Editing Language/Multiple-languages/Vocabulary/Mathematical-Analysis

{{Sci-Tech-Index-Intro}}

{| class="wikitable sortable"
! cmn-Hans-CN: 科目
! cmn-Latn.Pinyin-CN: kēmù
! deu-Latn-DE: Fach [n]
! eng-Latn-US: subject
! fra-Latn-FR: matière [f]
! jpn-Jpan-JP: 科目
! jpn-Hrkt-JP: か↑もく
! rus-Cyrl-RU: предме́т
|-
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| 
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|-
| 微积分学
| wēijīfēn xué
| Infinitesimalrechnung [f]
| calcul infinitésimal
| calculus
| 微分積分学
| び↑ぶんせきぶ↓んがく
| математи́ческий ана́лиз
|-
| 向量分析，向量微积分
| xiàngliàn fēnxī, xiàngliàng wēijīfēn
| Vektoranalysis [f]
| vector calculus, vector analysis
| analyse vectorielle
| ベクトル解析
| ベ↑クトルか↓いせき
| ве́кторный ана́лиз
|-
| 多元微积分，多变量微积分
| duōyuán wēijīfēn, duō biànliàng wēijīfēn
| multivariable Infinitesimalrechnung
| multivariable calculus, multivariate calculus
| calcul [m] à plusieurs variables [f pl]
| 多変数微分積分学，多変数解析学
| た↑へ↓んすうび↑ぶんせきぶ↓んがく, た↑へ↓んすうか↑いせき↓がく
| многоме́рный ана́лиз
|-
| 实分析，实数分析，实变函数论
| shí fēnxī, shíshù fēnxī, shí biàn hánshù lùn
| reelle Analysis
| analyse réelle
| real analysis
| 実解析，実関数論
| じ↑つか↓いせき, じ↑つかんすうろ↓ん
| тео́рия фу́нкций веще́ственной переме́нной, тео́рия фу́нкций действи́тельного переме́нного
|-
| 复分析
| fù fēnxī
| Funktionentheorie [f]
| complex analysis
| analyse [f] complexe
| 複素解析
| ふ↑くそか↓いせき
| ко́мпле́ксный ана́лиз, тео́рия фу́нкций ко́мпле́ксного переме́нного, тео́рия фу́нкций ко́мпле́ксной переме́нной
|-
| 泛函分析
| fànhán fēnxī
| Funktionalanalysis [f]
| functional analysis
| analyse fonctionnelle
| 関数解析学
| か↑んす↓うか↑いせき↓がく
| функциона́льный ана́лиз
|-
| 傅里叶分析
| Fùlǐyè fēnxī
| Fourier-Analysis [f], Fourier-Analyse [f]
| Fourier analysis
| analyse [f] de Fourier
| フーリエ解析
| フ↓ーリエかいせき
| ана́лиз Фурье́
|-
| 调和分析，谐波分析
| tiáohé fēnxī, xiébō fēnxī
| harmonische Analyse, harmonische Analysis
| harmonic analysis
| analyse [f] harmonique
| 調和解析
| ちょ↑うわか↓いせき
| гармони́ческий ана́лиз
|-
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|}

== Calculus ==
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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| backward difference || finite difference
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| central difference || finite difference
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| constant of integration || constant term <algebra>; antiderivative
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| continuous function || argument <mathematical logic>
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| definite integral || second fundamental theorem of calculus
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| derivative || function of a real variable <algebra>
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| difference quotient || finite difference
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| differentiable function || derivative; domain <mathematical logic>
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| differentiation || derivative
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| discontinuous function || argument <mathematical logic>
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| finite difference || function <mathematical logic>
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| forward difference || finite difference
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| indefinite integral || integral; antiderivative
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| indeterminate form || infinity <number theory>
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| infinitesimal, infinitesimal number || real number
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| integral || infinitesimal
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| 变限积分 || 
| biànxiàn jīfēn || 
| Integral [n]  mit variablen Grenzen [f pl] || 
| integral with variable limits || integral; limit; variable <mathematical logic>
| intégrale [f] avec limites [f pl] variables || 
| 可変限界との積分 || 
| か↑へんげ↓んかいとのせ↑きぶん  || 
| интегра́л с переме́нными преде́лами || 
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| integration by substitution, u-substitution, reverse chain rule, change of variables || integral; antiderivative
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| Leibniz integral rule || partial derivative
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| limit || function; sequence
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| linear approximation || affine transformation <geometry>
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| Lipschitz continuity || uniform continuity
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| maximum [pl:maxima] || function
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| minimum [pl:minima] || function
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| multiple integral || definite integral
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| partial derivative || derivative
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| second derivative, second order derivative || derivative
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| separation of variables, Fourier method || ordinary differential equation; partial differential equation
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| stationary point || differentiable function; derivative
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| Taylor polynomial || Taylor's theorem
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| total derivative || 
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|}

{| class="wikitable sortable"
|+ differential equation
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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| differential equation || 
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| delay differential equations || derivative
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| differential-algebraic system of equations || 
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| integro-differential equation || integral; derivative
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| ordinary differential equation || derivative; independent variable
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| partial differential equation || partial derivative; multivariable function
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| stochastic differential equation || stochastic process <probability theory>; differential equation
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| stochastic partial differential equation || stochastic process <probability theory>; partial differential equation
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|}

{| class="wikitable sortable"
|+ differentiation rule
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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| differentiation rule || 
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| chain rule || function composition; differentiable function
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| general Leibniz rule || product rule
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| inverse function rule || differentiable function; inverse function
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| power rule || differentiable function
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| product rule, Leibniz rule || differentiable function
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| quotient rule || differentiable function
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| reciprocal rule || multiplicative inverse; differentiable function
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|}

{| class="wikitable sortable"
|+ extremum
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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| extremum [pl:extrema] || function <mathematical logic>
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| absolute maximum, global maximum || maximum; domain <set theory>
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| absolute minimum, global minimum || minimum; domain <set theory>
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| relative maximum, local maximum || maximum; interval <topology>
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| relative minimum, local minimum || minimum; interval <topology>
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|-
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|}

{| class="wikitable sortable"
|+ integral transform
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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| integral transform || 
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| Abel transform || 
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| associated Legendre transform || 
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| Fourier cosine transform || 
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| Fourier sine transform || 
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| Fourier transform || 
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| Hankel transform || 
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| Hartley transform || 
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| Hermite transform || 
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| Hilbert transform || 
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| Jacobi transform || 
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| Laguerre transform || 
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| Laplace transform || 
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| Legendre transform || 
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|-
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| Mellin transform || 
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|-
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| Poisson kernel || 
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| Radon transform || 
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| two-sided Laplace transform, bilateral Laplace transform || 
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| Weierstrass transform || 
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| X-ray transform, John transform || 
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|}

{| class="wikitable sortable"
|+ notation for differentiation
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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| notation for differentiation || 
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|-
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|-
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| Euler's notation || differentiation
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|-
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| Lagrange's notation || differentiation
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|-
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| Leibniz's notation || differentiation
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|-
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| Newton's notation || differentiation
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|-
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|}

{| class="wikitable sortable"
|+ series
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
|  || 
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| series || addition <number theory>; infinity <number theory>
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|-
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|-
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| alternating series || series
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|-
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| convergent series || limit
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|-
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| divergent series || limit
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|-
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| Fourier series || series; periodic function; sine wave <trigonometry>
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|-
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| geometric series || series; ratio
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|-
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| Gregory's series || Taylor series; inverse trigonometric functions <trigonometry>
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|-
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| harmonic series || series
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|-
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| partial sum || series; sum <number theory>
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|-
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| Taylor series || series; derivative
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|-
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| telescoping series || series
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|-
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|}

{| class="wikitable sortable"
|+ theorem in calculus
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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| theorem in calculus || 
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| extreme value theorem || continuous function; maximum; minimum
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| first fundamental theorem of calculus || continuous function; closed interval <topology>; antiderivative
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| L'Hôpital's rule || limit; indeterminate form
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| Rolle's theorem, Rolle's lemma || stationary point
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| second fundamental theorem of calculus || interval <algebra>; antiderivative
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| Taylor's theorem || differentiable function; linear appriximation
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|}

{| class="wikitable sortable"
|+ mean value theorem
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
| 中值定理 || 
| zhōngzhí dìnglǐ || 
| Mittelwertsatz [m] || 
| mean value theorem || 
| théorème [m pl] des accroissements [m pl] finis || 
| 平均値の定理 || 
| へ↑いき↓んちの↑て↓いり || 
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|-
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|-
| 柯西中值定理 || 
| Kēxī zhōngzhí dìnglǐ || 
| cauchy'scher Mittelwertsatz || 
| Cauchy's mean value theorem, extended mean value theorem || 
| théorème [m] de la moyenne de Cauchy || 
| コーシーの平均値定理 || 
| コ↓ーシーのへ↑いきんちて↓いり || 
| теоре́ма Коши́ о сре́днем значе́нии || 
|-
| 积分第一中值定理 || 
| jīfēn dìyī zhōngzhí dìnglǐ || 
| erster Mittelwertsatz der Integralrechnung || 
| first mean value theorem for integrals || 
| première formule [f] de la moyenne || 
| 積分の第一平均値定理 || 
| せ↑きぶんのだ↓いい↑ちへいきんちて↓いり  || 
| пе́рвая теоре́ма о сре́днем || 
|-
| 拉格朗日中值定理 || 
| Lǎgélǎngrì zhǒngzhí dìnglǐ || 
| Mittelwertsatz [m] von Lagrange || 
| Lagrange's mean value theorem || 
| théorème [m] de la valeur moyenne de Lagrange || 
| ラグランジュの平均値の定理 || 
| ラ↑グラ↓ンジュのへ↑いき↓んちの↑て↓いり || 
| Фо́рмула коне́чных прираще́ний, теоре́ма Лагра́нжа о сре́днем значе́нии || 
|-
| 积分第二中值定理 || 
| jīfēn dì'èr zhōngzhí dìnglǐ || 
| zweiter Mittelwertsatz der Integralrechnung || 
| second mean value theorem for integrals || 
| deuxième formule [f] de la moyenne || 
| 積分の第二平均値定理 || 
| せ↑きぶんのだ↓いにへ↑いきんちて↓いり || 
| Втора́я теоре́ма о сре́днем || 
|-
| 罗尔定理 || 
| Luó'ěr dìnglǐ || 
| Satz [m] von Rolle || 
| Rolle's theorem, Rolle's lemma || 
| théorème [m] de Rolle, lemme [m] de Rolle || 
| ロルの定理 || 
| ロ↑ル↓の↑て↓いり || 
| теоре́ма Ро́лля || 
|-
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|}

=== Vector Calculus ===
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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|-
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|}

=== Multivariable Calculus ===
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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|-
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| differential operator || operator <algebra>; differentiation
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|-
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| triple product rule, cyclic chain rule, cyclic relation, cyclical rule, Euler's chain rule || partial derivative
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|-
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|}

== Real Analysis ==
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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|-
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|}

== Complex Analysis ==
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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|-
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| complex plane || Cartesian coordinate system <geometry>; complex number <number theory>
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|-
| 虚轴 || 
| xū zhóu || 
| imaginäre Achse || 
| imaginary axis || imaginary line
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|-
| 虚数线 || 
| xūshù xiàn || 
| imaginäre Linie || 
| imaginary line || line <geometry>; point <geometry>; imaginary number <arithmetic>
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|-
| 实轴 || 
| shí zhóu || 
| reelle Achse || 
| real axis || real line
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|-
| 实数线 || 
| shíshù xiàn || 
| reelle Linie || 
| real line || line <geometry>; point <geometry>; real number <arithmetic>
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|-
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|}

== Functional Analysis ==
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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|-
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|}

== Numerical Analysis ==
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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|-
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| truncation error || 
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|-
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|}

== Fourier Analysis ==
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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|-
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|}

== Harmonic Analysis ==
{| class="wikitable sortable"
! cmn-Hans-CN: 概念 !! cmn-Hans-CN: 前提
! cmn-Latn.Pinyin-CN: gàiniàn !! cmn-Latn.Pinyin-CN: qiántí
! deu-Latn-DE: Begriff [m] !! deu-Latn-DE: Voraussetzung [f]
! eng-Latn-US: concept !! eng-Latn-US: prerequisite
! fra-Latn-FR: concept [m] !! fra-Latn-FR: préalable [m]
! jpn-Jpan-JP: 概念 !! jpn-Jpan-JP: 前提
! jpn-Hrkt-JP: が↓いねん !! jpn-Hrkt-JP: ぜ↑んてい
! rus-Cyrl-RU: конце́пция !! rus-Cyrl-RU: предпосы́лка
|-
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|-
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|}

<center>
https://upload.wikimedia.org/wikipedia/commons/f/f5/LeibnizInfinitesimalTriangle.png
</center>

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