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MA 242.003 . Day 57 – April 8, 2013 Section 13.5: Review Curl of a vector field Divergence of a vector field. Section 13.5 Curl of a vector field. “A way to REMEMBER this formula”. “A way to REMEMBER this formula”. “A way to REMEMBER this formula”. “A way to REMEMBER this formula”.

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ma 242 003
MA 242.003
  • Day 57 – April 8, 2013
  • Section 13.5:
    • Review Curl of a vector field
    • Divergence of a vector field
slide15

Let F represent the velocity vector field of a fluid.

What we find is the following:

Example: F = <x,y,z> is diverging but not rotating

curl F = 0

slide16

All of these velocity vector fields are ROTATING.

What we find is the following:

F is irrotational at P.

Example: F = <x,y,z> is diverging but not rotating

curl F = 0

slide18

All of these velocity vector fields are ROTATING.

What we find is the following:

Example: F = <-y,x,0> has non-zero curl everywhere!

curl F = <0,0,2>

slide21

Differential Identity involving curl

Recall from the section on partial derivatives:

We will need this result in computing the

“curl of the gradient of f”

slide29

The Divergence of a vector field

Then div F can be written symbolically as:

slide30

The Divergence of a vector field

Then div F can be written symbolically as:

slide38

So the vector field

Is incompressible

slide39

So the vector field

Is incompressible

However the vector field

slide40

So the vector field

Is incompressible

However the vector field

Is NOT – it is diverging!