### 物理代写|理论力学作业代写Theoretical Mechanics代考|Vector-Valued Functions

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## 物理代写|理论力学作业代写Theoretical Mechanics代考|Parametrization of Space Curves

In physics space curves are typical examples of vector-valued functions. To start with we choose in the $E_{3}$ an arbitrary but fixed origin of coordinates $\mathcal{O}$. Then the momentary position $P$ of a ‘particle’ is determined by the position vector $\mathbf{r}=\overrightarrow{0 P}$ (Fig. 1.57). By a ‘particle’ we understand a physical body of mass $m$ but with negligible extension in all directions. Later we will introduce for it the term ‘mass point’. In course of time the particle will in general change its position, i.e. $\mathbf{r}$ will change direction and magnitude. In a time-independent, complete orthonormal system (CONS) $\mathbf{e}_{i}$ the components of the position vector become normal time-

dependent functions:
$$\mathbf{r}(t)=\sum_{j=1}^{3} x_{j}(t) \mathbf{e}{j} \equiv\left(x{1}(t), x_{2}(t), x_{3}(t)\right)$$
This is called the trajectory or the path line of the particle.
The set of space points the particle passes through over the time define the socalled
$$\text { space curve : }=\left{\mathbf{r}(t), t_{a} \leq t \leq t_{e}\right} \text {. }$$
One calls (1.202) a parametrization of the space curve (1.203). The independent parameter in this case is the time $t$. Of course there also exist other possibilities of parametrization as we will see later in this section. Furthermore, it is clear that different path lines may parametrize the same space curve. For example this is already true when one and the same space curve is run through in opposite directions or in different time intervals.
Examples

1. Circular motion in the $x z$-plane
Let the circle have the radius $R$ and let its center point be at the origin of coordinates (Fig. 1.58). Then a self-evident parametrization is via the angle $\varphi$ :
\begin{aligned} &M={\varphi ; \quad 0 \leq \varphi \leq 2 \pi}, \ &\mathbf{r}(\varphi)=R(\cos \varphi, 0, \sin \varphi) \end{aligned}
Another parametrization can use, e.g., the $x$ component $x_{1}$ :
\begin{aligned} &M=\left{x_{1} ; \quad-R \leq x_{1} \leq+R\right}, \ &\mathbf{r}\left(x_{1}\right)=\left(x_{1}, 0, \pm \sqrt{R^{2}-x_{1}^{2}}\right) \end{aligned}
where the plus sign holds for the upper, the minus sign for the lower half-plane.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Differentiation of Vector-Valued Functions

We consider a vector-valued function $\mathbf{a}(t)$ and look into the differential changes of the vector, i.e. the changes due to very small changes in time. Practically such a time interval, being determined by the measuring process, is of course always finite. Mathematically, however, an infinitely small time interval shall be considered. Furthermore, instead of time $t$ any other parameter can also be used in the following formulae. The vector-valued function $\mathbf{a}(t)$ in general has at different times (parameters) $t$ and $t+\Delta t$ different magnitudes and/or different directions. The magnitude of the difference vector
$$\Delta \mathbf{a}=\mathbf{a}(t+\Delta t)-\mathbf{a}(t)$$
will become smaller with decreasing time difference $\Delta t$, whereby its direction can change continuously in order to arrive for very small $\Delta t$ in the corresponding direction of the respective tangent.
Definition 1.4.2 Derivation of a Vector-Valued Function
$$\frac{d \mathbf{a}}{d t}=\lim _{\Delta t \rightarrow 0} \frac{\mathbf{a}(t+\Delta t)-\mathbf{a}(t)}{\Delta t} .$$
This definition clearly presumes that such a limiting vector does exist at all (Fig. 1.60). For time-derivatives sometimes one writes briefly:
$$\dot{\mathbf{a}}(t) \equiv \frac{d \mathbf{a}}{d t}$$

We represent $\mathbf{a}(t)$ in a time-independent basis system $\left{\mathbf{e}{i}\right}$ : $$\mathbf{a}(t)=\sum{j} a_{j}(t) \mathbf{e}{j} .$$ Then it holds: $$\mathbf{a}(t+\Delta t)-\mathbf{a}(t)=\sum{j}\left[a_{j}(t+\Delta t)-a_{j}(t)\right] \mathbf{e}{j} .$$ Therewith the differentiation of a vector-valued function is obviously and completely expressed in terms of derivatives of the time-dependent component functions: $$\dot{\mathbf{a}}(t)=\frac{d \mathbf{a}}{d i}=\sum{j} \dot{a}{j}(t) \mathbf{e}{j}$$
Correspondingly it holds also for all higher derivatives:
$$\frac{d^{n}}{d t^{n}} \mathbf{a}(t)=\sum_{j}\left(\frac{d^{n}}{d t^{n}} a_{j}(t)\right) \mathbf{e}_{j} ; \quad n=0,1,2, \ldots$$
Then it is not difficult to prove the following rules of differentiation
1) $\frac{d}{d t}[\mathbf{a}(t)+\mathbf{b}(t)]=\dot{\mathbf{a}}(t)+\dot{\mathbf{b}}(t)$,
2) $\frac{d}{d t}[f(t) \mathbf{a}(t)]=\dot{f}(t) \mathbf{a}(t)+f(t) \dot{\mathbf{a}}(t)$,
if $f(t)$ is a differentiable, scalar function,
3) $\frac{d}{d t}[\mathbf{a}(t) \cdot \mathbf{b}(t)]=\dot{\mathbf{a}}(t) \cdot \mathbf{b}(t)+\mathbf{a}(t) \cdot \dot{\mathbf{b}}(t)$,
4) $\frac{d}{d t}[\mathbf{a}(t) \times \mathbf{b}(t)]=\dot{\mathbf{a}}(t) \times \mathbf{b}(t)+\mathbf{a}(t) \times \dot{\mathbf{b}}(t)$.
In 4) we have to be very careful about the correct order of the factors.
Examples
a) velocity: $\mathbf{v}(t)=\dot{\mathbf{r}}(t)$
(always tangential to the path line),
acceleration: $\mathbf{a}(t)=\dot{\mathbf{v}}(t)=\ddot{\mathbf{r}}(t)$.
(1.215)

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Arc Length

The integration of vector-valued functions can also be transferred to the corresponding integration of parameter-dependent component functions:
$$\int_{t_{a}}^{t_{e}} \mathbf{a}(t) d t=\sum_{j=1}^{3} \mathbf{e}{j} \int{t_{a}}^{t_{e}} a_{j}(t) d t$$
If the basis vectors are parameter-independent they can be drawn in front of the integral Thus in cuch a case nne integrates the vertor hy integrating its components in the ordinary manner. However, is should be expressly indicaled that the so defined integral of course depends on the special choice of the parameters and therefore does not at all represent a genuine curve property. During the course of this book we will meet other integrals of totally different type. However, at this stage we will make do with (1.217).

From now on, temporarily, we want to concentrate ourselves exclusively on space curves and path lines as examples of vector-valued functions. Thereby we assume for the following that the curve under consideration is ‘smooth’.

Definition 1.4.3 A space curve is denoted as smooth, if there exists at least one continuously differentiable parametrization $\mathbf{r}=\mathbf{r}(t)$ for which at no point we have:
$$\frac{d \mathbf{r}}{d t}=0$$
For such smooth space curves it often appears convenient to use the so-called arc length $s$ as curve parameter.

Definition 1.4.4 The arc length $s$ is the length of the space curve, measured along the curved line starting from an arbitrarily chosen initial point.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Parametrization of Space Curves

r(吨)=∑j=13Xj(吨)和j≡(X1(吨),X2(吨),X3(吨))

\text { 空间曲线 : }=\left{\mathbf{r}(t), t_{a} \leq t \leq t_{e}\right} \text {. }\text { 空间曲线 : }=\left{\mathbf{r}(t), t_{a} \leq t \leq t_{e}\right} \text {. }

1. 在圆周运动X和-plane
让圆有半径R并使其中心点位于坐标原点（图 1.58）。然后一个不言而喻的参数化是通过角度披 :
米=披;0≤披≤2圆周率, r(披)=R(因⁡披,0,罪⁡披)
另一个参数化可以使用，例如，X零件X1:
\begin{aligned} &M=\left{x_{1} ; \quad-R \leq x_{1} \leq+R\right}, \ &\mathbf{r}\left(x_{1}\right)=\left(x_{1}, 0, \pm \sqrt {R^{2}-x_{1}^{2}}\right) \end{对齐}\begin{aligned} &M=\left{x_{1} ; \quad-R \leq x_{1} \leq+R\right}, \ &\mathbf{r}\left(x_{1}\right)=\left(x_{1}, 0, \pm \sqrt {R^{2}-x_{1}^{2}}\right) \end{对齐}
其中加号适用于上半平面，减号适用于下半平面。

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Differentiation of Vector-Valued Functions

Δ一种=一种(吨+Δ吨)−一种(吨)

d一种d吨=林Δ吨→0一种(吨+Δ吨)−一种(吨)Δ吨.

dnd吨n一种(吨)=∑j(dnd吨n一种j(吨))和j;n=0,1,2,…

1)dd吨[一种(吨)+b(吨)]=一种˙(吨)+b˙(吨),
2) dd吨[F(吨)一种(吨)]=F˙(吨)一种(吨)+F(吨)一种˙(吨),

3)dd吨[一种(吨)⋅b(吨)]=一种˙(吨)⋅b(吨)+一种(吨)⋅b˙(吨),
4) dd吨[一种(吨)×b(吨)]=一种˙(吨)×b(吨)+一种(吨)×b˙(吨).
4）我们必须非常小心因素的正确顺序。

a) 速度：在(吨)=r˙(吨)
（总是与路径线相切），

(1.215)

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Arc Length

∫吨一种吨和一种(吨)d吨=∑j=13和j∫吨一种吨和一种j(吨)d吨

drd吨=0

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