### 物理代写|理论力学代写theoretical mechanics代考|PHYC20014

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|理论力学代写theoretical mechanics代考|Space and Time

Space and time are two basic concepts which, according to Kant, inherently or innately determine the form of all experience in an a priori manner, thereby making possible experience as such: only in space and time can we arrange our sensations. [According to the doctrines of evolutionary cognition, what is innate to us has developed phylogenetically by adaption to our environment. This is why we only notice the insufficiency of these “self-evident”‘ concepts under extraordinary circumstances, e.g., for velocities close to that of light $\left(c_{0}\right)$ or actions of the order of Planck’s quantum $h$. We shall tackle such “weird” cases later-in electromagnetism and quantum mechanics. For the time being, we want to make sure we can handle our familiar environment.]

To do this, we introduce a continuous parameter $t$. Like every other physical quantity it is composed of number and unit (for example, a second $1 \mathrm{~s}=1 \mathrm{~min} / 60$ $-1 \mathrm{~h} / 3600$ ). The larger the unit, the smaller the number. Physical quantities do not depend on the unit-likewise equations between physical quantities. Nevertheless, the opposite is sometimes seen, as in: “We choose units such that the velocity of light $c$ assumes the value 1”. In fact, the concept of velocity is thereby changed, so that instead of the velocity $v$, the ratio $v / c$ is taken here as the velocity, and $c t$ as time or $x / c$ as length.

The zero time $(t=0)$ can be chosen arbitrarily, since basically only the time difference, i.e., the duration of a process, is important. A differentiation with respect to time $(\mathrm{d} / \mathrm{d} t)$ is often marked by a dot over the differentiated quantity, i.e., $\mathrm{d} x / \mathrm{d} t \equiv \dot{x}$.
In empty space every direction is equivalent. Here, too, we may choose the zero point freely and, starting from this point, determine the position of other points in a coordinate-free notation by the position vector $\mathbf{r}$, which fixes the distance and direction of the point under consideration. This coordinate-free type of notation is particularly advantageous when we want to exploit the assumed homogeneity of space. However, conditions often arise (i.e., when there is axial or spherical symmetry) which are best taken care of in special coordinates. We are free to choose a coordinate system. We only require that it determine all positions uniquely. This we shall treat in the next section.

## 物理代写|理论力学代写theoretical mechanics代考|Vector Algebra

From two vectors $\mathbf{a}$ and $\mathbf{b}$, their sum $\mathbf{a}+\mathbf{b}$ may be formed according to the construction of parallelograms (as the diagonal), as shown in Fig. 1.1. From this follows the commutative and associative law of vector addition:
$$\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}, \quad(\mathbf{a}+\mathbf{b})+\mathbf{c}=\mathbf{a}+(\mathbf{b}+\mathbf{c})$$
The product of the vectors a with a scalar (i.e., directionless) factor $\alpha$ is understood as the vector $\alpha \mathbf{a}=\mathbf{a} \alpha$ with the same (for $\alpha<0$ opposite) direction and with value $|\alpha| a$. In particular, a and $-\mathbf{a}$ have the same value, but opposite directions. For $\alpha=0$ the zero vector 0 results, with length 0 and undetermined direction.

The scalar product (inner product) $\mathbf{a} \cdot \mathbf{b}$ of the two vectors $\mathbf{a}$ and $\mathbf{b}$ is the product of their values times the cosine of the enclosed angle $\phi_{a b}$ (see Fig. 1.2 left):
$$\mathbf{a} \cdot \mathbf{b} \equiv a b \cos \phi_{a b}$$
The dot between the two factors is important for the scalar product-if it is missing, then it is the tensor product of the two vectors, which will be explained in Sect. 1.2.4 with $\mathbf{a} \cdot \mathbf{b} \mathbf{c} \neq \mathbf{a} \mathbf{b} \cdot \mathbf{c}$, if $\mathbf{a}$ and $\mathbf{c}$ have different directions, i.e., if $\mathbf{a}$ is not a multiple of $\mathbf{c}$. Consequently, one has
$$\mathbf{a} \cdot \mathbf{b}=\mathbf{b} \cdot \mathbf{a}$$
and
$$\mathbf{a} \cdot \mathbf{b}=0 \quad \Longleftrightarrow \quad \mathbf{a} \perp \mathbf{b} \text { or } a=0 \text { or } b=0 .$$
If the two vectors are oriented perpendicularly to each other $(\mathbf{a} \perp \mathbf{b})$, then they are also said to be orthogonal. Obviously, $\mathbf{a} \cdot \mathbf{a}=a^{2}$ holds. Vectors with value 1 are called unit vectors. Here they are denoted by e. Given three Cartesian, i.e., pairwise perpendicular unit vectors $\mathbf{e}{x}, \mathbf{e}{y}, \mathbf{e}{z}$, all vectors can be decomposed in terms of these: $$\mathbf{a}=\mathbf{e}{x} a_{x}+\mathbf{e}{y} a{y}+\mathbf{e}{z} a{z},$$
with the Cartesian components
$$a_{x} \equiv \mathbf{e}{x} \cdot \mathbf{a}, \quad a{y} \equiv \mathbf{e}{y} \cdot \mathbf{a}, \quad a{z} \equiv \mathbf{e}_{z} \cdot \mathbf{a} .$$

## 物理代写|理论力学代写theoretical mechanics代考|Trajectories

If a vector depends upon a parameter, then we speak of a vector function. The vector function $\mathbf{a}(t)$ is continuous at $t_{0}$, if it tends to $\mathbf{a}\left(t_{0}\right)$ for $t \rightarrow t_{0}$. With the same limit $t \rightarrow t_{0}$, the vector differential da and the first derivative da/d $t$ is introduced. These quantities may be formed for every Cartesian component, and we have
$$\begin{array}{ll} \mathrm{d}(\mathbf{a}+\mathbf{b})=\mathrm{d} \mathbf{a}+\mathrm{d} \mathbf{b}, & \mathrm{d}(\alpha \mathbf{a})=\alpha \mathrm{d} \mathbf{a}+\mathbf{a} \mathrm{d} \alpha \ \mathrm{d}(\mathbf{a} \cdot \mathbf{b})=\mathbf{a} \cdot \mathrm{d} \mathbf{b}+\mathbf{b} \cdot \mathrm{d} \mathbf{a}, & \mathrm{d}(\mathbf{a} \times \mathbf{b})=\mathbf{a} \times \mathrm{d} \mathbf{b}-\mathbf{b} \times \mathrm{d} \mathbf{a} \end{array}$$
Obviously, $\mathbf{a} \cdot \mathrm{d} \mathbf{a} / \mathrm{d} t=\frac{1}{2} \mathrm{~d}(\mathbf{a} \cdot \mathbf{a}) / \mathrm{d} t=\frac{1}{2} \mathrm{~d} a^{2} / \mathrm{d} t=a \mathrm{~d} a / \mathrm{d} t$ holds. In particular the derivative of a unit vector is always perpendicular to the original vector-if it does nôt vañish.

As an example of a vector function, we investigate $\mathbf{r}(t)$, the path of a point as a function of the time $t$. Thus we want to consider also the velocity $\mathbf{v}=\dot{\mathbf{r}}$ and the acceleration $\mathbf{a}=\ddot{\mathbf{r}}$ rather generally. The time is not important for the trajectories as geometrical lines. Therefore, instead of the time $t$ we introduce the path length $s$ as a parameter and exploit $\mathrm{d} s=|\mathrm{d} \mathbf{r}|=v \mathrm{~d} t$.

We now take three mutually perpendicular unit vectors $\mathbf{e}{\mathrm{T}}, \mathbf{e}{\mathrm{N}}$, and $\mathbf{e}{\mathrm{B}}$, which are attached to every point on the trajectory. Here $\mathbf{e}{\mathrm{T}}$ has the direction of $\mathbf{v}$ :
tangent vector $\quad \mathbf{e}{\mathrm{T}} \equiv \frac{\mathrm{d} \mathbf{r}}{\mathrm{d} s}=\frac{\mathbf{v}}{v}$ For a straight path, this vector is already sufficient for the description. But in general the path curvature $\quad \kappa \equiv\left|\frac{\mathrm{d} \mathbf{e}{\mathrm{T}}}{\mathrm{d} s}\right|=\left|\frac{\mathrm{d}^{2} \mathbf{r}}{\mathrm{d} s^{2}}\right|$
is different from zero. In order to get more insight into this parameter we consider a plane curve of constant curvature, namely, the circle with $s=R \varphi$. For $\mathbf{r}(\varphi)=\mathbf{r}{0}+$ $R\left(\cos \varphi \mathbf{e}{x}+\sin \varphi \mathbf{e}{y}\right)$, we have $\kappa=\left|\mathrm{d}^{2} \mathbf{r} / \mathrm{d}(R \varphi)^{2}\right|=R^{-1}$. Instead of the curvature $\kappa$, its reciprocal, the curvature radius $R \equiv \frac{1}{\kappa}$, can also be used to determine the curve. Hence as a further unit vector we have the normal vector $\quad \mathbf{e}{\mathrm{N}} \equiv R \frac{\mathbf{d} \mathbf{e}_{\mathrm{T}}}{\mathrm{d} s}=R \frac{\mathrm{d}^{2} \mathbf{r}}{\mathrm{d} s^{2}}$

## 物理代写|理论力学代写theoretical mechanics代考|Vector Algebra

$$\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}, \quad(\mathbf{a}+\mathbf{b})+\mathbf{c}=\mathbf{a}+(\mathbf{b}+\mathbf{c})$$

$$\mathbf{a} \cdot \mathbf{b} \equiv a b \cos \phi_{a b}$$

$$\mathbf{a} \cdot \mathbf{b}=\mathbf{b} \cdot \mathbf{a}$$

$$\mathbf{a} \cdot \mathbf{b}=0 \quad \Longleftrightarrow \quad \mathbf{a} \perp \mathbf{b} \text { or } a=0 \text { or } b=0$$

$$\mathbf{a}=\mathbf{e} x a_{x}+\mathbf{e} y a y+\mathbf{e} z a z$$

$$a_{x} \equiv \mathbf{e} x \cdot \mathbf{a}, \quad a y \equiv \mathbf{e} y \cdot \mathbf{a}, \quad a z \equiv \mathbf{e}_{z} \cdot \mathbf{a} .$$

## 物理代写|理论力学代写theoretical mechanics代考|Trajectories

$$\mathrm{d}(\mathbf{a}+\mathbf{b})=\mathrm{d} \mathbf{a}+\mathrm{d} \mathbf{b}, \quad \mathrm{d}(\alpha \mathbf{a})=\alpha \mathrm{d} \mathbf{a}+\mathbf{a d} \alpha \mathrm{d}(\mathbf{a} \cdot \mathbf{b})=\mathbf{a} \cdot \mathrm{d} \mathbf{b}+\mathbf{b} \cdot \mathrm{d} \mathbf{a}, \quad \mathrm{d}(\mathbf{a} \times \mathbf{b})=\mathbf{a} \times \mathrm{d} \mathbf{b}-\mathbf{b} \times \mathrm{d}$$

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