- #1
lennyleonard
- 23
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Hi everyone
I have a little problem in understanding the trasformation of vectors component when passing to a different coordinate system (abbreviated CS).
Theory says that the components of a vector in the first CS [tex]x[/tex] with component [tex](V^0,V^1,...,V^n)[/tex] will transform changing CS according to [tex]V^{\alpha'}=\Lambda^{\alpha'}_{\beta}B^{\beta}[/tex] where the primed (') letters refer to the new CS [tex]x'[/tex] and the Einsten notation is employed.
The matrix [tex]\Lambda[/tex] is the jacobian of the application [tex]x'(x)[/tex].
The problem I have is better expressed by an example.
Les's take the coordinate trasformation from cartesian to polar. We have:
[tex]x=rcos\theta\; ,\;y=rsin\theta[/tex] and [tex]r=\sqrt{x^2+y^2}\;,\;\theta=arctg\frac{y}{x}[/tex]
The jacobian we need is:
[tex]\left(\begin{array}{cc}\frac{\partial r}{\partial x} & \frac{\partial r}{\partial y}\\\frac{\partial \theta}{\partial x}&\frac{\partial \theta}{\partial y}\end{array}\right)[/tex]
and so:
[tex] \Lambda = \left(\begin{array}{cc}cos\theta & sin\theta\\ -\frac{1}{r}sin\theta & \frac{1}{r}cos\theta\end{array}\right)[/tex]
At this point I understand that if I have the vector [tex]\vec{V}=X\vec{e}_x,Y\vec{e}_y[/tex] and I want to know its components on the basis [tex]\vec{e}_r\vec{e}_{\theta}[/tex] all I have to do is calculate [tex]V^{r}=\Lambda^{r}_{\beta}B^{\beta}[/tex] and [tex]V^{\theta}=\Lambda^{\theta}_{\beta}B^{\beta}[/tex] where [tex]\beta=X,Y[/tex].
So I end up with: [tex]V^r=Xcos\theta+Ysin\theta[/tex] and [tex]V^{\theta}=-\frac{X}{r}sin\theta+\frac{Y}{r}cos\theta[/tex]; but i know that it should also be [tex]\vec{V}=r\vec{e}_r+\theta\vec{e}_{\theta}[/tex] and so: [tex]V^r=r[/tex] and [tex]V^{\theta}=\theta[/tex] but this is not true! In particular the [tex]V^{\theta}[/tex] I obtained above gets identically zero.
where is my mistake?
Tanks a lot for your answers!
I have a little problem in understanding the trasformation of vectors component when passing to a different coordinate system (abbreviated CS).
Theory says that the components of a vector in the first CS [tex]x[/tex] with component [tex](V^0,V^1,...,V^n)[/tex] will transform changing CS according to [tex]V^{\alpha'}=\Lambda^{\alpha'}_{\beta}B^{\beta}[/tex] where the primed (') letters refer to the new CS [tex]x'[/tex] and the Einsten notation is employed.
The matrix [tex]\Lambda[/tex] is the jacobian of the application [tex]x'(x)[/tex].
The problem I have is better expressed by an example.
Les's take the coordinate trasformation from cartesian to polar. We have:
[tex]x=rcos\theta\; ,\;y=rsin\theta[/tex] and [tex]r=\sqrt{x^2+y^2}\;,\;\theta=arctg\frac{y}{x}[/tex]
The jacobian we need is:
[tex]\left(\begin{array}{cc}\frac{\partial r}{\partial x} & \frac{\partial r}{\partial y}\\\frac{\partial \theta}{\partial x}&\frac{\partial \theta}{\partial y}\end{array}\right)[/tex]
and so:
[tex] \Lambda = \left(\begin{array}{cc}cos\theta & sin\theta\\ -\frac{1}{r}sin\theta & \frac{1}{r}cos\theta\end{array}\right)[/tex]
At this point I understand that if I have the vector [tex]\vec{V}=X\vec{e}_x,Y\vec{e}_y[/tex] and I want to know its components on the basis [tex]\vec{e}_r\vec{e}_{\theta}[/tex] all I have to do is calculate [tex]V^{r}=\Lambda^{r}_{\beta}B^{\beta}[/tex] and [tex]V^{\theta}=\Lambda^{\theta}_{\beta}B^{\beta}[/tex] where [tex]\beta=X,Y[/tex].
So I end up with: [tex]V^r=Xcos\theta+Ysin\theta[/tex] and [tex]V^{\theta}=-\frac{X}{r}sin\theta+\frac{Y}{r}cos\theta[/tex]; but i know that it should also be [tex]\vec{V}=r\vec{e}_r+\theta\vec{e}_{\theta}[/tex] and so: [tex]V^r=r[/tex] and [tex]V^{\theta}=\theta[/tex] but this is not true! In particular the [tex]V^{\theta}[/tex] I obtained above gets identically zero.
where is my mistake?
Tanks a lot for your answers!
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