Questions in category: 几何 (Geometry)

## 1. 初中数学中的不变量

Posted by haifeng on 2024-05-31 08:58:06 last update 2024-05-31 08:58:06 | Answers (0) | 收藏

1.  凸多边形的外角和等于 $360^{\circ}$.

## 2. 规范场理论(Gauge Theory)

Posted by haifeng on 2022-03-26 07:53:18 last update 2022-03-26 09:10:33 | Answers (0) | 收藏

D. S. Freed, K. K. Uhlenbeck:  Instantons and four-Manifolds.

H. B. Lawson, Jr.:  The theory of gauge fields in four-dimension.

K. Donaldson & Kronheimer:  Geometry of Four-Manifolds.

K. Kobayashi, K. Nomizu,  Foundations of Differential Geometry.

J. Milnor, J. Stasheff,  Characteristic classes.

N. Steenord, The topology of Fibre Bundles.

Aubin, Nonlinear Problems in Riemann manifolds.

Atiya,  K-theory.

## 3. 儒略日的计算

Posted by haifeng on 2021-06-28 07:18:21 last update 2021-06-28 09:44:42 | Answers (0) | 收藏

[分析]

Y:  Sunrise Time (LST)
Y2=X2-W2*4/1440

Z:  Sunset Time (LST)
Z2=X2+W2*4/1440

X:  Solar Noon (LST)
X2=(720-4*$b$4-V2+$B$5*60)/1440

V:  Equation of time (时差)  单位: minutes

$\begin{split} V2&=4\cdot\Bigl[U2\cdot\sin(2\cdot I2)-2\cdot K2\cdot\sin(J2)+4\cdot K2\cdot U2\cdot\sin(J2)\cdot\cos(2\cdot I2)\\ &\quad -\frac{1}{2}\cdot U2\cdot\sin(4\cdot I2)-\frac{5}{4}\cdot K2\cdot K2\cdot\sin(2\cdot J2)\Bigr] \end{split}$

I.  (Geom Mean Long Sun)  (deg)

I2=MOD(280.46646+G2*(36000.76983+G2*0.0003032),360)

J.  (Geom Mean Anom Sun)  (deg)

J2=357.52911+G2*(35999.05029-0.0001537*G2)

G.  Julian Century

G2=(F2-2451545)/36525

F.  Julian Day(儒略日)

$F2=D2+2415018.5+E2-B5/24$

$B$5 是 Time Zone (+ to E), E指 East.

D, Date,  D2=$B$7,  如: 2010/6/21

Julian Day

For example: The Julian day number for the day starting at 12:00 UT on January 1, 2000 was 2,451,545.

E.  Time (past local midnight). 从午夜开始经过的时间

$E(i+1)=E(i)+0.1/24,\quad E(2)=0.1/24$

240行, 24h/240=0.1h=6min

$\begin{split} F2&=D2+2415018.5+E2-\B\5/24\\ &=2010/6/21+2415018.5+0:06:00-1/24\\ &=2455368.75 \end{split}$

1899年12月29日  --- 2415018

Joseph Scliger 定义儒略周期为 7980年.  $7980=\text{l.c.m.}(19,15,28)$.

28年为一太阳周期(Solar cycle). 经过一太阳周期, 则星期的日序与月的日序会重复.

19年为一太阴周期, 或称默冬章(Metonic cycle), 因235朔望月=19回归年, 经过一太阴周期则阴历月年的日序重复.

15年为一小纪(indiction cycle), 此为罗马皇帝君士坦丁(Constantine)所颁, 每15年评定财产价值以供课税, 成为古罗马用的一个纪元单位.

>> lcm(28,19,15)
in> lcm(28,19,15)
7980

## 儒略日(Julian Date)的计算

\begin{aligned} a&=\bigl[(14-\text{month})/12\bigr]\\ y&=\text{year}+4800-a\\ m&=\text{month}+12a-3 \end{aligned}

$\text{JDN}=\text{day}+\bigl[(153m+2)/5\bigr]+365y+[y/4]-[y/100]+[y/400]-32045,$

$\text{JDN}=\text{day}+\bigl[(153m+2)/5\bigr]+365y+[y/4]-32083.$

$\text{MJD}=\text{JD}-2400000.5$

U
(var y)

R  (Obliq Corr)  (deg)

$R_2=Q_2+0.00256\cdot\cos(125.04-1934.136\cdot G_2)$

Q.  (Mean Obliq Ecliptic)   (deg)

Q2=23+(26+((21.448-G2*(46.815+G2*(0.00059-G2*0.001813))))/60)/60

K.  Eccent Earth Orbit   (eccentricity of earth)   地球的离心率

K2=0.016708634-G2*(0.000042037+0.0000001267*G2)

[待修改]

References:

https://blog.csdn.net/caolaosanahnu/article/details/7890008

Julian Day Number Calculations (numerical.recipes)

## 4. Integer multiplicity rectifiable currents

Posted by haifeng on 2020-09-22 11:04:52 last update 2020-09-22 11:22:27 | Answers (0) | 收藏

$T\in D_n(U)$ 称为一个 rectifiable current, 如果对所有的 $\omega\in D^n(U)$, 有

$T(\omega)=\int_M\langle\omega(x),\xi(x)\rangle\theta(x)\mathrm{d}H^n(x)$

$\theta$ 是局部 $H^n$-可积的正函数  (叫做重数函数).

$\xi:\ M\rightarrow\Lambda_n\mathbb{R}^p$ 是一个 $H^n$-可测函数, 满足下面的关系: 在几乎处处 $H^n$-可测的点 $x\in H$ 处(at $H^n$-a.e. $x\in H$), 有 $\xi(x)=\tau_1\wedge\cdots\wedge\tau_n$, 其中 $\{\tau_1,\ldots,\tau_n\}$ 是 $T_x M$ 的一个标准正交基.

Reference:

## 5. 求下列图形中绿色部分的面积.

Posted by haifeng on 2017-05-08 23:40:16 last update 2017-05-12 09:44:08 | Answers (2) | 收藏

$25\arccos\frac{1}{2\sqrt{2}}-100\arccos\frac{5}{4\sqrt{2}}+\frac{25}{2}\sqrt{7}$

## 6. 投影的计算

Posted by haifeng on 2015-09-13 15:27:40 last update 2015-09-13 15:27:40 | Answers (0) | 收藏

## 7. 球面的面积

Posted by haifeng on 2012-06-04 13:11:50 last update 2012-06-04 13:17:45 | Answers (0) | 收藏

$\text{Vol}(S^{2n-1})=\frac{2\pi^n}{(n-1)!}$

$\text{Vol}(S^{2n})=\frac{2^{n+1}\pi^n}{(2n-1)!!}$

$S^{2n-1}$ 的球坐标为

$\begin{cases}x_1&=\cos\theta_1\\ x_2&=\sin\theta_1\cos\theta_2\\ x_3&=\sin\theta_1\sin\theta_2\cos\theta_3\\ &\vdots\\ x_{2n-1}&=\sin\theta_1\sin\theta_2\cdots\sin\theta_{2n-2}\cos\theta_{2n-1}\\ x_{2n}&=\sin\theta_1\sin\theta_2\cdots\sin\theta_{2n-2}\sin\theta_{2n-1}\end{cases}$

$S^{2n}$ 的球坐标为

$\begin{cases}x_1&=\cos\theta_1\\ x_2&=\sin\theta_1\cos\theta_2\\ x_3&=\sin\theta_1\sin\theta_2\cos\theta_3\\ &\vdots\\ x_{2n}&=\sin\theta_1\sin\theta_2\cdots\sin\theta_{2n-1}\cos\theta_{2n}\\ x_{2n+1}&=\sin\theta_1\sin\theta_2\cdots\sin\theta_{2n-1}\sin\theta_{2n}\end{cases}$

## 8. 将教材《数据库系统基础教程》每章后面的参考文献整理, 输入数据库. test

Posted by huichun on 2011-04-02 19:39:24 last update 2012-09-01 19:32:44 | Answers (1) | 收藏