曲线积分

Curve Integral

Line Integral with Respect to Arc Length
对弧长的曲线积分，标量场

曲线形构件的质量，利用弧微分转换为定积分的计算

\int_{L} f (x, y) d s = lim_{λ \to 0} \sum_{i = 1}^{n} f (ξ_{i}, η_{i}) Δ s_{i}

$L$ 为积分弧段

\begin{array}{r} {\begin{cases} x = φ (t) \\ (α \leq t \leq β) \\ y = ψ (t) \end{cases} \int_{L} f (x, y) d s = \int_{α}^{β} f [φ (t), ψ (t)] \sqrt{φ^{' 2} (t) + ψ^{' 2} (t)} d t \end{array}

\begin{aligned} y & = ψ (x) (x_{0} \leq x \leq X) \int_{L} f (x, y) d s = \int_{x_{0}}^{X} f [x, ψ (x)] \sqrt{1 + ψ^{' 2} (x)} d x \\ x & = φ (y) (y_{0} \leq y \leq Y) \int_{L} f (x, y) d s = \int_{y_{0}}^{Y} f [φ (y), y] \sqrt{1 + φ^{' 2} (y)} d y \end{aligned}

Line Integral with Respect to the Coordinate
第二类曲线积分 向量场

变力沿曲线所作的功

\begin{array}{r} \int_{L} P (x, y) d x = lim_{λ \to 0} \sum_{i = 1}^{n} P (ξ_{i}, η_{i}) Δ x_{i} \\ \int_{L} Q (x, y) d y = lim_{λ \to 0} \sum_{i = 1}^{n} Q (ξ_{i}, η_{i}) Δ y_{i} \end{array}

转化为定积分进行计算

二维情形： $x = φ (t), y = ψ (t)$

\begin{aligned} \int_{L} F (x, y) \cdot d r & = \int_{L} P (x, y) d x + Q (x, y) d y = \int_{α}^{β} {P [φ (t), ψ (t)] φ^{'} (t) + Q [φ (t), ψ (t)] ψ^{'} (t)} d t \end{aligned}

三维情形： $x = φ (t), y = ψ (t), z = ω (t)$

\begin{aligned} \int_{Γ} A (x, y, z) \cdot d r = \int_{Γ} P (x, y, z) d x + Q (x, y, z) d y + R (x, y, z) d z \\ = \int_{α}^{β} {P [φ (t), ψ (t), ω (t)] φ^{'} (t) + Q [φ (t), ψ (t), ω (t)] ψ^{'} (t) + R [φ (t), ψ (t), ω (t)] ω^{'} (t)} d t \end{aligned}

\begin{array}{r} \int_{L} A \cdot d r = \int_{L} A \cdot τ d s \end{array}

\begin{array}{r} \int_{L} P d x + Q d y = \int_{L} (P \cos α + Q \cos β) d s \end{array}

\begin{array}{r} \int_{L} P d x + Q d y + R d z = \int_{L} (P \cos α + Q \cos β + R \cos γ) d s \end{array}