Rottor of a vector in a simple way

[slonopotam]

\vec{F}=(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}})
<br /> \vec{r}=(x,y,z)<br />

<br /> |r|=\sqrt{x^2+y^2+z^2}

\vec{F}=(\frac{x}{|r|},\frac{y}{|r|},\frac{z}{|r|})

so its F=\frac{r}{|r|}

i need to prove that F is a conservative field
where (x,y,z) differs (0,0,0)
so i need to show that rot f is 0
but for rottor i need a determinant
is there a way to do a rot on simpler way?

 

[kuruman]

You have a radial vector. Look up "curl in spherical coordinates" and apply it.

 

[gabbagabbahey]

The determinant method is usually the simplest way. In Cartesian coordinates, it's

\text{rot}\textbf{F}=\begin{vmatrix}\hat{x} &amp; \hat{y} &amp; \hat{z} \\ \partial_x &amp; \partial_y &amp; \partial_z \\ F_x &amp; F_y &amp; F_z \end{vmatrix}=\begin{vmatrix}\hat{x} &amp; \hat{y} &amp; \hat{z} \\ \partial_x &amp; \partial_y &amp; \partial_z \\ \frac{x}{\sqrt{x^2+y^2+z^2}} &amp; \frac{y}{\sqrt{x^2+y^2+z^2}} &amp; \frac{z}{\sqrt{x^2+y^2+z^2}} \end{vmatrix}

But since \textbf{F}=\frac{\textbf{r}}{r}=\textbf{e}_r, it is probably best to use spherical coordinates:

\text{rot}\textbf{F}=\begin{vmatrix}\textbf{e}_{r} &amp; r\textbf{e}_{\theta} &amp; r\sin\theta\textbf{e}_{\phi} \\ \partial_r &amp; \partial_{\theta} &amp; \partial_{\phi} \\ F_r &amp; r F_{\theta} &amp; r\sin\theta F_{\phi} \end{vmatrix}=\begin{vmatrix}\textbf{e}_{r} &amp; r\textbf{e}_{\theta} &amp; r\sin\theta\textbf{e}_{\phi} \\ \partial_r &amp; \partial_{\theta} &amp; \partial_{\phi} \\ 1 &amp; 0 &amp; 0 \end{vmatrix}

 

[slonopotam]

wow i how you came up with this kind of determinant
i am use the the first kind

 

[gabbagabbahey]

In any 3D Curvilinear coordinate system (u,v,w), the rotation (or 'curl') is given by

\text{rot}\textbf{F}=\begin{vmatrix}h_u\textbf{e}_{u} &amp; h_v\textbf{e}_{v} &amp; h_w\textbf{e}_{w} \\ \partial_u &amp; \partial_v &amp; \partial_w \\ h_u F_u &amp; h_v F_{v} &amp; h_w F_w \end{vmatrix}

where \textbf{e}_u, \textbf{e}_v, and \textbf{e}_w are unit vectors that point in the direction of increasing u, v and w respectively, and h_u, h_v, and h_w are scale factors given by

h_u\equiv \left| \frac{\partial \textbf{r}}{\partial u} \right|

h_v\equiv \left| \frac{\partial \textbf{r}}{\partial v} \right|

and

h_w\equiv \left| \frac{\partial \textbf{r}}{\partial w} \right|

This is derived in most multivariable calculus textbooks.