Lecture 48: Line Integrals with Respect to Arc Length
Graduate Entrance Examination Mathematics study notes: Lecture 48: Line Integrals with Respect to Arc Length. Original formulas, diagrams, and examples are retained.
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57.1 Curve Integral with Respect to Arc Length
57.1.1 Basic Concepts
#####Definition: Thelineintegralofthearclengthontheplane
description: $$\int_{L}f(x,y)ds=\lim_{\lambda\to0}\sum_{i=1}^{n}f(\xi_{i},\eta_{i})\Delta s_{i}$$
Explanation
- A binary function integrating along a two-dimensional curve segment;
- Divide the curve into n small ends, multiply the curve's function value by the length of the small arc segment, sum each segment and take the limit. If this limit exists, then the line integral exists;
#####Theorem: Propertiesoflineintegrals
description: $$\int_{L(AB)}f(x,y)ds=\int_{L(BA)}f(x,y)ds$$
Explanation
- Meaning: Line integral is unrelated to the direction of the path;
Inference
- 1. $\int_{L_{1}+L_{2}}f(x,y)ds=\int_{L_{1}}f(x,y)ds+\int_{L_{2}}f(x,y)ds$；
- 2. $\int_{L}[df(x,y)+\beta g(x,y)]ds=\alpha\int_{L}f(x,y)ds+\beta\int_{L}g(x,y)ds$；
- 3. $\int_{C}f(x,y)ds=\int_{C_{1}}f(x,y)ds+\int_{C_{2}}f(x,y)ds$；
- 4. $f(x,y)<=g(x,y) ; \int_{L}f(x,y)ds\leq\int_{L}g(x,y)ds$
57.2 Calculation of Curve Integrals
57.2.1 Basic Law
#####Theorem: Thefirsttypecalculatescurveintegrals: Direct method
Description: assume the parameter equation of L is $\begin{cases}x=\varphi(t),\\y=\psi(t),&\end{cases}(\alpha\leqslant t\leqslant\beta)$, then: $$\int_{L}f(x,y)\mathrm{d}s=\int_{\alpha}^{\beta}f(\varphi(t),\psi(t))\sqrt{\varphi'(t)^2+\psi'(t)^2}\mathrm{d}t$$
Explanation
- Note:
dsis the arc derivative of the curve;- The upper and lower limits are arc lengths from small to large;
#####Theorem: # Calculation of line integral of the first kind: Cartesian equations
description: $\text{ If }C:y=y(x),\quad a\leq x\leq b$, then:
$$\int_{}f(x,y)\mathrm{d}s=\int_{a}^{b}f(x,y(x))\sqrt{1+y'^2(x)}\mathrm{d}x$$
Explanation
- Equivalent to treating
xas a parameter;
#####Theorem: Thefirsttypecalculatescurveintegrals: Polar coordinate equation
description: $\text{ If }C:\rho=\rho (\theta)\quad\alpha\leq\theta\leq\beta$, then:
$$\int_{C}f(x,y)ds=\int_{\alpha}^{\beta}f(\rho(\theta)\cos\theta,\rho(\theta)\sin\theta)\sqrt{\rho^{2}(\theta)+\rho^{\prime2}(\theta)}d\theta $$
Explanation
57.2.2 Parity and Symmetry
#####Theorem: Parityoftheintegralcurve
description: $$\int_{C}f(x,y)\mathrm{d}s=\begin{cases}2\int_{C_{x>4}}f(x,y)\mathrm{d}s,&f(-x,y)=f(x,y)\\0,&f(-x,y)=-f(x,y)\end{cases}$$
Explanation
#####Theorem: Symmetry
description: General situation: $\int_{c}f(x,y)\mathrm{d}s=\int_{c}f(y,x)\mathrm{d}s$
Special: $\int_Cf(x)ds=\int_Cf(y)ds$