Chapter 16 – Vector Calculus 16.5 Curl and Divergence 1 Objectives: Understand the operations of...

1

Chapter 16 – Vector Calculus16.5 Curl and Divergence

16.5 Curl and Divergence

Objectives: Understand the

operations of curl and divergence

Use curl and divergence to obtain vector forms of Green’s Theorem

16.5 Curl and Divergence 2

Vector CalculusHere, we define two operations that:

◦ Can be performed on vector fields.◦ Play a basic role in the applications of

vector calculus to fluid flow, electricity, and magnetism.

Each operation resembles differentiation.

However, one produces a vector field whereas the other produces a scalar field.


Definition - CurlSuppose F = P i + Q j + R k is a vector field

on 3 and the partial derivatives of P, Q, and R all exist, then the curl of F is the vector field defined by:

curl R Q P R Q P

y z z x x y

F i j k


CurlAs a memory aid, let’s rewrite

Equation 1 using operator notation.

◦We introduce the vector differential operator (“del”) as:

x y z

i j k


Curl

curl

x y z

P Q R

R Q P R Q P

y z z x x y

i j k

F

i j k

F


CurlThus, the easiest way to

remember Definition 1 is by means of the symbolic expression

curl F F


Theorem 3If f is a function of three variables

that has continuous second-order partial derivatives, then

curl f 0


Conservative Vector FieldA conservative vector field is one for which

So, Theorem 3 can be rephrased as:

If F is conservative, then curl F = 0.◦ This gives us a way of verifying that a

vector field is not conservative.

fF


Theorem 4 If F is a vector field defined on all of 3

whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field.

NOTE: Theorem 4 is the 3-D version of Theorem 6 in Section 16.3

NOTE: This theorem says that it is true if the domain is simply-connected—that is, “has no hole.”


CurlThe reason for the name curl is that the curl

vector is associated with rotations. ◦ One connection is explained in Exercise 37.

◦ Another occurs when F represents the

velocity field in fluid flow (Example 3 in Section 16.1).


CurlParticles near (x, y, z) in the fluid tend to

rotate about the axis that points in the direction of curl F(x, y, z).

◦ The length of this curl vector is a measure of how quickly the particles move around the axis.


Irrotational CurlIf curl F = 0 at a point P, the fluid

is free from rotations at P.

F is called irrotational at P. ◦That is, there is no whirlpool or eddy

at P.


If curl F = 0, 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.

◦ We give a more detailed explanation in Section 16.8 as a consequence of Stokes’ Theorem.


Example 1 – pg. 1068 # 21Show that any vector field of the

form

where f, g, and h are differentiable functions, is irrotational.

( , , ) ( ) ( ) ( )x y z f x g y h z F i j k


Definition - Divergence If F = P i + Q j + R k is a vector field on 3

and ∂P/∂x, ∂Q/∂y, and ∂R/∂z exist, the divergence of F is the function of three variables defined by:

div P Q R

x y z

F


DivergenceIn terms of the gradient operator

the divergence of F can be written symbolically as the dot product of del and F:

x y z

i j k

div F F


Curl versus DivergenceObserve that:

◦ Curl F is a vector field.

◦ Div F is a scalar field.


Theorem 11If F = P i + Q j + R k is a vector field

on 3 and P, Q, and R have continuous second-order partial derivatives, then

div curl F = 0


DivergenceAgain, the reason for the name divergence

can be understood in the context of fluid flow.

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


Incompressible Divergence 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, F is said to be incompressible.


Example 2 – pg. 1068 # 22Show that any vector of the form

is incompressible.

( , , ) ( , ) ( , ) ( , )x y z f y z g x z h x y F i j k


Example 3 – pg. 1068Find the following for the given

vector field:a) The curlb) The divergence

2 2 22. ( , , )

8. ( , , ) , ,x xy xyz

x y z x yz xy z xyz

x y z e e e

F i j k

F


Gradient Vector FieldsAnother differential operator occurs when

we compute the divergence of a gradient vector field f.

◦ If f is a function of three variables, we have:

2 2 2

2 2 2

div f f

f f f

x y z


Laplace OperatorThis expression occurs so often

that we abbreviate it as 2f.

The operator is called the Laplace operator due to its relation to Laplace’s equation

2

2 2 22

2 2 20

f f ff

x y z


Green’s Theorem – Vector Form

Hence, we can now rewrite the equation in Green’s Theorem in the vector form using curl as equation 12:

curl C D

d dA F r F k



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 write a similar formula involving the normal component of F and the divergence. (see book for proof)



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.

div ,C D

ds x y dA F n F


Example 4 – pg. 1068Determine whether or not the vector field is

conservative. If it is conservative, find a function f such that F = f.

2 2 2 2 214. ( , , )

16. ( , , ) z z

x y z xyz x yz x y z

x y z e xe

F i j k

F i j k

Chapter 16 – Vector Calculus 16.5 Curl and Divergence 1 Objectives: Understand the operations of...

Documents

Transcript of Chapter 16 – Vector Calculus 16.5 Curl and Divergence 1 Objectives: Understand the operations of...