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## 物理代写|理论力学作业代写Theoretical Mechanics代考|Transformation of Variables, Jacobian Determinant

For our considerations so far we have presupposed, directly or at least indirectly, a Cartesian system of coordinates. However, in subsequent applications we shall use, as a rule, those coordinate systems which best fit the underlying problem with respect to its inherent symmetry. That will then not necessarily be the Cartesian coordinates. Therefore we consider in the following the principles for the transition from one set of coordinates to another one.

Let us inspect first, as an introductory example, plane polar coordinates by which the position of a point $P$ in the plane can almost always be defined as conveniently as by Cartesian coordinates $x_1, x_2$. In Fig. $1.74 r$ is the distance between $P$ and the origin of coordinates $\mathcal{O}$ and $\varphi$ is the angle between the straight line $\overline{O P}$ and the 1-axis.
The mapping
$$(r, \varphi) \Longrightarrow\left(x_1, x_2\right)$$
is described by the transformation formulae
\begin{aligned} &x_1=r \cos \varphi=x_1(r, \varphi), \ &x_2=r \sin \varphi=x_2(r, \varphi) \end{aligned}
One speaks of a two-dimensional point transformation which maps the $(r, \varphi)$ plane point by point onto the $\left(x_1, x_2\right)$-plane. We must reasonably require from the new coordinates that they catch each point of the plane. This is here obviously the case. However, it should also be guaranteed that each point $P \cong\left(x_1, x_2\right)$ is uniquely ascribed to a definite $(r, \varphi)$ pair. But here difficulties appear with $\left(x_1=0, x_2=0\right)$ since all pairs $(0, \varphi)$ are mapped on $(0,0)$. The mapping (1.357) is for $r=0$ not uniquely reversible, but otherwise for $r \neq 0$ :
\begin{aligned} r &=\sqrt{x_1^2+x_2^2}, \ \varphi &=\arctan \frac{x_2}{x_1} \end{aligned}
The trigonometric function arc tangent has to be restricted to the branch which delivers the values $0 \leq \varphi<2 \pi$. Hence the transformation (1.357) is almost always reversible.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Coordinate Lines

$\rho$ line: radial ray in the $x_1, x_2$ plane starting from the $z$-axis.
$\varphi$ line: circle in the $x_1, x_2$ plane with its center on the $z$-axis.
$z$ line: straight line parallel to the $x_3$ axis.
We derive the unit vectors:
\begin{aligned} &\frac{\partial \mathbf{r}}{\partial \rho}=(\cos \varphi, \sin \varphi, 0) \Longrightarrow b_\rho=1, \ &\frac{\partial \mathbf{r}}{\partial \varphi}=(-\rho \sin \varphi, \rho \cos \varphi, 0) \Longrightarrow b_{\varphi}=\rho, \ &\frac{\partial \mathbf{r}}{\partial z}=(0,0,1) \Longrightarrow b_z=1 . \end{aligned}
Therewith the unit vectors in cylindrical coordinates are given by:
\begin{aligned} &\mathbf{e}\rho=(\cos \varphi, \sin \varphi, 0) \ &\mathbf{e}{\varphi}=(-\sin \varphi, \cos \varphi, 0) \ &\mathbf{e}z=(0,0,1) \end{aligned} These are curvilinear-orthogonal and are oriented tangentially at the respective coordinate line. For the differential of the position vector according to (1.373) we have with cylindrical coordinates: $$d \mathbf{r}=d \rho \mathbf{e}\rho+\rho d \varphi \mathbf{e}_{\varphi}+d z \mathbf{e}_z$$

# 理论力学代写

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Transformation of Variables, Jacobian Determinant

$$(r, \varphi) \Longrightarrow\left(x_1, x_2\right)$$

$$x_1=r \cos \varphi=x_1(r, \varphi), \quad x_2=r \sin \varphi=x_2(r, \varphi)$$

$$r=\sqrt{x_1^2+x_2^2}, \varphi \quad=\arctan \frac{x_2}{x_1}$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Coordinate Lines

$\rho$ 线: 径向射线在 $x_1, x_2$ 飞机从 $z$-轴。
$\varphi$ 线: 圈在 $x_1, x_2$ 飞机的中心在 $z$-轴。
$z$ line: 平行于的直线 $x_3$ 轴。

$$\frac{\partial \mathbf{r}}{\partial \rho}=(\cos \varphi, \sin \varphi, 0) \Longrightarrow b_\rho=1, \quad \frac{\partial \mathbf{r}}{\partial \varphi}=(-\rho \sin \varphi, \rho \cos \varphi, 0) \Longrightarrow b_{\varphi}=\rho, \frac{\partial \mathbf{r}}{\partial z}=(0,0,1) \Longrightarrow b_z=1 \text {. }$$

$$\mathbf{e} \rho=(\cos \varphi, \sin \varphi, 0) \quad \mathbf{e} \varphi=(-\sin \varphi, \cos \varphi, 0) \mathbf{e} z=(0,0,1)$$

$$d \mathbf{r}=d \rho \mathbf{e} \rho+\rho d \varphi \mathbf{e}_{\varphi}+d z \mathbf{e}_z$$

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