Divergence and curl notation

For F : R3$ \to$R3, the formulas for the divergence and curl are

div F = $\displaystyle {\frac{{\partial F_1}}{{\partial x}}}$ + $\displaystyle {\frac{{\partial F_2}}{{\partial y}}}$ + $\displaystyle {\frac{{\partial F_3}}{{\partial z}}}$    
curl F = $\displaystyle \left(\vphantom{\frac{\partial F_3}{\partial y}-\frac{\partial F_... ... x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} }\right.$$\displaystyle {\frac{{\partial F_3}}{{\partial y}}}$ - $\displaystyle {\frac{{\partial F_2}}{{\partial z}}}$,$\displaystyle {\frac{{\partial F_1}}{{\partial z}}}$ - $\displaystyle {\frac{{\partial F_3}}{{\partial x}}}$,$\displaystyle {\frac{{\partial F_2}}{{\partial x}}}$ - $\displaystyle {\frac{{\partial F_1}}{{\partial y}}}$$\displaystyle \left.\vphantom{\frac{\partial F_3}{\partial y}-\frac{\partial F_... ... x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} }\right)$.    

These formulas are easy to memorize using a tool called the "del" operator, denoted by $ \nabla$. Think of $ \nabla$ as a "fake" vector composed of all the partial derivatives that we use just to help us remember the formulas:

$\displaystyle \nabla$ = $\displaystyle \left(\vphantom{\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}}\right.$$\displaystyle {\frac{{\partial }}{{\partial x}}}$,$\displaystyle {\frac{{\partial }}{{\partial y}}}$,$\displaystyle {\frac{{\partial }}{{\partial z}}}$$\displaystyle \left.\vphantom{\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}}\right)$.    

Although it doesn't make sense to just have the partial derivatives without them acting on a function, we won't worry about that. This is just notation.

Now, let's take the dot product of the $ \nabla$ vector with F = (F1, F2, F3):

$\displaystyle \nabla$ . F = $\displaystyle \left(\vphantom{\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}}\right.$$\displaystyle {\frac{{\partial }}{{\partial x}}}$,$\displaystyle {\frac{{\partial }}{{\partial y}}}$,$\displaystyle {\frac{{\partial }}{{\partial z}}}$$\displaystyle \left.\vphantom{\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}}\right)$ . (F1, F2, F3)    
  = $\displaystyle {\frac{{\partial }}{{\partial x}}}$F1 + $\displaystyle {\frac{{\partial }}{{\partial y}}}$F2 + $\displaystyle {\frac{{\partial }}{{\partial z}}}$F3    

If we think of each "multiplication" in the dot product as instead being the derivative of the corresponding F, then we have the formula for the divergence. So, if you can remember the del operator $ \nabla$ and how to take a dot product, you can easily remember the formula for the divergence

div F = $\displaystyle \nabla$ . F = $\displaystyle {\frac{{\partial F_1}}{{\partial x}}}$ + $\displaystyle {\frac{{\partial F_2}}{{\partial y}}}$ + $\displaystyle {\frac{{\partial F_3}}{{\partial z}}}$.    

This notation is also helpful because you will always know that $ \nabla$ . F is a scalar (since, of course, you know that the dot product is a scalar product).

The curl, on the other hand, is a vector. We know one product that gives a vector: the cross product. And, yes, it turns out that curl F is equal to $ \nabla$ x F. To see this, let's take the cross product of the $ \nabla$ vector with F.

$\displaystyle \nabla$ x F = $\displaystyle \left(\vphantom{\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}}\right.$$\displaystyle {\frac{{\partial }}{{\partial x}}}$,$\displaystyle {\frac{{\partial }}{{\partial y}}}$,$\displaystyle {\frac{{\partial }}{{\partial z}}}$$\displaystyle \left.\vphantom{\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}}\right)$ x (F1, F2, F3)    
  = $\displaystyle \left\vert\vphantom{ \begin{array}{ccc} \mathbf{i} & \mathbf{j} &... ...artial y} & \frac{\partial }{\partial z} F_1 & F_2 & F_3 \end{array} }\right.$$\displaystyle \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \frac{\... ...tial }{\partial y} & \frac{\partial }{\partial z} F_1 & F_2 & F_3 \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \ma... ...al y} & \frac{\partial }{\partial z} F_1 & F_2 & F_3 \end{array} }\right\vert$    
  = i$\displaystyle \left(\vphantom{\frac{\partial }{\partial y}F_3 - \frac{\partial }{\partial z}F_2}\right.$$\displaystyle {\frac{{\partial }}{{\partial y}}}$F3 - $\displaystyle {\frac{{\partial }}{{\partial z}}}$F2$\displaystyle \left.\vphantom{\frac{\partial }{\partial y}F_3 - \frac{\partial }{\partial z}F_2}\right)$ - j$\displaystyle \left(\vphantom{\frac{\partial }{\partial x} F_3 -\frac{\partial }{\partial z}F_1}\right.$$\displaystyle {\frac{{\partial }}{{\partial x}}}$F3 - $\displaystyle {\frac{{\partial }}{{\partial z}}}$F1$\displaystyle \left.\vphantom{\frac{\partial }{\partial x} F_3 -\frac{\partial }{\partial z}F_1}\right)$ + k$\displaystyle \left(\vphantom{\frac{\partial }{\partial x}F_2 - \frac{\partial }{\partial y}F_1}\right.$$\displaystyle {\frac{{\partial }}{{\partial x}}}$F2 - $\displaystyle {\frac{{\partial }}{{\partial y}}}$F1$\displaystyle \left.\vphantom{\frac{\partial }{\partial x}F_2 - \frac{\partial }{\partial y}F_1}\right)$    
  = $\displaystyle \left(\vphantom{\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}}\right.$$\displaystyle {\frac{{\partial F_3}}{{\partial y}}}$ - $\displaystyle {\frac{{\partial F_2}}{{\partial z}}}$$\displaystyle \left.\vphantom{\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}}\right)$i + $\displaystyle \left(\vphantom{\frac{\partial F_1}{\partial z} -\frac{\partial F_3}{\partial x}}\right.$$\displaystyle {\frac{{\partial F_1}}{{\partial z}}}$ - $\displaystyle {\frac{{\partial F_3}}{{\partial x}}}$$\displaystyle \left.\vphantom{\frac{\partial F_1}{\partial z} -\frac{\partial F_3}{\partial x}}\right)$j + $\displaystyle \left(\vphantom{\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}}\right.$$\displaystyle {\frac{{\partial F_2}}{{\partial x}}}$ - $\displaystyle {\frac{{\partial F_1}}{{\partial y}}}$$\displaystyle \left.\vphantom{\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}}\right)$k    

This is exactly the formula we gave above. So if you can use the rule that "multiplication" by $ {\frac{{\partial }}{{\partial x}}}$ is the same as taking the partial derivative with respect to x (and similar for the other derivatives), then you can remember the curl formula by

curl F = $\displaystyle \nabla$ x F.    

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Duane Nykamp
nykamp@math.umn.edu
2007-03-15
05514 hits since
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