**16.5** Curl and Divergence

### Curl

### Curl

**If F = P i + Q j + R k is a vector field on and the partial *** derivatives of P, Q, and R all exist, then the curl of F is the *
vector field on defined by

Let’s rewrite Equation 1 using operator notation. We introduce the vector differential operator ∇ (“del”) as

### Curl

It has meaning when it operates on a scalar function to
*produce the gradient of f :*

If we think of ∇ as a vector with components *∂/∂x, ∂/∂y, and *

*∂/∂z, we can also consider the formal cross product of ∇*
**with the vector field F as follows:**

### Curl

So the easiest way to remember Definition 1 is by means of the symbolic expression

### Example 1

**If F(x, y, z) = xz i + xyz j – y**^{2} **k, find curl F.**

Solution:

Using Equation 2, we have

*Example 1 – Solution *

**= (–2y – xy) i – (0 – x) j + (yz – 0) k**

**= –y(2 + x) i + x j + yz k**

cont’d

### Curl

*We know that the gradient of a function f of three variables *
is a vector field on and so we can compute its curl.

The following theorem says that the curl of a gradient
**vector field is 0.**

### Curl

**Since a conservative vector field is one for which F =** *∇f, *
Theorem 3 can be rephrased as follows:

**If F is conservative, then curl F = 0. **

This gives us a way of verifying that a vector field is not conservative.

### Curl

The converse of Theorem 3 is not true in general, but the
**following theorem says the converse is true if F is defined **
everywhere. (More generally it is true if the domain is

simply-connected, that is, “has no hole.”)

### Curl

*The reason for the name curl is that the curl vector is *
associated with rotations.

**Another occurs when F represents the velocity field in fluid **
*flow. Particles near (x, y, z) in the fluid tend to rotate about *
* the axis that points in the direction of curl F(x, y, z), and the *
length of this curl vector is a measure of how quickly the
particles move around the axis (see Figure 1).

### Curl

**If curl F = 0 at a point P, then the fluid is free from rotations ****at P and F is called irrotational at P. **

*In other words, there is no whirlpool or eddy at P. *

**If curl F = 0, then a tiny paddle wheel moves with the fluid **
but doesn’t rotate about its axis.

**If curl F ≠ 0, the paddle wheel rotates about its axis.**

### Divergence

### Divergence

**If F = P i + Q j + R k is a vector field on and **∂P/∂x,

* ∂Q/∂y, and ∂R/∂z exist, then the divergence of F is the *
function of three variables defined by

**Observe that curl F is a vector field but div F is a scalar **
field.

### Divergence

In terms of the gradient operator

∇ = (* ∂/∂x) i + (∂/∂y) j + (∂/∂z) k, the divergence of F can be *
written symbolically as the dot product of ∇ and F:

### Example 4

**If F(x, y, z) = xz i + xyz j – y**^{2} **k, find div F.**

Solution:

By the definition of divergence (Equation 9 or 10) we have
**div F = ∇** **F**

### Divergence

**If F is a vector field on** **, then curl F is also a vector field **
on . As such, we can compute its divergence.

The next theorem shows that the result is 0.

### Divergence

**If F(x, y, z) is the velocity of a fluid (or gas), then **

* div F(x, y, z) represents the net rate of change (with respect *
to time) of the mass of fluid (or gas) flowing from the point

*(x, y, z) per unit volume.*

**In other words, div F(x, y, z) measures the tendency of the ***fluid to diverge from the point (x, y, z).*

**If div F = 0, then F is said to be incompressible.**

### Divergence

*If f is a function of three variables, we have*

and this expression occurs so often that we abbreviate it as

∇^{2}*f. The operator*

∇^{2} = ∇ ∇

**is called the Laplace operator because of its relation to **
**Laplace’s equation**

### Divergence

We can also apply the Laplace operator ∇^{2} to a vector field
**F = P i + Q j + R k**

in terms of its components:

∇^{2}**F = ∇**^{2}**P i + ∇**^{2}**Q j + ∇**^{2}**R k**

### Vector Forms of Green’s Theorem

### Vector Forms of Green’s Theorem

The curl and divergence operators allow us to rewrite

Green’s Theorem in versions that will be useful in our later work.

*We suppose that the plane region D, its boundary curve C, *
*and the functions P and Q satisfy the hypotheses of *

Green’s Theorem.

**Then we consider the vector field F = P i + Q j.**

### Vector Forms of Green’s Theorem

Its line integral is

**and, regarding F as a vector field on with third **
component 0, we have

### Vector Forms of Green’s Theorem

Therefore

and we can now rewrite the equation in Green’s Theorem in the vector form

### Vector Forms of Green’s Theorem

Equation 12 expresses the line integral of the tangential
**component of F along C as the double integral of the **

**vertical component of curl F over the region D enclosed by ***C. We now derive a similar formula involving the normal *
**component of F.**

*If C is given by the vector equation*

**r(t) = x(t) i + y(t) j ***a ≤ t ≤ b*
then the unit tangent vector is

### Vector Forms of Green’s Theorem

*You can verify that the outward unit normal vector to C is *
given by

(See Figure 2.)

### Vector Forms of Green’s Theorem

Then, from equation

we have

### Vector Forms of Green’s Theorem

by Green’s Theorem.

But the integrand in this double integral is just the
**divergence of F.**

### Vector Forms of Green’s Theorem

So we have a second vector form of Green’s Theorem.

This version says that the line integral of the normal

**component of F along C is equal to the double integral of ****the divergence of F over the region D enclosed by C.**