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F16 PH 333 1.2.4-1.2.7

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Richard Datwyler

on 8 February 2019

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Transcript of F16 PH 333 1.2.4-1.2.7

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?
Full transcript