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数学
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圆锥曲线探究 · MA·MB 的取值范围
Conic investigation · range of MA·MB
取值范围由 Rust 计算核精确给出 · 转动弦看 MA·MB 在范围内变化
Exact range from the Rust kernel · rotate the chord to see MA·MB move within it
一次函数 y = kx + b
Linear function y = kx + b
k 是斜率,b 是 y 轴截距 · x 轴截距在 −b/k
k is the slope, b the y-intercept · x-intercept at −b/k
分式函数 y = a/(x−h) + k
Rational function y = a/(x−h) + k
竖直渐近线 x=h,水平渐近线 y=k
Vertical asymptote x=h, horizontal asymptote y=k
绝对值函数 y = a|x−h| + k
Absolute value y = a|x−h| + k
顶点在 (h, k) 的 V(或 ∧)形
A V (or ∧) with its vertex at (h, k)
平方根函数 y = a√(x−h) + k
Square-root function y = a√(x−h) + k
定义域 x ≥ h;从 (h, k) 出发,越升越缓
Defined for x ≥ h; starts at (h, k) and rises ever more slowly
二次函数 y = ax² + bx + c
Quadratic function y = ax² + bx + c
拖动 a、b、c —— 看顶点、根与判别式
Drag a, b, c — see the vertex, roots and discriminant
正弦函数 y = A·sin(Bx + C) + D
Sinusoid y = A·sin(Bx + C) + D
拖动 A、B、C、D —— 振幅、周期、相位、中线
Drag A, B, C, D — amplitude, period, phase, midline
指数函数 y = a·bˣ
Exponential function y = a·bˣ
b>1 增长,0<b<1 衰减;恒过 (0, a)
b>1 grows, 0<b<1 decays; passes through (0, a)
对数函数 y = a·log_b(x)
Logarithmic function y = a·log_b(x)
指数函数的反函数 · 定义域 x>0,恒过 (1, 0)
Inverse of y=bˣ · defined for x>0, always passes through (1, 0)
导数 = 切线斜率
Derivative = tangent slope
拖动 x₀ —— 切线斜率恰等于 f′(x₀);f′=0 处正是极值点
Drag x₀ — the tangent's slope equals f′(x₀); f′=0 at turning points
定积分 = 曲边梯形面积
Definite integral = area under the curve
n 个矩形面积之和 → n 越大越逼近精确的定积分
Sum of n rectangles → the exact integral as n grows
向量加法 r = a + b
Vector addition r = a + b
三角形法则(首尾相接)与平行四边形法则给出同一个合向量
Triangle rule (tip-to-tail) and the parallelogram give the same resultant
单位圆 · 三角函数定义
Unit circle · trig definitions
单位圆上的点坐标恰好是 (cosθ, sinθ)
On the unit circle, the point is exactly (cosθ, sinθ)
矩阵 · 线性变换
Matrix as a linear transformation
矩阵的两列就是 î、ĵ 的落点;行列式就是面积伸缩倍数
The columns say where î and ĵ go; det is the area scale factor
复数 · 复平面(阿甘图)
Complex numbers · the Argand plane
z = a + bi 是一个向量 · |z| 是长度,arg z 是角度
z = a + bi as a vector · |z| is its length, arg z its angle
概率树 · 贝叶斯定理
Probability tree · Bayes' theorem
沿分支相乘;后验概率 P(D|+) 可能低得出人意料
Multiply along branches; the posterior P(D|+) can be surprisingly low
参数方程 · 李萨如图形
Parametric curves · Lissajous figures
x、y 各自做正弦振动;两者的频率比决定图形形状
x and y each oscillate; their frequency ratio shapes the figure
微分方程 · 方向场
Differential equations · slope field
每一小段的斜率就是 f(x,y);解曲线沿着它们流动
Each little dash is the slope f(x,y); solutions flow along them
数列 · 等差与等比
Sequences · arithmetic & geometric
aₙ = a₁+(n−1)d 或 a₁·rⁿ⁻¹ · 及前 n 项和 Sₙ
aₙ = a₁+(n−1)d or a₁·rⁿ⁻¹ · with the partial sum Sₙ
二元一次方程组的解
Solving a 2×2 linear system
二元一次方程组 · 克拉默法则 D=a₁b₂−a₂b₁ · 交点/平行/重合图解
Two lines a₁x+b₁y=c₁ and a₂x+b₂y=c₂ · solved by Cramer's rule
向量点积与投影
Dot product & projection
a·b 与投影 · 夹角 θ · 锐角/钝角/垂直判定
a·b, |a|, |b|, angle θ and the projection of a onto b
圆的方程 · 位置与切线
Circle: equation, position & tangents
标准式 ↔ 一般式 · 点与圆位置 · 圆外一点切线
Standard ↔ general form · classify a point · draw the tangents
排列与组合
Permutations & combinations
nPr / nCr · 阶乘约分 · nCr=nC(n−r) 对称
nPr counts ordered selections, nCr unordered · nCr = nC(n−r)
骰子点数和分布
Dice-sum distribution
n 颗 f 面骰子求和 · 卷积精确分布 · E[X]、方差与正态近似
Exact distribution of the sum by convolution · normal approximation overlaid
傅里叶级数 · 波形合成与吉布斯现象
Fourier series · synthesis & Gibbs phenomenon
方波/锯齿/三角波 N 项合成 · 谐波叠加 · 吉布斯过冲 8.95%
Add harmonics to build a square / sawtooth / triangle wave; watch the overshoot near jumps
泰勒级数 · 多项式逼近
Taylor series · polynomial approximation
sin/cos/eˣ/ln(1+x)/1/(1−x) · N 阶多项式 · 收敛区间
Approximate a function by its degree-N Taylor polynomial about x₀; watch convergence and the interval of convergence
梯度下降 · 学习率、收敛与发散
Gradient descent · learning rate, convergence & divergence
等高线迭代 · 学习率过大发散 · 梯度向量
Iterate x ← x − η∇f over a contour surface; see convergence, slow crawl and divergence
2×2 特征值、特征向量与线性变换
2×2 eigenvalues, eigenvectors & the linear map
λ=(tr±√(tr²−4det))/2 · 不变方向 · 圆→椭圆变换
Eigenvectors are the invariant directions A only stretches by λ
二维向量场 —— 散度与旋度
2D vector field — divergence & curl
源/汇/旋转/鞍点 · div=∂P/∂x+∂Q/∂y · curl
Divergence measures spreading; curl measures swirling
马尔可夫链 —— 转移矩阵与稳态分布
Markov chain — transition matrix & stationary distribution
转移矩阵 · 状态图 · πP 迭代 → 稳态分布
Iterate πₜ₊₁ = πₜP and watch it converge to π with πP = π
Polar curves r = f(θ)
Polar curves r = f(θ)
r=f(θ) · 玫瑰线/心形线/螺线 · 极坐标→直角 · 描点
rose · cardioid · Archimedean spiral · limaçon — trace (r,θ)→(x,y)
Parametric curves x=f(t), y=g(t)
Parametric curves x=f(t), y=g(t)
x=f(t),y=g(t) · Lissajous/摆线/椭圆 · 随 t 描点
Lissajous · cycloid · ellipse — a point traces the curve as t advances
Conic sections by eccentricity e
Conic sections by eccentricity e
离心率 e · 圆→椭圆→抛物线→双曲线 · 焦点/准线
r = l/(1+e·cosθ) — one focus, one directrix, e morphs the whole family
Newton–Raphson · tangent root finding
Newton–Raphson · tangent root finding
x_{n+1}=x_n−f/f' · 切线交轴迭代 · 收敛/发散
Step through xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ): each tangent line's x-intercept is the next guess
L'Hôpital's rule · resolving 0/0
L'Hôpital's rule · resolving 0/0
0/0 · ∞/∞ · lim f/g=lim f'/g' · 导数比求极限
When f/g is the indeterminate form 0/0, the limit equals lim f'/g' — differentiate top and bottom
Hyperbolic functions · the unit hyperbola
Hyperbolic functions · the unit hyperbola
cosh/sinh/tanh · x²−y²=1 参数化 · cosh²−sinh²=1
cosh t = (eᵗ+e⁻ᵗ)/2, sinh t = (eᵗ−e⁻ᵗ)/2 — the point (cosh t, sinh t) rides the hyperbola x²−y²=1
2×2 matrix transform · linear map of the plane
2×2 matrix transform · linear map of the plane
单位方格→平行四边形 · 基向量像 · det=面积 · 旋转/剪切
M = [[a,b],[c,d]] sends e1 → (a,c), e2 → (b,d); det = ad − bc = signed area
Eigenvalues & eigenvectors of a 2×2 matrix
Eigenvalues & eigenvectors of a 2×2 matrix
Mv=λv · λ²−tr·λ+det=0 · 不旋转方向按 λ 缩放
M·v = λ·v · λ² − (tr)λ + det = 0 · λ = (tr ± √(tr² − 4·det)) / 2
Determinant as signed area
Determinant as signed area
det=ad−bc · 列向量平行四边形有向面积 · 共线为0
columns u = (a,c), v = (b,d) · det = ad − bc = signed area of the parallelogram
Vector cross product · a × b in 3D
Vector cross product · a × b in 3D
a×b · |a||b|sinθ · 右手法则 · =平行四边形面积
a × b ⟂ a and b, |a × b| = |a||b| sinθ = parallelogram area
nth roots of unity · solutions of zⁿ = 1
nth roots of unity · solutions of zⁿ = 1
z^n=1 · z_k=e^{2πik/n} · 单位圆正 n 边形 · 和为0
zₖ = e^(2πik/n) = (cos 2πk/n, sin 2πk/n), a regular n-gon on the unit circle
Gradient field · ∇f and level curves
Gradient field · ∇f and level curves
∇f · 最陡上升 · ⊥等高线 · 等高线+梯度箭头
∇f = (∂f/∂x, ∂f/∂y) points to steepest ascent, ⟂ the level curves
Green's theorem — circulation equals curl over area
Green's theorem — circulation equals curl over area
∮_C(P dx+Q dy)=∬_D(∂Q/∂x−∂P/∂y)dA · 环量=旋度积分
∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA, evaluated both ways
Stokes' theorem — circulation equals curl-flux
Stokes' theorem — circulation equals curl-flux
∮_∂S F·dr=∬_S(∇×F)·dS · 边界环量=旋度通量
∮_∂S F·dr = ∬_S (∇×F)·dS, and the flux is the same for any spanning surface
Divergence theorem — outward flux equals divergence over volume
Divergence theorem — outward flux equals divergence over volume
∯_S F·dS=∭_V(∇·F)dV · 外向通量=散度积分
∯_S F·dS = ∭_V (∇·F) dV, evaluated both ways
Jacobian · change of variables
Jacobian · change of variables
dA=|J|du dv · J=∂(x,y)/∂(u,v) · 网格面积缩放 · 极坐标
dA = |J| du dv, where J = ∂(x,y)/∂(u,v) is the Jacobian matrix
Lagrange multipliers · constrained optimization
Lagrange multipliers · constrained optimization
∇f=λ∇g · 等高线与约束相切 · 约束最优 · λ
Extremize f(x,y) subject to g(x,y)=0 by solving ∇f = λ∇g
Directional derivative · D_û f = ∇f · û
Directional derivative · D_û f = ∇f · û
D_û f=∇f·û · 沿梯度最大 · 垂直为零
The rate of change of f in direction û equals the projection of ∇f onto û
Line integrals · ∫_C f ds and work ∫_C F·dr
Line integrals · ∫_C f ds and work ∫_C F·dr
标量 ∫_C f ds · 向量 ∫_C F·dr 做功 · 沿曲线累加
Parametrise a curve C, step along it, and accumulate a Riemann sum
Surface integrals · area ∬_S f dS and flux ∬_S F·n̂ dS
Surface integrals · area ∬_S f dS and flux ∬_S F·n̂ dS
∬_S f dS · 通量 ∬_S F·n̂ dS · 面片求和
Tile a surface with small patches, sum dS or F·n̂ dS
Double integral = volume under a surface
Double integral = volume under a surface
∬_R f dA=曲面下体积 · 黎曼柱格 · 细分收敛
∬_R f(x,y) dA is the volume between z=f(x,y) and the region R
ε–δ definition of a limit
ε–δ definition of a limit
∀ε>0∃δ>0 · |x−a|<δ⟹|f−L|<ε · ε带/δ带
lim_{x→a} f(x) = L: for every ε > 0 there is a δ > 0 so that 0 < |x − a| < δ ⟹ |f(x) − L| < ε
2D Riemann sum — approximating a double integral
2D Riemann sum — approximating a double integral
N×N 网格采样求和 → ∬f dA · 分辨率细化收敛
∬_D f dA ≈ Σ f(sample) · ΔA over an N×N grid on the unit square; refine N and watch it converge
Taylor polynomial & Lagrange remainder
Taylor polynomial & Lagrange remainder
泰勒多项式+拉格朗日余项 · Rn=f⁽ⁿ⁺¹⁾(ξ)/(n+1)!·(x−a)ⁿ⁺¹
Pₙ(x) = Σ f⁽ᵏ⁾(a)/k! (x−a)ᵏ, with error controlled by Rₙ(x) = f⁽ⁿ⁺¹⁾(ξ)/(n+1)! (x−a)ⁿ⁺¹