Divergence and curl example

# Multivariable calculus and vector analysis

## A set of on-line readings

### Non-Math ML version

### Divergence and curl example

For F : R3 → R3, the formulas for the divergence and curl are

div F | = + + |  |
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

= 0, = x, = 1 |  |

we calculate that

div(F) = 0 + x + 1 = x + 1. |  |

Since

= -1, = y, |  |
= = = = 0, |  |

we calculate that

curl(F) = (0 - 0, 0 - 0,y + 1) = (0, 0,y + 1). |  |

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.

## Contents by lecture

### Part 1

### Part 2

### Part 3

### Part 4

### Part 5

### Part 6

### Part 7

### Part 8

### Part 9

### Part 10

### Part 11

### Part 12

### Part 13

- The idea of divergence and curl*
- Quiz 13: Divergence and curl quiz*
- Vector fields
- The components of the curl
- More details on the components of the curl
- Divergence and curl notation
- Divergence and curl example
- Subtleties about divergence
- Subtleties about curl

### Part 14

### Part 15

### Part 16

### Part 17

### Part 18

### Part 19

### Part 20

### Part 21

### Part 22

### Part 23

### Part 24

### Part 25

### Part 26

### Part 27