16.5 Curl and Divergence

Many ways to differentiate a vector field F(x, y, z) =( P(x,y,z)

Q(x,y,z)R(x,y,z)

). The nine scalar 1st-derivatives form

the total derivative or Jacobian matrix:

DF =

Px

Py

Pz

Qx

Qy

Qz

Rx

Ry

Rz

Entries of DF can combined in various ways: curl and divergence are two such combinations that haveuseful physical interpretations.

Curl

Definition. If F = Pi + Qj + Rk is a three-dimensional vector field then the curl of F is the vector field

curl F = F =(

Ry Q

z

)i +(

Pz R

x

)j +(

Qx P

y

)k

defined wherever all partial derivatives exist.1

Differential Operators and Notation Nabla or Del is the differential operator

= i x

+ j

y+ k

z=

xyz

The gradient of f is the action of the operator on f :

grad f = f =(

i

x+ j

y+ k

z

)f =

fx

i + fy

j + fz

k

The cross product notation for curl now makes sense:

PQ

R

=xy

z

PQ

R

=

i j kx y zP Q R

=y zQ R

i x zP R j + x yP Q

k = This construction is easier to remember than the formula in the definition and is most simple incolumn-vector notation.

Example If F = (x2 3y)i + xzj + (x + yz)k then

F =

xyz

x2 3yxz

x + yz

=z x1

z + 3

1Some authors will treat two-dimensional vector fields and state that if F = Pi + Qj is such then its curl is the scalar

quantity Qx Py , familiar from Greens Theorem.

1

Curl and Conservatism Recall Section 16.3 where a vector field F = Pi + Qj + Rk on a simply-connected region is conservative if and only if

Qx

=Py

,Ry

=Qz

,Pz

=Rx

This says precisely that all parts of the curl vanish.

Theorem. If F has continuous first-derivatives on a simply-connected region of R3, then curl F = 0 Fis conservative

One direction of this theorem can be written as follows:

Corollary. If f has continuous second-derivatives, then curl( f ) = 0 (often written f = 0).

Rotating fields: Interpretation of Curl Curl measures the tendency of objects to rotate.

2

2

y

4 2 2 4x 2

2

y

4 2 2 4x

(xi + yj) = 0 (yi xj) = 2kNo rotation Rotating flow

Definition. A vector field F is said to be irrotational if curl F = 0 everywhere.

Local rotation The picture is too simplistic: curl measures local rotation. The fact that an object inthe vector field F = yi xj will travel in circles round the origin is irrelevant to curl. It is the fact thatthe object will also rotate so that it changes the direction it is facing.

Theorem. Suppose that a small paddle with vertical axis is positioned at the point (a, b) in a fluid with ve-locity field F = Pi + Qj. Assuming no friction and that the paddle rotates freely, the paddle will rotate withangular velocity

=12

(Qx

(a, b) Py

(a, b))

k

More generally, place a paddle at position r and with axis nin a fluid flow F. Then the paddle will rotate around n withangular speed

=12(curl F) n rad/s

2

The proof is a little tricky and (at least in 3D) requires Stokes Theorem. It is much easier to considera. . .

Duck Race! Suppose that the water in a river of width 4 has velocity v =(1 14 y2

)i. The curl of

the flow is

v =xy

z

1 14 y20

0

= 00

12 y

Release several ducks and assume that they move with the water.

There are two aspects to the motion of the ducks:

1. Linear. The ducks follow the direction of flow to the right. The faster ducks are closer tomidstream.

2. Rotational. The side of a duck closer to midstream will move faster than the side nearer the edgeof the river. This will cause most ducks to rotate. According to the Theorem, and interpretingthe right hand rule, a duck with y = 1 will rotate with angular velocity 12 v = 14 k, i.e.counter-clockwise at 0.25 radians per second.

Combine the two and you get the motion in the animation.2 If youve ever been white-water raftingthis explains why the guide always wants to stay in the center of the river: you travel fastest and youdont spin!

2Click it to make it move after opening the pdf in Acrobat on your desktop.

3

Global Rotation 6= Local Rotation It is important to distinguish between local rotation (changingthe direction an object faces) and global rotation (vector fields making objects travel in loops). Com-pare the following three duck races. In all cases the ducks will travel counter-clockwise around theorigin (global rotation). Their local rotation changes each time.

F1 =1

(x2 + y2)1/2

yx0

F1 =

1(x2 + y2)1/2

k

Ducks have local rotation counter-clockwise.Because of the global rotation of the vector field, thereare more arrows on the side of the duck further fromthe origin. Since all arrows have the same length, theduck rotates counter-clockwise.

F2 =1

x2 + y2

yx0

F2 = 0

Ducks have zero local rotation.The fact that the speed of flow is faster nearer the originexactly balances the the fact that more arrows are on theoutside of the duck.

F3 =1

(x2 + y2)3/2

yx0

F3 =

1(x2 + y2)3/2

k

Ducks have local rotation clockwise.Now the speed of flow is so much greater nearer theorigin that the duck spins the opposite way.

4

Circulation: Greens Theorem Revisited

Suppose that F = Pi + Qj is a vector field in two dimensions.Then ( F) k = Qx Py , forms part of Greens Theorem,which we can now rewrite:

CF dr =

D

Qx P

ydA

=

D( F) k dA

Recall that

C F dr =

C F T ds where T is the unit tangentvector of C. This is often called the circulation of the vector fieldF around C: how vigorously F pushes a particle round C.

TC

Interpreting curl as causing local rotation, the double integral

D F k dA may be interpreted

as the sum of all the infinitessimal rotations inside C.

To summarize, Greens Theorem says: sum of infinitessimal rotations = total circulation.

Divergence

Definition. If F = Pi + Qj + Rk is a three-dimensional vector field then the divergence of F is the scalarfield

div F = F = Px

+Qy

+Rz

defined wherever all partial derivatives exist.3

Notation The dot product notation is completely natural when is viewed as a differential opera-tor

F =(

i

x+ j

y+ k

z

)PQ

R

=xy

z

PQ

R

= Px

+Qy

+Rz

Example

F =xy

z

x2 3yxz

x + yz

= x

(x2 3y) + y

xz +

z(x + yz) = 2x + y

The following easy result is analogous to the fact that f = 0.

Theorem. If F = Pi + Qj + Rk has continuous second-derivatives, then div curl F = 0 (often written F = 0)

3Similarly div F = Px +Qy for a two-dimensional vector field.

5

Proof.

div curl F =

xyz

Ry QzPz RxQx Py

= 2Rxy

2Q

xz+

2Pyz

2R

yx+

2Qzx

2P

zy= 0

Laplacians

We have f = 0 and F = 0. Only two other combinations make sense:4 ( F) and f . The first is not commonly used, but the latter is very important in Physics.

Definition. If f is a scalar function then its Laplacian is the expression

2 f = f = div grad f = 2P

x2+

2Qy2

+2Rz2

If F = Pi + Qj + Rk we define

2F = 2Pi +2Qj +2Rk

Example 2(3x2y3z + ex2y2) = 6y3z + 18x2yz + 4(x2 + y2 1)ex2y2

Physics applications Among the many places where Laplacians appear in real life are the follow-ing.

1. Minimal surfaces: Suppose that a wire in the shape of acurve C, parameterized by r(t) =

( cos tsin t

cos 2t

)for 0 t 2

is dipped in soap. In the absence of gravity, the soapfilm formed will be the saddle-surface with equation z =y2 x2. This satisfies Laplaces equation

2z = 2z

x2+

2zy2

= 0

A surface formed by a soap bubble has minimal area out of all surfaces with boundary C.

2. The Heat Equation: The temperature T(x, y, z; t) at position (x, y, z) and time t satisfies the partialdifferential equation Tt = k2T (k > 0 constant).

3. The Wave Equation: The height u(x, y; t) of a wave at position (x, y) and time t satisfies the PDE2ut2 = c

22u (c constant), where c is the speed of the wave.

4Think about what is a vector field and what is a scalar. . .

6

Expanding Fields: Interpretation of Divergence Divergence measures the tendency of a vectorfield to make things expand.

2

2

y

4 2 2 4x

2

2

y

4 2 2 4x

(yi xj) = 0 (xi + yj) = 2No expansion Expanding flow

Definition. A vector field F is incompressible if div F = 0 everywhere.

Local expansion Again the picture is too simplistic: con-sider local expansion which, in three-dimensions, meansincrase in volume.

Theorem. Place an infinitessimally balloon in a vector field F.Assume that the balloon drifts with the vector field along a pathparameterized by r(t) and that the surface of the balloon moveswith the vector field. Then the balloons volume V(t) satisfies thedifferential equation

1V(t)

dVdt

= div F(r(t))

The larger the divergence, the greater the rate of volume increase. For a two-dimensional vector field,replace the balloon with a tiny oil drop and volume with area. A proof will have to wait until wehave the divergence theorem.

Example The vector field F =( x

yz

)= r has constant divergence 3. It follows that an infinitessimal

balloon released into the field will see its volume satisfy V (t) = 3V(t). This is the natural growthequation, with solution

V(t) = V0e3t

where V0 is the initial volume of the balloon.

7

Global Expansion 6= Local Expansion Similarly to our discussion of local rotation and curl, wedistinguish between global expansion (particles seeming to travel away from something) and localexpansion (increase in volume). In the following animations, pretend that the duck is an oil-dropmoving with the field. Divergence effects the volume of the duck (seen here as area).

F1 =1r

xy0

, r = x2 + y2 F1 =

1r

< 0

Duck/oil-drop shrinks as it moves towards origin.The arrows bring all points in towards the center andreduce volume.

F2 =1r2

xy0

, F2 = 0Ducks volume unchanged.The vector field gets stronger nearer the origin. Theduck stetches out, perfectly balancing the squashing ofthe vector field as it moves nearer the origin.

F3 =1r3

xy0

, F3 = 1r3 > 0Ducks volume increases!The increase in speed is so great near the origin, that theduck gets longer faster than it narrows.

8

Greens Theorem, version III We can also interpret Greens Theorem using divergence of a 2-dimensional vector field.

Suppose that r(t) =(

x(t)y(t)

), a t b parameterizes a positively-oriented, simple, closed curve. Its

outward pointing normal vector n must be the unit tangent vector T = 1|r| r rotated 90 clockwise. That

is,

n(t) =1|r(t)|

(y(t)x(t)

)We now compute a new line integral, and convert to a doubleintegral using Greens Theorem.

CF n ds =

ba

(PQ

) 1|r|

(y

x) r dt

=

CQ dx + P dy

=

D

Px

+Qy

dA =

Ddiv F dA

n

C

F n measures how much vector field F points across the boundary C. The integral

CF n ds there-

fore measures the total flow rate of F out of C. This is called the flux of the vector field F out of C.

Since divergence measures the tendency of a vector field to cause expansion, the integral

Ddiv F dA

is the sum of all the infinitessimal expansions inside C.

Greens Theorem can therefore be interpreted: sum of infinitessimal expansions = total outwardflow.5

Summary

Nabla = i x + j y + k z is a differential operator

Gradient f = fx i + fy j +

fz k is a vector field

Measures direction and magnitude of greatest increase in f

Curl F =(

Ry Qz

)i +(

Pz Rx

)j +(

Qx Py

)k is a vector field

Measures local rotation of F in 3D12 curl F n is the angular speed of flow around axis n

Divergence F = Px + Qy + Rz is a scalar fieldMeasures local rate of change of volumediv F > 0 = expansion, div F < 0 = contraction

Identities f = 0, F = 0

5This is precisely the Divergence Theorem in 2-dimensions.

9

Curl and Divergence

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