y$ 平面仅为惯例）。当需要绘制另一个函数 $g (x)$ 时，这种“函数名称与坐标名称的混用”可能引发歧义。

It should be obvious that if $y$ replaces $f$ it cannot replace $g$ that is different than $f$ .
显然，若 $y$ 被用作替代 $f$ 的函数名称，则不能再用于替代与 $f$ 不同的 $g$ 。

For that reason it is a good thing to have coordinates with symbols which are not function names.
因此，建议使用非函数名称的符号表示坐标。

Definitions vs. equations or conditions

定义与方程/条件

Another problem: we often write function definitions the same way as conditions to be met or equations to solve. Compare the two:
另一个问题在于：我们常以相同的形式表示函数定义、需满足的条件或待求解的方程。对比以下两式：

$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$

$\cos(x) = \frac{1}{2}$

The former may be treated as non-geometric definition of $\cos$ function. The latter is just the equation to solve for $x$ . We have some experience and often feel the difference, but a person (say: Bob) completely unaware of $\cos$ will be confused. Bob may find every $x$ that satisfies
前者可视为余弦函数 $\cos$ 的非几何定义，后者仅为求解 $x$ 的方程。我们凭借经验通常能区分二者，但对于完全不了解 $\cos$ 函数的人（例如 Bob）而言，可能会产生困惑。Bob 可能会求解出所有满足下式的 $x$ ：

$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = \frac{1}{2}$

and still will not be able to tell what number $\cos(x)$ is equal to for any other $x$ .
但即便如此，他仍无法确定对于其他任意 $x$ ， $\cos(x)$ 的取值为何。

It’s worse than that! Bob cannot tell what number $\cos(x)$ is equal to even for $x$ being his solution, because he cannot be sure that either equation defines the function (we know it’s the first one, Bob doesn’t). To clarify that, let’s see what happens when I change $\cos$ to $\sin$ only:
更严重的是，即使对于他求解出的 $x$ ，Bob 也无法确定 $\cos(x)$ 的具体数值——因为他无法判断哪个方程是函数的定义（我们知道是第一个，但 Bob 不知道）。为进一步说明，仅将 $\cos$ 替换为 $\sin$ ，观察以下两式：

$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$

$\sin(x) = \frac{1}{2}$

We know by experience that neither of above defines $\sin$ . Yet these are legitimate equations to solve either separately or as a system (with empty set solution). Bob (not knowing about $\sin$ ) may only assume that one of the equations is a definition – this will be wrong, his set of solutions will not be empty.
根据经验，我们知道上述两式均非正弦函数 $\sin$ 的定义，但它们都是合法的方程，可单独求解或作为方程组求解（方程组的解集为空集）。Bob（不了解 $\sin$ 函数）可能会假设其中一个方程是定义——这显然是错误的，且他求解出的解集不会为空。

That’s why I like the notation $\equiv \dots$ or the word “def” above the equality sign, or the explicit statement (“let us define…”) – just to cut out possible ambiguity.
因此，我倾向于使用符号 $\equiv \dots$ 、在等号上方标注“def”（定义），或通过明确表述（“我们定义……”）来表示函数定义，以消除潜在歧义。

I’ve got the impression that you meant $\equiv 2x + 4$ because there is no other expression in your example that you may want to define as $y (x)$ .
我的理解是，你想表达的是 $\equiv 2x + 4$ ，因为在你的示例中没有其他表达式需要定义为 $y (x)$ 。

Summary

总结

Is there a difference between $y = 2 x + 4$ and $y (x) = 2 x + 4$ ?
$y = 2 x + 4$ 与 $y (x) = 2 x + 4$ 是否存在区别？

My answer is: in general it may be. The second form is more likely to be read as a definition of a function, yet any form may or may not be intended to be a definition. Both may be equations to solve, when $y (x)$ is defined elsewhere ( $y$ may be a given number or a parameter not depending on $x$ , still it can be formally written as $y (x)$ ). The second form states that there is some function $y (x)$ ; the first one may mention a function or a variable (coordinate) $y$ . The coordinate (not function) interpretation allows the first form to be a condition for points (i.e. $(x, y)$ pairs) – as it may be in your example – that leaves room for another conditions for another sets of points.
我的回答是：通常情况下可能存在区别。 第二种形式更可能被解读为函数定义，但两种形式均可能并非用于定义函数。当 $y (x)$ 在其他地方已被定义时（ $y$ 可能是给定的常数或与 $x$ 无关的参数，仍可形式上表示为 $y (x)$ ），两者都可能是待求解的方程。第二种形式明确表明存在某个函数 $y (x)$ ；第一种形式中的 $y$ 可能表示函数，也可能表示变量（坐标）。若按坐标（非函数）解读，第一种形式可表示点（即 $(x, y)$ 对）需满足的条件——如你的示例所示——这为其他点集的条件预留了空间。

Is there a difference between $y = 2 x + 4$ and $\equiv 2x + 4$ ?
$y = 2 x + 4$ 与 $\equiv 2x + 4$ 是否存在区别？

Yes. The second form defines a function for sure. The first one may have another meaning (explained above).
是。第二种形式明确定义了一个函数，第一种形式则可能具有其他含义（如上文所述）。

Is there a difference between $\equiv 2x + 4$ and $\equiv 2x + 4$ ?
$\equiv 2x + 4$ 与 $\equiv 2x + 4$ 是否存在区别？

There is a subtle one: from the second form we know the independent variable is $x$ ; it may be $x$ xor $y$ in the first one.
存在细微区别：第二种形式明确独立变量为 $x$ ；第一种形式的独立变量可能是 $x$ ，也可能是 $y$ （二者互斥）。

Is there a difference between $y (x)$ and $f (x)$ ?
$y (x)$ 与 $f (x)$ 是否存在区别？

No. In a sense: you can name your function with any unused symbol. But if $y$ is already in use (e.g. to name a different function, coordinate, parameter) then you cannot freely rename $f$ to $y$ .
不存在。 从某种意义上说，可使用任意未被占用的符号命名函数。但如果 $y$ 已被占用（例如用作其他函数、坐标或参数的名称），则不能随意将 $f$ 重命名为 $y$ 。

answered Jul 1, 2016 at 0:54
Kamil Maciorowski
This answer was very illuminating to me, especially since it was on a topic that I thought I was comfortable with. I particularly enjoyed your 2nd section, highlighting the important difference between equality and definitions/the distinctness of objects themselves. It seems like $:=$ “⊂” $=$ . In other words, the definition symbol can serve as an equality sign. But the equality sign should not be used to stand for definitions/distinctness. I will probably start using the notation $f :=$ , $\equiv$ , or $\mapsto$ more frequently for defining functions. Thanks again
这个回答对我很有启发，尤其是涉及到一个我本以为已经熟悉的主题。我特别喜欢你的第二部分，强调了相等关系与定义/对象本身独特性之间的重要区别。似乎 $:=$ “包含于” $=$ ，换句话说，定义符号可视为一种相等符号，但相等符号不应被用于表示定义/独特性。我可能会更频繁地使用 $f :=$ 、 $\equiv$ 或 $\mapsto$ 等符号来定义函数。再次感谢！

– DWade64
Commented Oct 3, 2018 at 13:34

总结

一、积分（Integration）与求和（Summation）

共性：均为数学中的“累积类运算”，通过数值累加获得结果。
差异：
- 适用对象：积分针对连续函数，求和针对离散值；
- 运算本质：积分累加无穷多个无穷小量，求和累加有限/可数个离散项；
- 符号与结果：积分用 $\int$ 表示，结果为曲线下面积等连续累积量；求和用 $\Sigma$ 表示，结果为数列总和等离散累积量；
应用场景：积分多用于微积分、物理（面积/体积/变化率）、经济学；求和多用于代数、离散数学、计算机科学（数据集统计）、金融学。

二、 $y (x)$ 与 $f (x)$

$y (x)$ 与 $f (x)$ 的关系：无本质区别，均为函数符号，可任意选用未占用的符号命名函数；若 $y$ 已用作坐标/其他参数，则不可再替代其他函数（如 $g (x)$ ）。
符号歧义辨析：
- $y = 2 x + 4$ ：可能是坐标点满足的条件（笛卡尔平面中），也可能是函数定义（需结合语境）；
- $y (x) = 2 x + 4$ ：更倾向于明确的函数定义，直接表明 $y$ 是关于 $x$ 的函数；
- 定义与方程的区别：函数定义需明确标识（如 $\equiv \dots$ 、 $\dots$ ），避免与待求解方程（如 $\cos(x) = 1/2$ ）混淆。