Divergence and curl example

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

$${\frac{{\partial F_1}}{{\partial x}}} {\frac{{\partial F_2}}{{\partial y}}} {\frac{{\partial F_3}}{{\partial z}}}$$ div F

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

(The formula for curl was somewhat motivated in an earlier reading.)

Given these formulas, there isn't a whole lot to computing the divergence and curl. Just "plug and chug," as they say.

Example

Calculate the divergence and curl of F = (- y, xy, z).

Solution: Since

$${\frac{{\partial F_1}}{{\partial x}}} {\frac{{\partial F_2}}{{\partial y}}} {\frac{{\partial F_3}}{{\partial z}}}$$

we calculate that

Since

$${\frac{{\partial F_1}}{{\partial y}}} {\frac{{\partial F_2}}{{\partial x}}}$$

$${\frac{{\partial F_1}}{{\partial z}}} {\frac{{\partial F_2}}{{\partial z}}} {\frac{{\partial F_3}}{{\partial x}}} {\frac{{\partial F_3}}{{\partial y}}}$$

we calculate that

Good things we can do this with math. If you can figure out the divergence or curl from the picture of the vector field (below), you doing better than I can.

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