MA 242.003

• Day 55 – April 5, 2013• Section 13.3: The fundamental theorem for line

integrals– An interesting example

• Section 13.5: Curl of a vector field

x = cos(t)y = sin(t)

t = 0 .. Pi

x = cos(t)y = - sin(t)

t = 0 .. Pi

D open Means does not contain its boundary:

D open Means does not contain its boundary:

D simply-connected means that each closed curve in D contains only points in D.

D simply-connected means that each closed curve in D contains only points in D.

Simply connected regions “contain no holes”.

D simply-connected means that each closed curve in D contains only points in D.

Simply connected regions “contain no holes”.

Section 13.5Curl of a vector field

Section 13.5Curl of a vector field

Section 13.5Curl of a vector field

Section 13.5Curl 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”

(see maple for sketch)

(see maple for sketch)

All of these velocity vector fields are ROTATING.

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 = <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>

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”