# Curl of a vector field: good remembering technique

• Nov 29th 2007, 09:20 PM
shilz222
Curl of a vector field: good remembering technique
We all know that $\displaystyle \text{curl} \ \bold{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \bold{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \bold{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \bold{k}$.

Here is a good technique to remember this:

Write down $\displaystyle P, Q, R$.

Put your left index finger on $\displaystyle R$. Now put your left middle finger on $\displaystyle Q$. 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 $\displaystyle \text{curl} \ \bold{F}$.
• Nov 29th 2007, 09:26 PM
Jhevon
Quote:

Originally Posted by shilz222
We all know that $\displaystyle \text{curl} \ \bold{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \bold{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \bold{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \bold{k}$.

Here is a good technique to remember this:

Write down $\displaystyle P, Q, R$.

Put your left index finger on $\displaystyle R$. Now put your left middle finger on $\displaystyle Q$. 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 $\displaystyle \text{curl} \ \bold{F}$.

i don't think i get your technique...:(:confused:

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

where $\displaystyle \bold{F} = \left< P, Q, R \right>$ and $\displaystyle \nabla = \left< \frac {\partial}{\partial x}, \frac {\partial}{\partial y}, \frac {\partial}{\partial z} \right>$
• Nov 29th 2007, 09:32 PM
shilz222
Its like playing the piano. Let your index finger, your middle finger, and your crown finger (left hand) represent $\displaystyle R,Q,P$ 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
• Nov 30th 2007, 12:35 AM
shilz222
I practiced it on the piano. I called it the "curl" piece. Choose 3 piano keys, and let them represent $\displaystyle P, Q, R$.
• Nov 30th 2007, 02:07 AM
shilz222
Actually this is also a good method of remembering anything involving cross products (that way you do not have to do the calculation).
• Nov 30th 2007, 07:57 AM
ThePerfectHacker
Why not just use determinants? If you start playing the paino during an exam it will look a little strange.
• Nov 30th 2007, 09:05 AM
topsquark
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? (Itwasntme) )

-Dan
• Nov 30th 2007, 09:18 AM
Plato
$\displaystyle \begin{array}{l} \nabla = \left\langle {\partial x,\partial y,\partial z} \right\rangle \\ curl(F) = \nabla \times F = \left| {\begin{array}{*{20}c} i & j & k \\ {\partial x} & {\partial y} & {\partial z} \\ P & Q & R \\ \end{array}} \right| \\ \end{array}$
• Nov 30th 2007, 10:36 AM
shilz222
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).
• Dec 1st 2007, 07:17 PM
Jhevon
Quote:

Originally Posted by shilz222
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?
• Dec 1st 2007, 08:12 PM
shilz222
Ok you have $\displaystyle P, Q, R$ written down. Put your index finger (left hand) on $\displaystyle R$ and your middle finger on $\displaystyle Q$. Thats $\displaystyle \frac{\partial R}{\partial y}$. Note that the middle finger is on $\displaystyle Q$ which corresponds to $\displaystyle y$. Now tap your middle finger on $\displaystyle Q$ and your index finger on $\displaystyle R$. That's $\displaystyle \frac{\partial Q}{\partial z}$.

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