Calc 3 Lecture Notes Section 14.5 Page 2 of 7

Section 14.5: Curl and Divergence

Big idea: The operations of curl and divergence can be used to tell us about the nature of a vector field.

Big skill: You should be able to compute curl and divergence, and know what they have to say about the nature of a vector field.

Definition 5.1: Curl of a Vector Field

The curl of the vector field is the vector field

,

defined at all points at which the partial derivatives exist.

Alternative notation:

Recall from section 14.4 (Green’s Theorem) that a vector field is conservative if .

Also recall from the notes at the end of section 14.3 that another criteria for a three-component vector field to be conservative is:

We can anticipate a zero curl to be an indicator of a conservative vector field…

Practice:

Compute the curl of


Definition 5.2: Divergence of a Vector Field

The divergence of the vector field is the scalar function

,

defined at all points at which the partial derivatives exist.

Alternative notation:

Practice:

Compute the divergence of

Compute the divergence and curl of the following vector fields.

Uniform Vector Field
Example: F(x, y) = á1, 0ñ. /
Example: F(x, y, z) = á1, 2, -1ñ.
Radial Vector Field
Example: F(x, y) = áx, yñ. /
Example:
Rotational (Circular) Vector Field
Example: F(x, y) = á-y, xñ. /
Example: /

Vocabulary (arising from considering the vector field as fluid flow lines):

·  When the curl is non-zero, the fluid tends to rotate about an axis parallel to the direction of the curl.

·  When the curl is zero, the fluid does not tend to rotate, and thus is called “irrotational.”

·  When the divergence is positive, there is a “source”; more fluid flows into a region than flows out.

·  When the divergence is negative, there is a “sink”; more fluid leaves a region than flows into it.

·  When the divergence is zero, we call the flow “source-free” or “incompressible” because the amount of fluid that flows into a region is equal to amount that flows out .

Practice:

Determine which of the following expressions are scalars, vector fields, or undefined, and expand the defined expressions, given that f is a scalar function and F is a vector field:, , .

Theorem 5.1: A Conservative Field is Irrotational

If a conservative vector field has components with continuous first-order partial derivatives throughout an open region , then .

Note: we proved this when we showed, since a conservative vector field is the gradient of a scalar function.


Theorem 5.2: An Irrotational Vector Field is Conservative

If a vector field has components with continuous first-order partial derivatives throughout an open region , then F is conservative if and only if .

Note: our unit radial vector field does not meet the conditions of this theorem, since it is undefined when x = y = 0 (eben though we found a potential function in the homework).

Conservative Vector Fields
If the vector field has components with continuous first partial derivatives on , then the following five statements are equivalent (i.e., they are either all true or all false)
1.  is conservative in D.
2.  is a gradient field in D. (i.e., ).
3.  is independent of path in D.
4.  for every piecewise-smooth closed curve C lying in D.
5.  .

Alterative Forms of Green’s Theorem using the Curl and Divergence:

Where
and /

Gradient, Curl, and Divergence in Cylindrical Coordinates

(comes from …)

For a vector field


Gradient, Curl, and Divergence in Spherical Coordinates

(comes from …)

For a vector field