微積分極限

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

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

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

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 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

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