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Curl in terms of "fractional" notation

  1. Jun 17, 2015 #1
    See these equations:
    [tex]
    \left | \frac{d \vec{f}}{d \vec{r}} \right | = \left | \frac{d(f_1,f_2)}{d(x,y)} \right | =
    \frac{\partial f_1 \wedge \partial f_2}{\partial x \wedge \partial y}
    =
    \left |
    \begin{bmatrix}
    \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\
    \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \\
    \end{bmatrix}
    \right | = \frac{\partial f_1}{\partial x}\frac{\partial f_2}{\partial y} - \frac{\partial f_1}{\partial y} \frac{\partial f_2}{\partial x}
    [/tex]

    [tex]
    \vec{\nabla} \cdot \vec{f} = \text{tr}\left ( \frac{d \vec{f}}{d \vec{r}} \right )
    =
    \text{tr}\left ( \begin{bmatrix}
    \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\
    \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \\
    \end{bmatrix} \right )
    =
    \frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y}
    [/tex]
    Realize how in the second equation I could to write the divergence in terms of df/dr. Now, what I want do is express the curl through of notation df/dr. Note that although of the first equation recall the expression of curl, is not the curl... Do you can help me with this?
     
  2. jcsd
  3. Jun 17, 2015 #2

    DEvens

    User Avatar
    Education Advisor
    Gold Member

    I am a little vague on your first equation. It appears that you have defined the norm of a matrix to be its determinant. Is that what you intended? Also, the form in the middle of your first equation where you have a wedge product on the top and a wedge product on the bottom is kind of weird.

    Your second equation seems unduly cumbersome. It seems you are calculating (or at least referencing) the off-diagonal elements only to drop them. Why not just use the fact that you have a dot product defined in a perfectly straight forward fashion.

    The curl needs three dimensions. You can use a determinant as a sort of memory aid if you want. The first row in the matrix consists of the unit vectors in the three coordinate directions. The second row consists of the partial derivative operators for x, y, and z. And the third row consists of the components of the vector field you will take the curl of. Then you take the determinant of this matrix.
     
  4. Jun 17, 2015 #3
    I'll rewrite the 1st equation for you understand:
    [tex]
    \text{det} \left ( \frac{d \vec{f}}{d \vec{r}} \right ) = \text{det} \left ( \frac{d(f_1,f_2)}{d(x,y)} \right ) =
    \frac{\partial f_1 \wedge \partial f_2}{\partial x \wedge \partial y}
    =
    \text{det} \left (
    \begin{bmatrix}
    \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\
    \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \\
    \end{bmatrix}
    \right ) = \frac{\partial f_1}{\partial x}\frac{\partial f_2}{\partial y} - \frac{\partial f_1}{\partial y} \frac{\partial f_2}{\partial x}
    [/tex]
     
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