Riemann surface
matlab
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质点运动：质点在直线和曲线上的运动，如加速度、速度和位移的计算和分析。
力和运动：力的作用和性质，如牛顿定律的应用、力的合成和分解等。
动量和冲量：动量的定义和性质，冲量的计算和应用，如动量守恒定律和碰撞问题的分析等。
能量和功：能量的转化和守恒，功的计算和应用，如机械能守恒和功率的计算等。
旋转运动：刚体的旋转运动，如角速度、角加速度和力矩的计算和分析。
弹性力学：弹簧和弹性体的力学性质和分析，如胡克定律和弹性势能的计算等。
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Problem 1.1
In terms of the standard basis set ${\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}}, \boldsymbol{a}=2 \boldsymbol{i}-\boldsymbol{j}-2 \boldsymbol{k}, \boldsymbol{b}=3 \boldsymbol{i}-4 \boldsymbol{k}$ and $\boldsymbol{c}=\boldsymbol{i}-5 \boldsymbol{j}+3 \boldsymbol{k}$
(i) Find $3 \boldsymbol{a}+2 \boldsymbol{b}-4 \boldsymbol{c}$ and $|\boldsymbol{a}-\boldsymbol{b}|^2$.
(ii) Find $|\boldsymbol{a}|,|\boldsymbol{b}|$ and $\boldsymbol{a} \cdot \boldsymbol{b}$. Deduce the angle between $\boldsymbol{a}$ and $\boldsymbol{b}$.
(iii) Find the component of $\boldsymbol{c}$ in the direction of $\boldsymbol{a}$ and in the direction of $\boldsymbol{b}$.
(iv) Find $\boldsymbol{a} \times \boldsymbol{b}, \boldsymbol{b} \times \boldsymbol{c}$ and $(\boldsymbol{a} \times \boldsymbol{b}) \times(\boldsymbol{b} \times \boldsymbol{c})$.
(v) Find $\boldsymbol{a} \cdot(\boldsymbol{b} \times \boldsymbol{c})$ and $(\boldsymbol{a} \times \boldsymbol{b}) \cdot \boldsymbol{c}$ and verify that they are equal. Is the set ${\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}}$ right- or left-handed?
(vi) By evaluating each side, verify the identity $\boldsymbol{a} \times(\boldsymbol{b} \times \boldsymbol{c})=(\boldsymbol{a} \cdot \boldsymbol{c}) \boldsymbol{b}-(\boldsymbol{a} \cdot \boldsymbol{b}) \boldsymbol{c}$.
Solution
(i)
$$
\begin{aligned}
& 3 \boldsymbol{a}+2 \boldsymbol{b}-4 \boldsymbol{c}=3(2 \boldsymbol{i}-\boldsymbol{j}-2 \boldsymbol{k})+2(3 \boldsymbol{i}-4 \boldsymbol{k})-4(\boldsymbol{i}-5 \boldsymbol{j}+3 \boldsymbol{k}) \
& =8 i+17 \boldsymbol{j}-26 \boldsymbol{k} \text {. } \
& |\boldsymbol{a}-\boldsymbol{b}|^2=(\boldsymbol{a}-\boldsymbol{b}) \cdot(\boldsymbol{a}-\boldsymbol{b}) \
& =(-\boldsymbol{i}-\boldsymbol{j}+2 \boldsymbol{k}) \cdot(-\boldsymbol{i}-\boldsymbol{j}+2 \boldsymbol{k}) \
& =(-1)^2+(-1)^2+2^2=6 \text {. } \
&
\end{aligned}
$$
(ii)
$$
\begin{aligned}
|\boldsymbol{a}|^2 & =\boldsymbol{a} \cdot \boldsymbol{a} \
& =(2 \boldsymbol{i}-\boldsymbol{j}-2 \boldsymbol{k}) \cdot(2 \boldsymbol{i}-\boldsymbol{j}-2 \boldsymbol{k}) \
& =2^2+(-1)^2+(-2)^2=9 .
\end{aligned}
$$
Hence $|a|=3$.
$$
\begin{aligned}
|\boldsymbol{b}|^2 & =\boldsymbol{b} \cdot \boldsymbol{b} \
& =(3 \boldsymbol{i}-4 \boldsymbol{k}) \cdot(3 \boldsymbol{i}-4 \boldsymbol{k}) \
& =3^2+(-4)^2=25 .
\end{aligned}
$$
Hence $|\boldsymbol{b}|=5$.
$$
\begin{aligned}
\boldsymbol{a} \cdot \boldsymbol{b} & =(2 \boldsymbol{i}-\boldsymbol{j}-2 \boldsymbol{k}) \cdot(3 \boldsymbol{i}-4 \boldsymbol{k}) \
& =(2 \times 3)+((-1) \times 0)+((-2) \times(-4)) \
& =14 .
\end{aligned}
$$
Problem 1.3
$A B C D E F$ is a regular hexagon with centre $O$ which is also the origin of position vectors. Find the position vectors of the vertices $C, D, E, F$ in terms of the position vectors $\boldsymbol{a}, \boldsymbol{b}$ of $A$ and $B$.
FIGURE $1.2 A B C D E F$ is a regular hexagon.
Solution
(i) The position vector $c$ is represented by the line segment $\overrightarrow{O C}$ which has the same magnitude and direction as the line segment $\overrightarrow{A B}$. Hence
$$
\boldsymbol{c}=\boldsymbol{b}-\boldsymbol{a}
$$
(ii) The position vector $d$ is represented by the line segment $\overrightarrow{O D}$ which has the same magnitude as, but opposite direction to, the line segment $\overrightarrow{O A}$. Hence
$$
d=-a
$$
(iii) The position vector $e$ is represented by the line segment $\overrightarrow{O E}$ which has the same magnitude as, but opposite direction to, the line segment $\overrightarrow{O B}$. Hence
$$
e=-b
$$
(iv) The position vector $f$ is represented by the line segment $\overrightarrow{O F}$ which has the
same magnitude as, but opposite direction to, the line segment $\overrightarrow{A B}$. Hence
$$
e=-(b-a)=\boldsymbol{a}-\boldsymbol{b} .
$$
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