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

Section 13.5Curl 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:

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

curl F = 0

F is irrotational at P.

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)