appendix b; a quick look at the del operator

7/21/2019 Appendix B; A Quick Look at the Del Operator

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443

Appendix BA Quick Look at the Del Operator

We use the del operatorto take the gradient of a scalar function, sayf(x,y,z):

=

+

+

f i

f

xj

f

yk

f

z

.

If we factor out the functionf, the gradient off looks like

=

+

+

f i

xj

yk

zf .

The term in parentheses is called deland is written as

=

+

+

.i

xj

yk

z

By itself, has no meaning. It is meaningful only when it acts on a scalarfunction. The term operateson scalar functions by taking their derivativesand combining them into the gradient. We say that is a vector operator act-

ing on scalar functions, and we call it the del operator.Since resembles a vector, we consider all the ways that we can act on vec-

tors and see how the del operator acts in each case.

Vectors Del

Operation Result Operation Result

Multiply by a scalar a Aa Operate on a scalar f f

Dot product withanother vector B AB Dot product withvector F(x, y, z) F

Cross product withanother vector B

ABCross product with

vector F(x, y, z) F


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444 Introduction to Engineering Mechanics: A Continuum Approach

Divergence

Lets first compute the form of the divergence in regular Cartesian coordi-nates. If we let a random vector

F F i F j F kx y z= + +

,

then we define

divF F ix

jy

kz

F ix= =

+

+

++ +( )= +

+

F j F k Fx

Fy

Fz

y zx y z .

Like any dot product, the divergence is a scalar quantity. Also note that, ingeneral, div Fis a function and changes in value from point to point.

Physical Interpretation of the Divergence

The divergence quantifies how much a vector field spreads out, or diverges,from a given point P. For example, the figure on the left (Figure B.1) haspositive divergence at P, since the vectors of the vector field are all spread-ing as they move away from P. The figure in the center has zero divergenceeverywhere since the vectors are not spreading out at all. This is also easyto compute, since the vector field is constant everywhere and the derivativeof a constant is zero. The field on the right has negative divergence since thevectors are coming closer together instead of spreading out.

FIGURE B.1Vector Fields

In the context of continuum mechanics, the divergence has a particularlyinteresting meaning. For solids, if the vector field of interest is the displace-ment vector U, the divergence of this vector tells us about the overall changein volumeof the solid. See equation (3.5) and homework problem (3.2), bothin Chapter 3, in this textbook. When we have =U 0 we know that the

volume of a given solid body remains constant, and we can call the solid


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Appendix B: A Quick Look at the Del Operator 445

incompressible. For fluids, we use the velocity vector Vto talk about the defor-mation kinematics. The divergence of the velocity vector tells us about thevolumetric strain rate, and when we have =V 0 we say that the flow isincompressible. This generally allows us to neglect changes in fluid densityand say that density remains constant (Chapter 8, equation 8.9).

Example

Calculate the divergence of

F xi yj zk= + +

.

=

+

+

= + + =F

xx

yy

zz( ) ( ) ( ) .1 1 1 3

This is the vector field shown on the left om Figure B.1. Its divergence isconstant everywhere.

Curl

We can also compute the curl in Cartesian coordinates. Again, let

F F i F j F kx y z= + + ,

and calculate

curlF F

i j k

x y z

F F F

i F

y

F

x y z

z= =

=

yy x z

zj

F

z

F

x

+

++

.k

F

x

F

y

y x

Not surprisingly, the curl is a vector quantity.

Physical Interpretation of the Curl

The curl of a vector field measures the tendency of the vector field to swirl.Consider the illustrations in Figure B.2. The field on the left, called F, has curl

with positivek component. To see this, use the right hand rule. Place your


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446 Introduction to Engineering Mechanics: A Continuum Approach

right hand at P. Point your fingers toward the tail of one of the vectors of F.Now curl your fingers around in the direction of the tip of the vector. Stickyour thumb out. Since it points toward the +zaxis (out of the page), the curlhas a positive k component.

The second vector field G has no visible swirling tendency at all so wewould expect =G 0 . The third vector field doesnt look like it swirlseither, so it also has zero curl.

FIGURE B.2Vector Fields

Examples

Example 1

Compute the curl of F yi xj= + .

=

=F

i j k

x y zy x

k

.

0

2

This is the vector field on the left in Figure B.2. As you can see, the analytical

approach demonstrates that the curl is in the positive k direction, as expected.

Example 2

Compute the curl of H xi yj zk= + + , orH(r) = r.

=

=H

i j k

x y z

x y z

.0


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Appendix B: A Quick Look at the Del Operator 447

This, as youve probably guessed, is the vector field on the far right inFigure B.2.

Laplacian

The divergence of the gradient appears so often that it has been given a spe-

cial name: the Laplacian. It is written as 2 or and, in Cartesian compo-nents, has the form

=

+

+

22

2

2

2

2

2f

f

x

f

y

f

z.

It operates on scalar functions and produces a scalar result. When we takethe Laplacian of a vector field,

F F i F j F k

x y z

= + + ,

we get

= + +

2 2 2 2F F i F j F kx y z( ) ( ) ( ) .


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appendix b; a quick look at the del operator

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