微积分双语常用词汇

MATHEMATICAL TERMS (Part 1) calculus 微积分definition 定义theorem 定理lemma 引理corollary 推论prove 证明proof 证明show 证明solution 解formula 公式if and only if ( iff ) 当且仅当∀ x∈X for all x∈X∃x∈X there exists an x∈Xsuch that 使得given 已知set 集合finite set 有限集infinite set 无限集interval 区间open interval 开区间closed interval 闭区间neighborhood 邻域number 数natural number 自然数integer 整数odd number 奇数even number 偶数real number 实数rational number 有理数irrational number 无理数positive number 正数negative number 负数mapping 映射function 函数monotone function 单调函数increasing function 增函数decreasing function 减函数bounded function 有界函数odd function 奇函数even function 偶函数periodic function 周期函数composite function 复合函数inverse function 反函数domain 定义域range 值域variable 变量independent variable 自变量dependent variable 因变量sequence 数列convergent sequence 收敛数列divergent sequence 发散数列bounded sequence 有界数列decreasing sequence 递减数列increasing sequence 递增数列limit 极限one-sided limit 单侧极限left-hand limit 左极限right-hand limit 右极限The Squeeze Theorem 夹挤定理infinity 无穷大infinitesimal 无穷小equivalent infinitesimal 等价无穷小infinitesimal of higher order 高阶无穷小order of infinitesimal 无穷小的阶infinitesimals of the same order 同阶无穷小increment 增量continuous function 连续函数continuity 连续性f(x) is continuous at x 在x 连续f(x) is discontinuous at x 在x 间断discontinuity 间断点discontinuity of the first (second) kind 第一（二）类间断点removable discontinuity 可去间断点jump discontinuity 跳跃间断点infinite discontinuity 无穷间断点intermediate value 介值The Intermediate Value Theorem 介值定理zero point 零点The Zero Point Theorem 零点定理root 根equation 方程uniform continuity 一致连续derivative 导数rate of change 变化率velocity 速度instantaneous velocity 瞬时速度tangent (line) 切线normal (line) 法线slope 斜率left-hand derivative 左导数right-hand derivative 右导数f(x) is differentiable at x f(x) 在x 处可导(可微) differentiation 求导The Chain Rule 链式法则differentiation formulas 求导公式implicit function 隐函数explicit function 显函数implicit differentiation 隐函数求导logarithm 对数Logarithmic differentiation 对数求导法parameter 参数parametric equation 参数方程parametric curve 参数曲线hyperbolic function 双曲函数hyperbolic sine 双曲正弦hyperbolic cosine 双曲余弦hyperbolic tangent 双曲正切hyperbolic cotangent 双曲余切MATHEMATICAL TERMS (Part 2)differential 微分differential quotient 微商approximate value 近似值error 误差relative error 相对误差absolute error 绝对误差invariance of differential form 微分形式不变性higher derivative 高阶导数first derivative 一阶导数second derivative 二阶导数third derivative 三阶导数nth derivative n 阶导数twice differentiable 二阶可导acceleration 加速度mean value 中值The Mean Value Theorem 中值定理Rolle’s Theorem 罗尔定理Lagrange’s Mean Value Theorem 拉格朗日中值定理Cauchy’s Mean Value Theorem 柯西中值定理equality 等式inequality 不等式indeterminate form 不定型 (未定式)indeterminate form of type 00 ( ∞∞ ) 00 ( ∞∞ )型未定式L’Hospital’s Rule 洛必达法则Taylor’s formula 泰勒公式polynomial 多项式nth-degree polynomial n 次多项式remaind 余项Lagrange’s form of remainder 拉格朗日型余项Peano’s form of remainder 皮亚诺型余项Maclaurin formula 麦克劳林公式Taylor polynomial 泰勒多项式Mauclaurin polynomial 麦克劳林多项式polynomial approximation 多项式逼近accuracy 精确度margin 边际marginal cost 边际成本marginal revenue 边际收益elasticity 弹性density 密度mass 质量extreme value 极值local maximum value 极大值local minimum value 极小值(absolute) maximum 最大值(absolute) minimum 最小值stationary point 驻点（稳定点）critical point 临界点The First (Second) Derivative Test （极值的）一（二）阶判别法convex 凸的convex curve 凸曲线concave 凹的concave curve 凹曲线convex function 凸函数point of inflection 拐点asymptote 渐近线horizontal asymptote 水平渐近线vertical asymptote 垂直渐近线slant asymptote 斜渐近线curve sketching 作图sketch a curve 作图curvature 曲率The bisection method 二分法The secant method 弦位法Newton’s method 牛顿（切线）法The tangent method 切线法differential calculus 微分学integral 积分integral calculus 积分学definite integral 定积分indefinite integral 不定积分partition 分割Riemann sum 黎曼和integral sign 积分符号integrand 被积函数upper (lower) limit of integration 积分上（下）限integration 积分（求积）integrable 可积的f(x) is integrable on [a, b]integrable function 可积函数integrability 可积性sufficient condition 充分条件necessary condition 必要条件piecewise continuous 分段连续property 性质The mean value theorem of integral 积分中值定理The fundamental theorem of calculus 微积分基本定理Newton-Leibniz formulaprimitive function (anti-derivative) 原函数（反导数）The substitution rule for integration 换元积分法The inverse of the chain rule 反链式法（凑微分法）integration by parts 分部积分法rational function 有理函数fraction 分式irreducible fraction 最简分式partial fraction 部分分式partial fraction decomposition 部分分式分解MATHEMATICAL TERMS (Part 3)vector 矢量free vector 自由矢量zero vector 零矢量magnitude of a vector 矢量的模unit vector 单位矢量scalar product 数量积dot product 点积vector product 矢量积cross product 叉积a is perpendicular (orthogonal) tob a 与b 垂直coordinate 坐标coordinate system 坐标系coordinate axis 坐标轴x-axis x 轴coordinate plane 坐标面direction angle 方向角direction cosine 方向余弦rectangular coordinate system 直角坐标系octant 卦限the first octant 第一卦限variable 变量function of two (three) variables 二（三）元函数function of several variables 多元函数independent variable 自变量dependent variable 因变量domain 定义域range 值域set of points 点集neighborhood 邻域interior point 内点boundary point 边界点bound 边界open set 开集closed set 闭集connected set 连通集region 区域open region 开区域closed region 闭区域bounded region 有界区域unbounded region 无界区域cluster point 聚点double limit 二重极限iterated limit 累次极限continuity 连续性increment 增量total increment 全增量partial increment 偏增量partial derivative 偏导数partial derivative of f(x,y) with respect to x ( y ) f(x,y)关于x（y）的偏导数higher partial derivative 高阶偏导数mixed partial derivative 混合偏导数Laplace equation 拉普拉斯方程total differential 全微分differentiable 可微chain rule 链式法则implicit function 隐函数implicit differentiation 隐函数微分法Jacobian determinant 雅可比行列式curve 曲线space curve 空间曲线tangent vector 切矢tangent line 切线normal plane 法平面surface 曲面normal vector 法矢normal line 法线tangent plane 切平面sphere 球面cylinder 柱面cone 锥面directional derivative 方向导数gradient 梯度gradient vector 梯度矢量f delflevel curve 等值线level surface 等值面local extremum 极值local maximum 极大值local minimum 极小值extreme value 最值absolute maximum (minimum) 最大（最小）值stationary point (critical point) 驻点（临界点）conditional extremum 条件极值Lagrange multiplier 拉格朗日乘数method of Lagrange multiplier 拉格朗日乘数法objective function 目标函数constraint 约束条件method of least square 最小二乘法field 场scalar field 数量场vector field 矢量场gradient field 梯度场potential field 势场potential function 势函数conservative field 保守场gravitational field 引力场force field 力场velocity field 速度场MATHEMATICAL TERMS (Part 4)multiple integral 重积分double integral 二重积分iterated integral 累次积分region 区域region of integration 积分区域type X (Y) region X（Y）型区域order of integration 积分秩序reverse the order of integration 交换积分秩序polar coordinates 极坐标double integrals in polar coordinates 极坐标下的二重积分volume 体积lamina 平面薄片mass 质量density 密度moment about x-axis 关于x 轴的（静）力矩center of mass 重心moment of inertia 转动惯量surface 曲面area of a surface 曲面的面积triple integral 三重积分rectangle 矩形rectangular coordinates 直角坐标系cylinder 柱面cylindrical coordinates 柱面坐标系sphere 球面spherical coordinates 球面坐标系change of variables in multiple integrals 重积分的变量替换Jacobian determinant 雅可比行列式line integral 曲线积分line integral with respect to arc length 对弧长的曲线积分（第一型）line integral with respect to x ( y ) 对坐标x（y）曲线积分（第二型）line integral of a vector field 向量场的曲线积分smooth curve 光滑曲线piecewise smooth curve 逐段光滑曲线oriented curve 有向曲线orientation of a curve 曲线的方向work 功the line integral is independent of path 曲线积分与路径无关connected region 连通区域simply-connected region 单连通区域closed curve 闭曲线Green’s theorem 格林定理（公式）positive orientation of a curve 曲线的正向Fundamental theorem for line integrals 曲线积分的基本定理surface integral 曲面积分surface integral of a scalar field 数量场的曲面积分（第一型）surface integral of a vector field 向量场的曲面积分（第二型）orientable surface 可定向曲面oriented surface 有向曲面Möbius strip 莫比乌斯带Klein bottle 克莱因瓶one-sided surface 单侧曲面two-sided surface 双侧曲面closed surface 闭曲面flux 流量、通量electric flux 电通量divergence 散度rotation (curl) 旋度Gauss’ theorem 高斯定理（公式）The divergence theorem 散度定理（公式）Stokes’ theorem 斯托克斯定理（公式）curl theorem 旋度定理（公式）circulation of v around L v 沿L 的环流量Hamilton operator 哈密顿算子harmonic field 调和场。