微積分極限

Limits

• Definition: The limit of a function f(x) as x approaches a is L, denoted as lim x→a f(x) = L, if for every ε > 0, there exists a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ.
• Properties:
  • Linearity: lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
  • Homogeneity: lim x→a [c * f(x)] = c * lim x→a f(x)
  • Sum and product of limits: lim x→a [f(x) * g(x)] = lim x→a f(x) * lim x→a g(x)
• Types of limits:
  • One-sided limits (left-hand and right-hand limits)
  • Infinite limits
  • Limits at infinity

Derivatives

• Definition: The derivative of a function f(x) at a point x = a is denoted as f'(a) and represents the rate of change of the function with respect to x at x = a.
• Rules of differentiation:
  • Power rule: If f(x) = x^n, then f'(x) = nx^(n-1)
  • Product rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
  • Quotient rule: If f(x) = u(x) / v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
  • Chain rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)
• Geometric interpretation: The derivative represents the slope of the tangent line to the graph of the function at a point.
• Applications:
  • Finding maximum and minimum values
  • Determining the rate of change of physical quantities

Integrals

• Definition: The definite integral of a function f(x) from a to b, denoted as ∫[a,b] f(x) dx, represents the area under the curve of f(x) between a and b.
• Types of integrals:
  • Definite integrals
  • Indefinite integrals (antiderivatives)
• Basic integration rules:
  • Constant multiple rule: ∫[a,b] c * f(x) dx = c * ∫[a,b] f(x) dx
  • Sum rule: ∫[a,b] [f(x) + g(x)] dx = ∫[a,b] f(x) dx + ∫[a,b] g(x) dx
• Substitution method: ∫[a,b] f(g(x)) * g'(x) dx = ∫[g(a),g(b)] f(u) du
• Applications:
  • Finding area under curves
  • Finding volumes of solids
  • Solving physical problems involving accumulation

Multivariable Calculus

• Functions of multiple variables: Functions that take in multiple inputs (x, y, z, ...) and produce a single output.
• Partial derivatives: Derivatives of a function with respect to one of its variables, while keeping the other variables constant.
• Gradient: A vector of partial derivatives of a function, representing the direction of the maximum rate of change.
• Double and triple integrals: Extensions of definite integrals to functions of multiple variables.
• Applications:
  • Optimization problems
  • Physics and engineering applications (e.g., electromagnetism, fluid dynamics)

Vector Calculus

• Vector fields: Functions that assign a vector to each point in space.
• Gradient field: A vector field that represents the direction of the maximum rate of change of a scalar function.
• Divergence: A measure of the rate of change of a vector field's magnitude.
• Curl: A measure of the rotation of a vector field.
• Line integrals: Integrals of vector fields along curves.
• Surface integrals: Integrals of vector fields over surfaces.
• Stokes' theorem: Relates the line integral of a vector field around a closed curve to the surface integral of the curl of the vector field over the enclosed surface.
• Gauss' divergence theorem: Relates the flux of a vector field through a closed surface to the divergence of the vector field within the enclosed volume.

极限

• 极限的定义：函数 f(x) 在 x 接近 a 时的极限是 L，记作 lim x→a f(x) = L，若对于任意的 ε > 0，存在 δ > 0，使得当 0 < |x - a| < δ 时，有 |f(x) - L| < ε。
• 极限的性质：
  • 线性：lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
  • 同质：lim x→a [c * f(x)] = c * lim x→a f(x)
  • 积和极限：lim x→a [f(x) * g(x)] = lim x→a f(x) * lim x→a g(x)
• 极限的类型：
  • 一侧极限（左极限和右极限）
  • 无限极限
  • 无穷大极限

导数

• 导数的定义：函数 f(x) 在点 x = a 处的导数是 f'(a)，表示函数关于 x 的变化率。
• 导数规则：
  • 幂规则：如果 f(x) = x^n，则 f'(x) = nx^(n-1)
  • 乘积规则：如果 f(x) = u(x)v(x)，则 f'(x) = u'(x)v(x) + u(x)v'(x)
  • 商率规则：如果 f(x) = u(x) / v(x)，则 f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
  • 链式规则：如果 f(x) = g(h(x))，则 f'(x) = g'(h(x)) * h'(x)
• 几何解释：导数表示函数图形在点的切线坡度。
• 应用：
  • 求函数的最大值和最小值
  • 确定物理量的变化率

积分

• 定义：函数 f(x) 在区间 [a, b] 上的定积分，记作 ∫[a,b] f(x) dx，表示函数图形在 a 和 b 之间的梯形面积。
• 积分类型：
  • 定积分
  • 不定积分（反导数）
• 基本积分规则：
  • 常数倍规则：∫[a,b] c * f(x) dx = c * ∫[a,b] f(x) dx
  • 和规则：∫[a,b] [f(x) + g(x)] dx = ∫[a,b] f(x) dx + ∫[a,b] g(x) dx
• 替换法：∫[a,b] f(g(x)) * g'(x) dx = ∫[g(a),g(b)] f(u) du
• 应用：
  • 求函数图形下的面积
  • 求实体的体积
  • 解决物理问题中的累积问题

多元微积分

• 多元函数：函数可以处理多个输入（x, y, z,...）和产生一个输出。
• 部分导数：函数关于某个变量的导数，而保持其他变量不变。
• 梯度：函数的部分导数向量，表示变化率的最大方向。
• 双重积分和三重积分：定积分的多元扩展。
• 应用：
  • 优化问题
  • 物理和工程应用（例如 电磁学、流体动力学）

矢量微积分

• 矢量场：将每个点配对一个矢量的函数。
• 梯度场：一个矢量场，表示标量函数的最大变化率方向。
• 散度：一个矢量场的大小变化率。
• 旋度：一个矢量场的旋转程度。
• 线积分：矢量场沿曲线的积分。
• 表面积分：矢量场在曲面的积分。
• 斯托克斯定理：将矢量场的线积分与旋度的表面积分相关联。

了解微積分極限的定義、properties和類型。涵蓋線性、同次性、和積分的性質。