**Here we go again!!**

1.2.4-1.2.7

1.2.4 Divergence (not the movie or book)

1.2.5 Curl (not the olympic event)

1.2.6 Product rule

1.2.7 Second derivatives

Divergence

This means spreading out.

As we look at a vector field, and we move around, do the vectors change?

important note. As you move around in

a

direction, does the vector change in that

direction

Another good description, is lines in and lines out.

**By the numbers**

again think as you move through a direction does it change in that direction.

Try to construct a picture of a divergence in 2D where the x gives positive and the y gives negative, and the net is zero (hint, use the equation)

Math practice

This is a leading practice for another day,

but compute the divergence of the electric field of a point charge. (let q = 1.11*10^-10)

**Curl**

This is also well named, how much does the vector 'curl' around an area.

**practice**

compute the curl of

Product rule

need to practice with del operator

**Prove this is true.**

this one is straight from your text but is a better proof

Second Derivatives

five options (of meaning)

divergence of a gradient

curl of gradient

gradient of divergence

Divergence of curl

curl of curl

**Gradients**

divergence of gradient, we call this the Laplacian

(note: we can let a Laplacian act on a vector but that is a bit different)

Curl of gradient is always zero

why?

Divergence

because the divergence produces a scalar, you can only take the gradient of it.

Gradient of Divergence

returns a vector,

rarely happens in a physics

it is not the same as the Laplacian

Curl

Divergence of Curl

why?

Curl of Curl

already did this one

already did this one two

returns a vector

Summary of 2nd derivatives

2 are zero.

1 is the other two

leaving just

Laplacian

Grad of Div

( and it is seldom used)

Practice

Take the Laplacian of

recall

Summary

1.2.4 Divergence

1.2.5 Curl

1.2.6 Product rule

1.2.7 2nd Derivatives

"I didn't quite understand the Second Derivatives section."

"Can we go over why the curl of a gradient is always zero one more time?"

"how to calculate divergence?,

Can you do another example of calculating the divergence? "

"I don't understand the meaning of second derivatives of gradient, divergence, or curl."

"Why are the curl of gradient and the divergence of a curl always zero?"

Can you work through problem 1.25 in class?