16.5 Curl and Divergence

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 – y² 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 – y² 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

∇²f. 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:

∇²F = ∇²P i + ∇²Q j + ∇²R 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.