# 数学用語の英単語と例文｜【微分積分学編】

その他

## 実数

### 英例文

#### 例１

$\Q$, $\R$はそれぞれ有理数全部の集合，実数全部の集合を表す．

— $\Q$ and $\R$ represent the set of all rational numbers and that of all real numbers, respectively.

#### 例２

$X\subset\R$が$\R$で稠密であるとは，任意の$a\in\R$に対して，ある$\epsilon>0$が存在して，$U_{\epsilon}(a)$（すなわち$a$の$\epsilon$近傍）と$X$の共通部分が空でないことをいう．

— We say that $X\subset\R$ is dense in $\R$, if for all $a\in\R$ there exists $\epsilon>0$ such that the intersection of  $U_{\epsilon}(a)$ (i.e., $\epsilon$-neighborhood of $a$) and $X$ isn’t the empty set.

#### 例３

$A\subset\R$とする．ある$m\in\R$が存在して，任意の$a\in A$に対して$a\le m$が成り立つとき，$A$は上に有界であるという．

— Let $A\subset\R$. If there exists $m\in\R$ such that for all $a\in A$ we have $a\le m$, $A$ is said to be bounded above.

#### 例４

— If a non-empty set $A\subset\R$ is bounded above, supremum $\sup{A}$ always exists in $\R$ (continuity of real numbers).

## 数列

### 英単語

(狭義)単調増加(strictly) monotonically increasing
(狭義)単調減少(strictly) monotonically decreasing
コーシー列Cauchy sequence

### 英例文

#### 例１

— The real sequence $\{x_n\}$ converges to $x$ as $n\to\infty$.

#### 例２

— The superior of each real sequence $\{x_n\}$ always exists in extended real number $\R\cup\{\pm\infty\}$.

#### 例３

— Any bounded complex sequence $\{x_n\}$ has an convergent subsequence (Bolzano-Weierstrass theorem).

#### 例４

— Assume that real sequence $\{x_n\}$ is bounded above and monotonically increasing. Then it converges to the supremum $\sup_{n\in\N}x_n$.

#### 例５

もし級数$\sum_{n=1}^{\infty}x_n$が収束すれば，$\{x_n\}$は$0$に収束する

— If the series $\sum_{n=1}^{\infty}x_n$ converge, $\{x_n\}$ converges to 0.

## 微分

### 英単語

($n$階)導関数($n$th) derivative

### 英例文

#### 例１

— If a function $f:\R\to\R$ is differentiable at $a$, it is continuous at $a$.

#### 例２

$I$を開区間とする．$I$上の導関数$\dfrac{df}{dx}$が連続であれば，関数$f$を$C^1(I)$級であるという．

— Let $I$ be an open interval. If a first derivative $\dfrac{df}{dx}$ on $I$ exists and is continuous, the function $f$ is said to be of class $C^1(I)$.

#### 例３

$a$で極値をとる可微分関数$f$に対して，微分係数$\dfrac{df}{dx}(a)$は$0$である．

— For a differentiable function $f$ which attains an extrema at $a$, the differential coefficient $\dfrac{df}{dx}(a)$ is $0$.

## 積分

### 英単語

リーマン積分Riemann integral

### 英例文

#### 例１

リーマン積分は有界閉区間$[a,b]$上の積分であり，$a\to-\infty$や$b\to\infty$とした場合は広義積分と呼ばれる．

— the Riemann integral is an integral on a bounded closed interval $[a,b]$, and it is called improper integral as $a\to-\infty$ or $b\to\infty$.

#### 例２

— For continuous function $f:R\to\R$, its primitive function $F$ and indefinite integral $\int_{a}^{x} f(t)\,dt$ are equal, up to addition by a constant (fundamental theorem of calculus).

#### 例３

$\cos{x}$の不定積分は$\sin{x}+C$である．ここで$C$は積分定数である．

— A indefinite integral of $\cos{x}$ is $\sin{x}+C$, where $C$ is a constant of integration.

## 関数列

### 英例文

#### 例１

$f_n(x)=\frac{n+1}{n}x$で定まる関数列$\{f_n\}$は各点収束するが，一様収束しない．

— The sequence of functions $\{f_n\}$ defined as $f_n(x)=\frac{n+1}{n}x$ converge pointwisely, but doesn’t converge uniformly.

#### 例２

— When the sequence of functions $\{f_n\}$ on closed bounded interval $I$ converge to the limit function $f$, it is integrable term by term.