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”
“A way to REMEMBER this formula”
“A way to REMEMBER this formula”
(continuation of example)
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:
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>
(See Maple worksheet for the calculation)
Differential Identity involving curl
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
The Divergence of a vector field
The Divergence of a vector field
The Divergence of a vector field
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:
The Divergence of a vector field
The Divergence of a vector field
So the vector field
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!
Differential Identity involving div
Differential Identity involving div
Differential Identity involving div
Proof:
(continuation of proof)