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Rottor of a vector in a simple way

  1. Aug 13, 2009 #1
    [tex]\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}})[/tex]
    [tex]
    \vec{r}=(x,y,z)
    [/tex]

    [tex]
    |r|=\sqrt{x^2+y^2+z^2}[/tex]

    [tex]\vec{F}=(\frac{x}{|r|},\frac{y}{|r|},\frac{z}{|r|})[/tex]

    so its [tex]F=\frac{r}{|r|}[/tex]

    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?
     
  2. jcsd
  3. Aug 13, 2009 #2

    kuruman

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    You have a radial vector. Look up "curl in spherical coordinates" and apply it.
     
  4. Aug 13, 2009 #3

    gabbagabbahey

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

    [tex]\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}[/tex]

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

    [tex]\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}[/tex]
     
  5. Aug 13, 2009 #4
    wow i how you came up with this kind of determinant
    i am use the the first kind
     
  6. Aug 13, 2009 #5

    gabbagabbahey

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

    [tex]\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}[/tex]

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

    [tex]h_u\equiv \left| \frac{\partial \textbf{r}}{\partial u} \right|[/tex]

    [tex]h_v\equiv \left| \frac{\partial \textbf{r}}{\partial v} \right|[/tex]

    and

    [tex]h_w\equiv \left| \frac{\partial \textbf{r}}{\partial w} \right|[/tex]

    This is derived in most multivariable calculus textbooks.
     
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