解析几何学
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特别注意
本书中微分算符与教科书不同。
在本书中，全微分算符为$\Lambda$，偏微分算符为$\lambda$，一阶求导算符为$\dfrac{\Lambda}{\lambda} ≡ \dfrac{\Lambda^{1} }{\lambda^{1} }$，高阶求导算符为$\dfrac{\Lambda^{n} }{\lambda^{n} } ≡ \dfrac{\Lambda}{\lambda} \dfrac{\Lambda^{n-1} }{\lambda^{n-1} }$。
一阶微分同等定理。
$\Lambda f(x) = \lambda f(x)$
$\dfrac{\Lambda f^{⇵}(x)}{\lambda x} \mathop{====}\limits_{x={‘}f^{⇵}(y)}^{y=f^{⇵}(x)} \left[ \dfrac{\Lambda {‘}f^{⇵}(y)}{\lambda y} \right]^{-1}$
$\dfrac{\Lambda g(f(x))}{\lambda x} = \dfrac{\Lambda g(f(x))}{\lambda f(x)} · \dfrac{\Lambda f(x)}{\lambda x} \mathop{===}\limits^{y=f(x)} \dfrac{\Lambda g(y)}{\lambda y} · \dfrac{\Lambda f(x)}{\lambda x}$
二变元全微分定理。
$\dfrac{\Lambda^{2} f(x,y)}{\lambda^{1} x · \lambda^{1} y} = f_{y,x} = \dfrac{\Lambda^{2} f(x,y)}{\lambda^{1} y · \lambda^{1} x}$
$\Lambda^{1} f(x,y) = f_{x} · \lambda x + f_{y} · \lambda y$
$\Lambda^{2} f(x,y) = \Lambda [f_{x} · \lambda x + f_{y} · \lambda y] = f_{x,x} · \lambda^{2} x + f_{x,y} · \lambda^{1} x · \lambda^{1} y + f_{y,x} · \lambda^{1} y · \lambda^{1} x + f_{y,y} · \lambda^{2} y$
典例：函数$f(x) = x^{3}$。
| $\int \Lambda f(x) = f(x)$ | $\Lambda^{0} f(x) ≡ f(x)$ | $\lambda^{0} x ≡ 1$ | $\lambda h(x) = \dfrac{\Lambda h(x)}{\lambda x} · \lambda x$ | ||
|---|---|---|---|---|---|
| $\dfrac{\Lambda^{-2} f(x)}{\lambda^{-2} x} = \dfrac{x^{5} }{20}$ | $\dfrac{\Lambda^{-1} f(x)}{\lambda^{-1} x} = \dfrac{x^{4} }{4}$ | $\dfrac{\Lambda^{0} f(x)}{\lambda^{0} x} = x^{3}$ | $\dfrac{\Lambda^{1} f(x)}{\lambda^{1} x} = 3 · x^{2}$ | $\dfrac{\Lambda^{2} f(x)}{\lambda^{2} x} = 6 · x^{1}$ | $\dfrac{\Lambda^{3} f(x)}{\lambda^{3} x} = 6$ |
| $\Lambda^{-2} f(x) = \dfrac{x^{5} }{20} · \lambda^{-2} x$ | $\Lambda^{-1} f(x) = \dfrac{x^{4} }{4} · \lambda^{-1} x$ | $\Lambda^{0} f(x) = x^{3} · \lambda^{0} x$ | $\Lambda^{1} f(x) = 3 · x^{2} · \lambda^{1} x$ | $\Lambda^{2} f(x) = 6 · x^{1} · \lambda^{2} x$ | $\Lambda^{3} f(x) = 6 · \lambda^{3} x$ |
| $\int \Lambda^{-1} f(x) = \Lambda^{-2} f(x)$ | $\int \Lambda^{0} f(x) = \Lambda^{-1} f(x)$ | $\int \Lambda^{1} f(x) = \Lambda^{0} f(x)$ | $\int \Lambda^{2} f(x) = \Lambda^{1} f(x)$ | $\int \Lambda^{3} f(x) = \Lambda^{2} f(x)$ | |
| $\int \dfrac{x^{4} }{x} · \lambda^{-1} x = \dfrac{x^{5} }{20} · \lambda^{-2} x$ | $\int x^{3} · \lambda^{0} x = \dfrac{x^{4} }{4} · \lambda^{-1} x$ | $\int 3 · x^{2} · \lambda^{1} x = x^{3} · \lambda^{0} x = x^{3}$ | $\int 6 · x^{1} · \lambda^{2} x = 3 · x^{2} · \lambda^{1} x = \lambda^{1} x^{3}$ | $\int 6 · \lambda^{3} x = 6 · x^{1} · \lambda^{2} x$ |
单变元隐式方程
$0 = F(y(x),x)$
$\dfrac{\Lambda y(x)}{\lambda x} = -\dfrac{\Lambda F(y,x)}{\lambda x} · \left[ \dfrac{\Lambda F(y,x)}{\lambda y} \right]^{-1}$
$\dfrac{\Lambda^2 y(x)}{\lambda^2 x} = -\left[ \dfrac{\Lambda^2 F(y,x)}{\lambda^2 y} · \left[ \dfrac{\Lambda y}{\lambda x} \right]^{2} + \dfrac{\Lambda^2 F(y,x)}{\lambda y · \lambda x} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda^2 F(y,x)}{\lambda x · \lambda y} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda^2 F(y,x)}{\lambda^2 x} \right] · \left[ \dfrac{\Lambda F(y,x)}{\lambda y} \right]^{-1}$
| $\dfrac{\Lambda 0}{\lambda x}$ | $\dfrac{\Lambda F(y,x)}{\lambda y} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda F(y,x)}{\lambda x}$ |
| $\dfrac{\Lambda^2 0}{\lambda^2 x}$ | $\left[ \dfrac{\Lambda^2 F(y,x)}{\lambda^2 y} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda^2 F(y,x)}{\lambda y · \lambda x} \right] · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda F(y,x)}{\lambda y} · \dfrac{\Lambda^2 y}{\lambda^2 x} + \dfrac{\Lambda^2 F(y,x)}{\lambda x · \lambda y} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda^2 F(y,x)}{\lambda^2 x}$ |
| $\dfrac{\Lambda y(x)}{\lambda x}$ | $-\dfrac{\Lambda F(y,x)}{\lambda x} · \left[ \dfrac{\Lambda F(y,x)}{\lambda y} \right]^{-1}$ |
| $\dfrac{\Lambda^2 y(x)}{\lambda^2 x}$ | $-\left[ \dfrac{\Lambda^2 F(y,x)}{\lambda^2 y} · \left[ \dfrac{\Lambda y}{\lambda x} \right]^{2} + \dfrac{\Lambda^2 F(y,x)}{\lambda y · \lambda x} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda^2 F(y,x)}{\lambda x · \lambda y} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda^2 F(y,x)}{\lambda^2 x} \right] · \left[ \dfrac{\Lambda F(y,x)}{\lambda y} \right]^{-1}$ |
单变元参数方程
$\left\lbrace\begin{aligned}
y &= y(t)
x &= x(t)
\end{aligned}\right.$
$y(x) = y[{‘}x(t)]$
$\dfrac{\Lambda y(x)}{\lambda x} = \dfrac{\Lambda y(t)}{\lambda t} · \left[ \dfrac{\Lambda x(t)}{\lambda t} \right]^{-1}$
$\dfrac{\Lambda^{2} y(x)}{\lambda^{2} x} = \left[ \dfrac{\Lambda^{2} y}{\lambda^{2} t} · \dfrac{\Lambda x}{\lambda t} - \dfrac{\Lambda y}{\lambda t} · \dfrac{\Lambda^{2} x}{\lambda^{2} t} \right] · \left[ \dfrac{\Lambda x}{\lambda t} \right]^{-3}$
| $\dfrac{\Lambda y(x)}{\lambda x}$ | $\dfrac{Λ y(t)}{λ t} · \dfrac{Λ t}{λ x(t)} = \left[ \dfrac{\Lambda y(t)}{\lambda t} \right] · \left[ \dfrac{\Lambda x(t)}{\lambda t} \right]^{-1}$ |
|---|---|
| $\dfrac{\Lambda^{2} y(x)}{\lambda^{2} x}$ | $\dfrac{\Lambda}{\lambda x} \dfrac{\Lambda y(x)}{\lambda x} = \dfrac{\Lambda}{\lambda t} \left[ \dfrac{\Lambda y}{\lambda t} · \left[ \dfrac{\Lambda x}{\lambda t} \right]^{-1} \right] · \dfrac{\Lambda t}{\lambda x} = \left[ \dfrac{\Lambda^{2} y}{\lambda^{2} t} · \left[ \dfrac{\Lambda x}{\lambda t} \right]^{-1} + \dfrac{\Lambda y}{\lambda t} · (-1) · \left[ \dfrac{\Lambda x}{\lambda t} \right]^{-2} · \dfrac{\Lambda^{2} x}{\lambda^{2} t} \right] · \dfrac{\Lambda t}{\lambda x} = \left[ \dfrac{\Lambda^{2} y}{\lambda^{2} t} · \dfrac{\Lambda x}{\lambda t} - \dfrac{\Lambda y}{\lambda t} · \dfrac{\Lambda^{2} x}{\lambda^{2} t} \right] · \left[ \dfrac{\Lambda x}{\lambda t} \right]^{-3}$ |
平面极角坐标系
$\left\lbrace\begin{aligned}
y &= r(\phi) · \sin \phi
x &= r(\phi) · \cos \phi
\end{aligned}\right.$
$y(x) = y[{‘}x(\phi)]$
$\tan α = \dfrac{\Lambda y(x)}{\lambda x} = \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} · \tan \phi + r(\phi) \right] · \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} - r(\phi) · \tan \phi \right]^{-1}$
$\tan ν = \tan (α - \phi) = r(\phi) · \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} \right]^{-1}$
| $\tan α = \dfrac{\Lambda y(x)}{\lambda x}$ | $\dfrac{\Lambda y(\phi)}{\lambda \phi} · \left[ \dfrac{\Lambda x(\phi)}{\lambda \phi} \right]^{-1} = \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} · \sin \phi + r(\phi) · \cos \phi \right] · \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} · \cos \phi - r(\phi) · \sin \phi \right]^{-1} = \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} · \tan \phi + r(\phi) \right] · \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} - r(\phi) · \tan \phi \right]^{-1}$ |
|---|---|
| $\tan ν = \tan (α - \phi)$ | $\dfrac{\tan α - \tan \phi}{1 + \tan α · \tan \phi} = \dfrac{\left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} · \tan \phi + r(\phi) \right] · \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} - r(\phi) · \tan \phi \right]^{-1} - \tan \phi}{1 + \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} · \tan \phi + r(\phi) \right] · \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} - r(\phi) · \tan \phi \right]^{-1} · \tan \phi} = \dfrac{\dfrac{\Lambda r(\phi)}{\lambda \phi} · \tan \phi + r(\phi) - \dfrac{\Lambda r(\phi)}{\lambda \phi} · \tan \phi + r(\phi) · \tan^{2} \phi}{\dfrac{\Lambda r(\phi)}{\lambda \phi} - r(\phi) · \tan \phi + \dfrac{\Lambda r(\phi)}{\lambda \phi} · \tan^{2} \phi + r(\phi) · \tan \phi} = r(\phi) · \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} \right]^{-1}$ |
平面图形的面积
平面曲线所包围图形的面积。注意：当$\left\lbrace\begin{aligned}
y(α) &= y(β)
x(α) &= x(β)
\end{aligned}\right.$时平面曲线形成闭环。
$\left\lbrace\begin{aligned}
A &= \left| \int\limits_{x_α}^{x_β} y(x) · \lambda x \right| = \left| \int\limits_{t_α}^{t_β} y(x(t)) · {^1}x(t) · \lambda t \right|
A &= \left| \int_{y_α}^{y_β} x(y) · \lambda y \right| = \left| \int\limits_{t_α}^{t_β} x(y(t)) · {^1}y(t) · \lambda t \right|
A &= \left| \dfrac{1}{2} · \int\limits_{α}^{β} r^{2}(\phi) · \lambda\phi \right|
\end{aligned}\right.$
旋转体截面的面积，旋转体的体积。
$A = π·f^2(x)$
$V = \int\limits_{x_α}^{x_β} π·f^2(x)·\lambda x$
空间曲线的长度
$L_3 = \int\limits_{t_α}^{t_β} \sqrt{[{^1}x(t)]^2 + [{^1}y(t)]^2 + [{^1}z(t)]^2} · \lambda t$
$L_3 = \int\limits_{x_α}^{x_β} \sqrt{1 + [{^1}y(x)]^2 + [{^1}z(x)]^2} · \lambda x$
$L_2 \mathop{=======}\limits_{x(\phi)=r(\phi)·\cos \phi}^{y(\phi)=r(\phi)·\sin \phi} \int\limits_{\phi_{α} }^{\phi_β} \sqrt{[^{1}r(\phi)]^2 + [r(\phi)]^2} · \lambda \phi$
| $L_3 = \int\limits_{t_α}^{t_β} \sqrt{[\lambda x(t)]^2 + [\lambda y(t)]^2 + [\lambda z(t)]^2}$ | ||
|---|---|---|
| $L_3 = \int\limits_{t_α}^{t_β} \sqrt{\left[ \dfrac{\Lambda x(t)}{\lambda t} \right]^2 + \left[ \dfrac{\Lambda y(t)}{\Lambda t} \right]^2 + \left[ \dfrac{\Lambda z(t)}{\lambda t} \right]^2} · \lambda t = \int\limits_{t_α}^{t_β} \sqrt{[{^1}x(t)]^2 + [{^1}y(t)]^2 + [{^1}z(t)]^2} · \lambda t$ | ||
| $L_3 = \int\limits_{x_α}^{x_β} \sqrt{\left[ \dfrac{\Lambda x}{\lambda x} \right]^2+\left[ \dfrac{\Lambda y(x)}{\lambda x} \right]^2+\left[ \dfrac{\Lambda z(x)}{\lambda x} \right]^2} · \lambda x = \int\limits_{x_α}^{x_β} \sqrt{1 + [{^1}y(x)]^2 + [{^1}z(x)]^2} · \lambda x$ | ||
| $L_2 \mathop{=======}\limits_{x(\phi)=r(\phi)·\cos \phi}^{y(\phi)=r(\phi)·\sin \phi} \int\limits_{\phi_α}^{\phi_β} \sqrt{[{^1}x(\phi)]^2 + [{^1}y(\phi)]^2} · \lambda \phi = \int\limits_{\phi_{α} }^{\phi_β} \sqrt{[^{1}r(\phi)]^2 + [r(\phi)]^2} · \lambda \phi$ | $⇐$ | $[{^1}x(\phi)]^2 + [{^1}y(\phi)]^2 \mathop{============}\limits_{ {^1}x(\phi) = {^1}r(\phi) · \cos \phi - r(\phi) · \sin \phi}^{ {^1}y(\phi) = {^1}r(\phi) · \sin \phi + r(\phi)·\cos \phi} [{^1}r(\phi)]^2 + [r(\phi)]^2$ |
旋转曲面的面积
平面光滑正则曲线$\left\lbrace\begin{aligned}
y &= y(t)
x &= x(t)
\end{aligned}\right.$绕$x$轴旋转所得曲面的面积。
$A \mathop{=}\limits_{}^{} \int\limits_{t_α}^{t_β} 2·π·y(t) · \sqrt{[{^1}x(t)]^2 + [{^1}y(t)]^2} · \lambda t$
$A = \int\limits_{x_α}^{x_β} 2 ·π·y(x) ·\sqrt{1 + [{^1}y(x)]^2} · \lambda x$
抛物体
$\left\lbrace\begin{aligned}
y(t) &= v_y · t - \dfrac{1}{2} · g · t^2
x(t) &= v_x · t
\end{aligned}\right.$
$y(x) = \dfrac{v_{y} }{v_x} · x - \dfrac{1}{2} · \dfrac{g}{v_x^2} · x^2$
$v(t) = \sqrt{\left[ \dfrac{\Lambda y(t)}{\lambda t} \right]^{2} + \left[ \dfrac{\Lambda x(t)}{\lambda t} \right]^{2} } = \sqrt{v_{x}^{2} + (v_{y} - g · t)^{2} }$
$\tan α = \dfrac{\Lambda y(x)}{\lambda x} = \dfrac{v_{y} - g · t}{v_{x} }$
| $\dfrac{\Lambda y(t)}{\lambda t}$ | $v_{y} - g · t$ |
|---|---|
| $\dfrac{\Lambda x(t)}{\lambda t}$ | $v_{x}$ |
| $v(t) = \sqrt{\left[ \dfrac{\Lambda y(t)}{\lambda t} \right]^{2} + \left[ \dfrac{\Lambda x(t)}{\lambda t} \right]^{2} }$ | $\sqrt{v_{x}^{2} + (v_{y} - g · t)^{2} }$ |
| $\tan α = \dfrac{\Lambda y(x)}{\lambda x}$ | $\dfrac{\Lambda y(t)}{\lambda t} · \left[ \dfrac{\Lambda x(t)}{\lambda t} \right]^{-1} = \dfrac{v_{y} - g · t}{v_{x} }$ |
椭圆形
$\left\lbrace\begin{aligned}
y(\phi) &= r_y · \sin \phi
x(\phi) &= r_x · \cos \phi
\end{aligned}\right.$
$\dfrac{x^2}{r_x^2} + \dfrac{y^2}{r_y^2} = 1$
$\dfrac{\Lambda y(x)}{\lambda x} = \dfrac{-r_{y} }{r_{x} · \tan \phi}$
$\dfrac{\Lambda^{2} y(x)}{\lambda^{2} x} = \dfrac{-r_{y} }{r_{x} · \sin^{3} \phi}$
| $\dfrac{\Lambda y(x)}{\lambda x}$ | $\dfrac{\Lambda y(\phi)}{\lambda \phi} · \left[ \dfrac{\Lambda x(\phi)}{\lambda \phi} \right]^{-1} = \dfrac{+r_{y} · \cos \phi} {-r_{x} · \sin \phi} = \dfrac{-r_{y} }{r_{x} · \tan \phi}$ |
|---|---|
| $\dfrac{\Lambda^{2} y(x)}{\lambda^{2} x}$ | $\left[ \dfrac{\Lambda^{2} y}{\lambda^{2} t} · \dfrac{\Lambda x}{\lambda t} - \dfrac{\Lambda y}{\lambda t} · \dfrac{\Lambda^{2} x}{\lambda^{2} t} \right] · \left[ \dfrac{\Lambda x}{\lambda t} \right]^{-3} = \left[ (-r_{y} · \sin \phi) · (-r_{x} · \sin \phi) - (+r_{y} · \cos \phi) · (-r_{x} · \cos \phi) \right] · [-r_{x} · \sin \phi]^{-3} = \dfrac{-r_{y} }{r_{x} · \sin^{3} \phi}$ |
椭圆的面积。
$A = π·r_x·r_y$
| $A = \int\limits_{0}^{2·π} y(x(\phi)) · {^1}x(\phi) · \mathrm{d} \phi = \int\limits_{0}^{2·π} r_{y} ·\sin \phi · \dfrac{\Lambda}{\lambda \phi} [r_x · \cos \phi] · \lambda \phi$ |
|---|
| $A = r_x·r_{y}·\int\limits_{0}^{2·π} \sin^2 \phi · \lambda \phi = r_x·r_y · \int_{0}^{2·π} \dfrac{1 - \cos 2 ·t}{2} · \lambda t = r_x·r_y·\left[ \dfrac{1}{2} · t - \dfrac{\sin 2 ·t}{4} \right]_{0}^{2·π} = π·r_x·r_y$ |
旋轮线
$\left\lbrace\begin{aligned}
y(t) &= r · (1 - \cos t)
x(t) &= r · (t - \sin t)
\end{aligned}\right.$
$x(y) = r · {‘}\cos \left( 1 - \dfrac{y}{r} \right) - \sqrt{y · (2 · r - y)}$
$\dfrac{\Lambda y(x)}{\lambda x} = \dfrac{1}{\tan \frac{t}{2} }$
$\dfrac{\Lambda^{2} y(x)}{\lambda^{2} x} = \dfrac{-1}{r · (1 - \cos t)^{2} }$
| $\dfrac{\Lambda y(x)}{\lambda x}$ | $\dfrac{\Lambda y(t)}{\lambda t} · \left[ \dfrac{\Lambda x(t)}{\lambda t} \right]^{-1} = \dfrac{r · \sin t}{r · (1 - \cos t)} = \dfrac{2 · \sin \frac{t}{2} · \cos \frac{t}{2} }{2 · \sin^{2} \frac{t}{2} } = \dfrac{1}{\tan \frac{t}{2} }$ |
| $\dfrac{\Lambda^{2} y(x)}{\lambda^{2} x}$ | $\left[ \dfrac{\Lambda^{2} y}{\lambda^{2} t} · \dfrac{\Lambda x}{\lambda t} - \dfrac{\Lambda y}{\lambda t} · \dfrac{\Lambda^{2} x}{\lambda^{2} t} \right] · \left[ \dfrac{\Lambda x}{\lambda t} \right]^{-3} = \dfrac{[r · \cos t] · [r · (1 - \cos t)] - [r · \sin t] · [r · \sin t]}{[r · (1 - \cos t)]^{3} } = \dfrac{-1}{r · (1 - \cos t)^{2} }$ |
旋轮线单拱的面积。
$A = 3·π·r^2$
| $A = \int\limits_{0}^{2·π·r} y(x) · \lambda x = \int\limits_{0}^{2·π} y(t) · {^1}x(t) · \lambda t = \int\limits_{0}^{2·π} r·(1 - \cos t) · r · (1 - \cos t) · \lambda t $ |
|---|
| $A = r^2 · \int\limits_{0}^{2·π} \left( 1 - 2·\cos t + \dfrac{1 + \cos 2 · t}{2} \right) · \lambda t = r^2 · \left[ \dfrac{3}{2} · t - 2 · \sin t + \dfrac{\sin 2 · t}{4} \right]_{0}^{2·π} = 3·π·r^2$ |
旋轮线单拱的长度。
$L = 8 · r$
| $L = \int\limits_{0}^{2·π} \sqrt{[{^1}x(t)]^2 + [^{1}y(t)]^2} · \lambda t = \int\limits_{0}^{2·π} \sqrt{r^2 · (1-\cos t)^2 + r^2 · \sin^2 t} · \lambda t = \int\limits_{0}^{2·π} \sqrt{2·r^2·(1-\cos t)} · \lambda t = 2 · r · \int\limits_{0}^{2·π} \sin \dfrac{t}{2} · \lambda t = -4 · r · \left[ \cos \dfrac{t}{2} \right]_{0}^{2·π} = 8 · r$ |
|---|
螺旋线
$r(\phi) = β^{α · \phi}$
$\tan ν = \dfrac{1}{α · \ln β}$
| $\tan κ$ | $r(\phi) · \left[ \dfrac{\Lambda r(\phi)}{\lambda \phi} \right]^{-1} = β^{α·\phi} · [ β^{α·\phi} · \ln β · α ]^{-1} = \dfrac{1}{α · \ln β}$ |
|---|---|
抛物线
$y^2 = 2 · p · x$
$\tan α = \dfrac{\Lambda y(x)}{\lambda x} = \dfrac{p}{y}$
$\dfrac{\Lambda^2 y(x)}{\lambda^2 x} = -\dfrac{p^2}{y^3}$
| $\dfrac{\Lambda y(x)}{\lambda x}$ | $-\dfrac{\Lambda F(y,x)}{\lambda x} · \left[ \dfrac{\Lambda F(y,x)}{\lambda y} \right]^{-1} = -(-2 · p) · [2 · y]^{-1} = \dfrac{p}{y}$ |
|---|---|
| $\dfrac{\Lambda^2 y(x)}{\lambda^2 x}$ | $-\left[ \dfrac{\Lambda^2 F(y,x)}{\lambda^2 y} · \left[ \dfrac{\Lambda y}{\lambda x} \right]^{2} + \dfrac{\Lambda^2 F(y,x)}{\lambda y · \lambda x} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda^2 F(y,x)}{\lambda x · \lambda y} · \dfrac{\Lambda y}{\lambda x} + \dfrac{\Lambda^2 F(y,x)}{\lambda^2 x} \right] · \left[ \dfrac{\Lambda F(y,x)}{\lambda y} \right]^{-1} = -\left[ 2 · \dfrac{p^2}{y^2} + 0 + 0 + 0 \right] · [2 · y]^{-1} = -\dfrac{p^2}{y^3}$ |
平直线
$\left\lbrace\begin{aligned}
& 0 = +p · y + q · x + r
& \tan α = -\dfrac{q}{p}
& \tan κ = +\dfrac{p}{q}
& 0 = -q · y + p · x + r_{κ}
\end{aligned}\right.
\left\lbrace\begin{aligned}
& y = +k · x + c
& \tan α = +k
& \tan κ = -\dfrac{1}{k}
& y = -\dfrac{1}{k} · x + c_{κ}
\end{aligned}\right.
\left\lbrace\begin{aligned}
& \dfrac{y - y_0}{y_1 - y_0} = +\dfrac{x - x_0}{x_1 - x_0}
& \tan α = +\dfrac{y_1 - y_0}{x_1 - x_0}
& \tan κ = -\dfrac{x_1 - x_0}{y_1 - y_0}
& \dfrac{y - y_{κ} }{x_1 - x_0} = -\dfrac{x - x_{κ} }{y_1 - y_0}
\end{aligned}\right.
\left\lbrace\begin{aligned}
& y - y_0 = +t · (x - x_0)
& \tan α = +t
& \tan κ = -\dfrac{1}{t}
& y - y_{κ} = -\dfrac{1}{t} · (x - x_{κ})
\end{aligned}\right.$
$\left\lbrace\begin{aligned}
& r_{κ} = +q · y_0 - p · x_0
& y_{κ} = \dfrac{-p · r + q · r_{κ} }{p^2 + q^2} = \dfrac{-p·r + q^2·y_0 -p·q·x_0}{p^2 + q^2}
& x_{κ} = \dfrac{-q · r - p · r_{κ} }{p^2 + q^2} = \dfrac{-q·r-p·q·y_0+p^2·x_0}{p^2 + q^2}
& \mathrm{D}[\langle x_0,y_0 \rangle; 0=p·y+q·x+r] = \sqrt{(y_{κ}-y_0)^2+(x_{κ}-x_0)^2} = \dfrac{|p·y_0+q·x_0+r|}{\sqrt{p^2 + q^2} }
& \mathrm{D}[0=p·y+q·x+r_1;0=p·y+q·x+r_2] = \dfrac{|p·y_1+q·x_1+r_2|}{\sqrt{p^2+q^2} } = \dfrac{|r_1-r_2|}{\sqrt{p^2 + q^2} }
\end{aligned}\right.$
双扭线
$(x^2 + y^2)^2 = a^2 · (x^2 - y^2)$
$r^2 = a^2 · \cos (2 · \phi)$
| $r^4 = (x^2+y^2)^2 = a^2 · (x^2 - y^2) = a^2 · (r^2·\cos^2 \phi - r^2 ·\sin^2 \phi) = a^2 · r^2 · \cos (2 · \phi)$ | $⇒$ | $r^2 = a^2 · \cos (2 · \phi)$ |
|---|---|---|
双扭线的面积。
$A = a^2$
| $A = 4 · \dfrac{1}{2} · \int\limits_{0}^{\frac{π}{4} } r^{2} (\phi) · \lambda \phi$ |
|---|
| $A = 2 · \int\limits_{0}^{\frac{π}{4} } a^2 · \cos (2·\phi) · \lambda \phi = a^2 · [\sin (2·\phi)]_{0}^{\frac{π}{4} } = a^2$ |
椭球体
$\dfrac{x^2}{r_x^2} + \dfrac{y^2}{r_y^2} + \dfrac{z^2}{r_z^2} = 1$
$A_x = π·r_y·r_z·\left( 1 - \dfrac{x^2}{r_x^2} \right)$
$V = \dfrac{4}{3}·π·r_x·r_y·r_z$
| $A_x = π·r_y·\sqrt{1 - \dfrac{x^2}{r_x^2} }·r_z·\sqrt{1-\dfrac{x^2}{r_x^2} } = π·r_y·r_z·\left( 1 - \dfrac{x^2}{r_x^2} \right)$ | $⇐$ | $\dfrac{y^2}{r_y^2 · \left( 1 - \dfrac{x^2}{r_x^2} \right)} + \dfrac{z^2}{r_z^2 · \left( 1 - \dfrac{x^2}{r_x^2} \right)} = 1$ |
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| $V = \int\limits_{-r_x}^{+r_x} A_x · \lambda x = \int\limits_{-r_x}^{+r_x} π·r_y·r_z·\left( 1 - \dfrac{x^2}{r_x^2} \right) \lambda x = π·r_y·r_z·\left[ x - \dfrac{x^3}{3·r_x^2} \right]_{-r_x}^{+r_x} = \dfrac{4}{3} ·π·r_x·r_y·r_z$ |
圆球体
$x^2 + y^2 = r^2$
圆球体表面积。
$A=4·π·r^2$
| $A = \int\limits_{x_α}^{x_β} 2 ·π·y(x) ·\sqrt{1 + [{^1}y(x)]^2} · \lambda x$ | $⇐$ | $y(x) = \sqrt{r^2 - x^2}$ |
|---|---|---|
| $A = \int\limits_{-r}^{+r} 2·π·\sqrt{r^2-x^2} ·\sqrt{1+\dfrac{x^2}{r^2 - x^2} } ·\lambda x = \int\limits_{-r}^{+r} 2·π·r · \lambda x = 2·π·r·\left[ x \right]_{-r}^{+r} = 4·π·r^2$ | $⇐$ | ${^1}y(x) = \dfrac{-x}{\sqrt{r^2 - x^2} }$ |
圆锥体
$A_x = π·\dfrac{r^2}{h^2} ·x^2$
$V = \dfrac{1}{3} · π ·r^2 · h$
| $A_x = π·\left( \dfrac{r}{h} ·x \right)^2 = π·\dfrac{r^2}{h^2} ·x^2$ | |
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| $V = \int\limits_{0}^{h} A_x · \lambda x = \int\limits_{0}^{h} π·\dfrac{r^2}{h^2} ·x^2 · \lambda x = π·\dfrac{r^2}{h^2} ·\left. \dfrac{x^3}{3} \right | _{0}^{h} = \dfrac{1}{3} · π·r^2 ·h$ |
圆环体
圆环体截面方程，其中$r_{O}$为圆心到原点的距离，$r_{A}$为圆的半径。
$x^2 + (y-r_O)^2 = r_A^2$
$A_l = π·r_A^2$
$V = 2·π^2·r_{A}^2·r_{O}$
| $V = \int\limits_{0}^{2·π·r_{O} } A_l · \lambda l = \int\limits_{0}^{2·π·r_{O} } π·r_A^2 · \lambda l = 2·π^2·r_A^2·r_O$ |
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悬链线
$y = \dfrac{ә^{x} + ә^{-x} }{2}$
悬链线从$x=0$到$x=x_0$的弧长。
$L = \dfrac{ә^{x_0} - ә^{-x_0} }{2}$
| $L = \int\limits_{0}^{x_0} \sqrt{1 + [{^1}y(x)]^2} · \lambda x = \int\limits_{0}^{x_0} \sqrt{1 + \left[ \dfrac{ә^{x} - ә^{-x} }{2} \right]^2} · \lambda x = \int\limits_{0}^{x_0} \dfrac{ә^{x}+ә^{-x} }{2} · \lambda x = \left[ \dfrac{ә^x - ә^{-x} }{2} \right]_{0}^{x_0} = \dfrac{ә^{x_0} - ә^{-x_0} }{2}$ |
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心形线
$r = a · (1 + \cos \phi)$
心形线的周长。
$L = 8 · a$
| $L = \int\limits_{0}^{2·π} \sqrt{[{^1}r(\phi)]^2 + [r(\phi)]^2} · \lambda \phi = \int\limits_{0}^{2·π} \sqrt{a^2·(-\sin \phi)^2 + a^2 · (1 + \cos \phi)^2} · \lambda \phi$ | ||
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| $L = a · \int\limits_{0}^{2·π} \sqrt{2·(1 + \cos \phi)} · \lambda \phi = 2 ·a · \int\limits_{0}^{2·π} \left | \cos \dfrac{\phi}{2} \right | · \lambda \phi = 2·a·\int\limits_{0}^{π} \cos \dfrac{\phi}{2}·\lambda \phi - 2·a·\int\limits_{π}^{2·π} \cos \dfrac{\phi}{2} · \lambda \phi = 4·a·\left[\sin\dfrac{\phi}{2}\right]0^{π} - 4·a·\left[ \sin \dfrac{\phi}{2} \right]{π}^{2·π} = 8·a$ |
圆柱螺线
$\left\lbrace\begin{aligned}
z(t) &= h· t
y(t) &= r · \sin t
x(t) &= r · \cos t
\end{aligned}\right.$
圆柱螺线单螺距的弧长。
$L = 2·π·\sqrt{r^2+h^2}$
| $L = \int\limits_{0}^{2·π} \sqrt{[{^1}x(t)]^2 + [{^1}y(t)]^2 + [{^1}z(t)]^2} · \lambda t = \int\limits_{0}^{2·π} \sqrt{(-r·\sin t)^2 + (r·\cos t)^2 + h^2} · \lambda t = \sqrt{r^2+h^2} · \int\limits_{0}^{2·π} \lambda t = 2·π·\sqrt{r^2+h^2}$ |
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