Rottor of a vector in a simple way

[slonopotam]

$$\vec{F}=(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}})$$
$$\vec{r}=(x,y,z)$$

$$|r|=\sqrt{x^2+y^2+z^2}$$

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

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

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

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

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

 

[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

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

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.