MA 242.003

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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
• Day 57 – April 8, 2013
• Section 13.5:
• Review Curl of a vector field
• Divergence of a vector field

Section 13.5Curl of a vector field

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

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

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>

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”

The Divergence of a vector field

Then div F can be written symbolically as:

The Divergence of a vector field

Then div F can be written symbolically as:

So the vector field

Is incompressible

So the vector field

Is incompressible

However the vector field

So the vector field

Is incompressible

However the vector field

Is NOT – it is diverging!