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Math Help - Curl of a vector field: good remembering technique

  1. #1
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    Curl of a vector field: good remembering technique

    We all know that  \text{curl} \ \bold{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \bold{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \bold{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \bold{k} .

    Here is a good technique to remember this:

    Write down  P, Q, R .

    Put your left index finger on  R . Now put your left middle finger on  Q . Then go the opposite direction (like playing the piano). This is the first term. Do the same for the other terms. Keep playing this (as if you were playing the piano).

    This is how I remember  \text{curl} \ \bold{F} .
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by shilz222 View Post
    We all know that  \text{curl} \ \bold{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \bold{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \bold{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \bold{k} .

    Here is a good technique to remember this:

    Write down  P, Q, R .

    Put your left index finger on  R . Now put your left middle finger on  Q . Then go the opposite direction (like playing the piano). This is the first term. Do the same for the other terms. Keep playing this (as if you were playing the piano).

    This is how I remember  \text{curl} \ \bold{F} .
    i don't think i get your technique...

    like almost everyone else, i remember curlF by \mbox{curl} \bold{F} = \nabla \times \bold{F}

    where \bold{F} = \left< P, Q, R \right> and \nabla = \left< \frac {\partial}{\partial x}, \frac {\partial}{\partial y}, \frac {\partial}{\partial z} \right>
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  3. #3
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    Its like playing the piano. Let your index finger, your middle finger, and your crown finger (left hand) represent  R,Q,P respectively. Tap your index finger, then tap your middle finger. Then do the opposite. Now tap your crown finger, then tap your index finger. Now do the opposite. Now tap your middle finger, then tap your crown finger. Now do the opposite. Keep "practicing" this.

    So we have: P Q R
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  4. #4
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    I practiced it on the piano. I called it the "curl" piece. Choose 3 piano keys, and let them represent  P, Q, R .
    Last edited by shilz222; November 30th 2007 at 02:07 AM.
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    Actually this is also a good method of remembering anything involving cross products (that way you do not have to do the calculation).
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    Why not just use determinants? If you start playing the paino during an exam it will look a little strange.
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    Forum Admin topsquark's Avatar
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    I've always found that proctoring an Intro Physics test just after mid-semester is rather fun. The students have hit the cross-product by then and during the exam they're pulling their tendons and writhing around in their seats trying to apply the right hand rule to figure out the direction.

    (Okay, so what if I'm a sadist? )

    -Dan
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    \begin{array}{l}<br />
 \nabla  = \left\langle {\partial x,\partial y,\partial z} \right\rangle  \\ <br />
 curl(F) = \nabla  \times F = \left| {\begin{array}{*{20}c}<br />
   i & j & k  \\<br />
   {\partial x} & {\partial y} & {\partial z}  \\<br />
   P & Q & R  \\<br />
\end{array}} \right| \\ <br />
 \end{array}
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  9. #9
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    Yes I know thats the conventional way of doing it. TPH, I don't mean actually playing the piano. I mean tapping on the desk. Your speed of calculating the curl will improve a lot (at least it did for me).
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by shilz222 View Post
    Yes I know thats the conventional way of doing it. TPH, I don't mean actually playing the piano. I mean tapping on the desk. Your speed of calculating the curl will improve a lot (at least it did for me).
    i get what you are saying now. your method helps you to know the positions of P,Q and R. but how do you remember the positions of the partial derivatives with this method? some sort of triangle perhaps? or just y --> z --> x --> y?
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  11. #11
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    Ok you have  P, Q, R written down. Put your index finger (left hand) on  R and your middle finger on  Q . Thats  \frac{\partial R}{\partial y} . Note that the middle finger is on  Q which corresponds to  y . Now tap your middle finger on  Q and your index finger on  R . That's  \frac{\partial Q}{\partial z} .

    So our first term is:  \frac{\partial R}{\partial y} - \frac{\partial R}{\partial y} . Keep doing this.
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