Divergence Curl Vector Feilds

8/12/2019 Divergence Curl Vector Feilds

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MA156 - Mathematical Methods for Physical Sciences

Conservative vector fields

Contents

1 Introduction 2

2 Gradient and directional derivative 2

2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Conservative fields 5

3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Properties of conservative fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Potentials of conservative vector fields . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.1 Integral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.2 Differential method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Stokes theorem 10

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 The curl of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Stokes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.4 Stokes theorem and conservative fields. . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Divergence and Divergence theorem 15

6 Physical Applications of the divergence theorem 16

6.1 Divergence and the sources of vector fields . . . . . . . . . . . . . . . . . . . . . . 16

6.2 Divergence and Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.3 Divergence and electrostatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18


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MA156 - Conservative vector fields 2

1 Introduction

Vector fields are extremely useful to describe forces. Some of these, like the gravitational or the

electrostatic force, have a very important property: it is possible to associate an energy to them. A

stone held on the top of a mountain has a certaingravitational potential energythat can be transformed

into kinetic energy by letting it drop. It is clear that the vector field that describes the gravitational

force is somewhat peculiar, it must have some additional properties with respect to a generic vector

field. The purpose of these notes is to explain very briefly what are the characteristics that differentiate

the gravitational field from a generic vector field to which it is not possible to associate a potential

energy.

In order to do this we must introduce an approach to vector fields that is the complement of

what we have done until now. To understand this point consider, for the moment, a real function of

one real variable, f(x), witha x b. The study of the properties of this function can be carriedout in two ways: the integral of a functionf(x) over an intervala x brequires the knowledgeof the function over the entire interval and gives global properties of the function, for example its

average value. The derivative of a function, instead, requires only a local knowledge of the function

and gives only local information: knowing the first derivative of a function at a point x0 allows us toapproximate the function in a small neighbourhood ofx0but does not gives us any information on the

values of the function away from that point. The global (integration) and the local (differentiation)approaches are not unrelated: as a matter of fact the fundamental theorem of calculus states that the

derivative is (very roughly) the inverse of the integral.

The same two approaches can be used to study vector fields. Until now we have used the global

approach and we have defined the line and surface integrals of vector fields. In order to study the

properties of conservative vector fields we must also introduce the local approach and try to define

differentiation operations that can either produce a vector field by acting on a scalar function f(x,y,z)(such operation is called the gradient) or that act directly on vector fields (these two operations are

called thecurland thedivergenceof the vector field).

We are now going to introduce the gradient of a scalar function and use it to define a special

class of vector fields calledconservative vector fields. We then show that these fields possess all the

properties of the gravitational force field, namely that it is possible to associate a potential energyto them. Finally we will use the two other differential operations on vector fields, the curl and the

divergence, to find some easy methods to identify whether a vector field is or is not conservative and

to write conservation laws as partial differential equations.

2 Gradient and directional derivative

2.1 Definition

Given a function of two or more variables, f(x, y)for example, we define the directional derivativeoffat the pointx0 = (x0, y0)in the direction of theunitvectorn as

fn

= lim0

f(x0+n) f(x0)

.

The geometrical interpretation of the directional derivative is that it represents the slope of the graph of

f(x, y)when we move from the point(x0, y0)in the direction indicated by the vector n (see Figure1).

Example Evaluate the directional derivative off(x, y) = sin(x+y2)in the direction ofv =i + 2jat(0, 0).


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f

x

y

n

z

f(x,y)

Figure 1: The directional derivative in the direction of the unit vectorn is the slope of the graph of the functionin the direction ofn. The gradient off, f, points in the direction of maximum slope. The l ine orthogonal tothe gradient is a level line of the graph.

In order to compute the directional derivative we need a unit vector,

n= v

|v| = 1

5(i + 2j) .

The partial derivative off(x, y)at the origin in the direction ofn is

f

n

= lim0

f[(0, 0) +n] f(0, 0)

= lim0

f(/

5, 2/

5) f(0, 0)

=

lim0

sin(/

5 + 4/52) 0

= 1

5.

The geometrical interpretation of the directional derivative suggests that to describe the derivative

of a function of two or more variables we need two pieces of information: the slope of the graph

and the direction along which this slope is measured. In other words, a complete description of the

derivative of a function of two or more variables entails the use of a vector. It turns out that a most

sensible and useful vector is the gradientof the function defined as

f=f

xi +

f

yj+

f

zk (1)

for a function f(x,y ,z). The symbol is called gradornabla and you can think of it as a vectoroperator

=

xi +

yj+

zk

that acts on the function f(x,y ,z)to give Equation (1).


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Example - Evaluate sin(x+y2).

sin(x+y2) =

xsin(x+ y2)i +

ysin(x+y2)j = cos(x+y2)i + 2y cos(x+y2)j.

2.2 Properties

The knowledge of the gradient of a function allows us to compute all the directional derivativesin a

straightforward manner. Provided that the functionfis differentiable then the directional derivativeoffin the direction of the unit vector n is given by

f

n= f n, |n| = 1. (2)

This relation allows us to find the geometrical meaning of the gradient of a function: it points in

the direction ofsteepest ascent. To show this we can use (2) to write the directional derivative of a

functionfas

f

n = f n= |f| |n| cos() = |f| cos(),where is the angle between the gradient and n. The directional derivative, i.e. the slope, is maximalfor = 0, i.e. whennis parallel to f.

On the contrary the surface is level in the directions orthogonal to the gradient: ifnis orthogonal

to f then

n f= 0 = fn

= 0.

To summarise, the knowledge of the gradient allows us to know all the directional derivatives, the

direction of steepest ascent and the level lines of the function.

Exercise 1- Show that (f g) =fg+gf.Exercise 2 - Show that the gradient of a function of the radial distance from the origin, f(r), withr=

x2 +y2 +z2, is equal to

f(r) = df

drr,

whereris a unit vector in the direction of the radius,

r= xi +yj+zk

x2 +y2 +z2.

Use this result to show that the gradient of the function (r) = 1/ris equal to

= rr2

.


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3 Conservative fields

3.1 Definition

If(x,y ,z)is a differentiable function defined in a domain D it is possible to evaluate its gradient atevery point of the domainD. The gradient of, is a vector, function of the coordinates (x,y,z).In other words it is a vector field. More formally:

Definition Let (x,y ,z) be a differentiable function in a domain D. The vector fieldF(x,y,z)defined by

F = (x,y,z)is called aconservativevector field inD. The function is called a potentialfor F inD.

Remark- We will see that a conservative vector field can be used to represent a force with an energy

associated to it. The potentialof the vector field is the potential energy of the force. The minus signused in the definition of conservative field is just a matter of convention: this choice of sign implies

that a body moves under the action of the force from a point of high potential to a point with a lower

value of the potential function.

Example The gravitational force field of a point mass,

F = kmr2

r,

is conservative since we can write

F =

1r

.

Remark - There are infinitely many potentials that correspond to the same vector field, but they all

differ by a constant. Both1(x,y,z) and2 1+ C, whereCis a constant, produce the samevector field:

1= 2= F.This is consistent with the physical interpretation of the potential as the potential energy of a force:

the only physically important quantity is not the energy associated to a point, but the energy difference

between different points: this is independent of the value of the arbitrary constant,

2(r1) 2(r2) = [1(r1) +C] [1(r2) + C] =1(r1) 1(r2).The following theorem completes the analogy between conservative vector fields and forces with

a potential energy. We know form physics that the work done by the gravitational force on a body

that moves from a point with height h1 to a point with height h2 is independent of the path takento go from one to the other. Another way of stating this property is that in the absence of friction a

ball rolling down a slide will climb up to exactly the same height it started from (see Figure 2). The

mathematical equivalent of these statements is that the line integral of a conservative vector field is

independent of the path and depends only on the end points.

Theorem Let Fbe a conservative vector field on a domain D and let be a potential for it, F =. LetA andB be any two points inD and letcbe any curve inD that joins them. Then

c

F ds= (A) (B).


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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

hv

Figure 2: In the absence of friction the ball reaches the same height it started from and its energy is continuallytransformed from potential energy to kinetic energy in the downward part of the slope and from kinetic energy

to potential energy in the upward part of it.

Example- The function(x, y) = xyis the potential of the conservative vector field F = =yi+ xj. Call r(t) the arc of circle of radius 1 centred at the origin that joins A = (1, 0) withB = (1/

2, 1/

2):

r(t) = cos(t)i + sin(t)j, 0 t /4.The line integral ofFalong the pathr is

r

F ds= /40

F[r(t)] drdt

dt=

/40

cos(2t) dt=1

2.

The difference of potentials between starting and ending point isr

F ds= (A) (B) =(0, 0)

12

, 1

2

=

1

2.

3.2 Properties of conservative fields

Given a potential it is straightforward to construct a conservative field: we just need to evaluate the

gradient of the potential. The reverse problem, given a vector field can we ascertain whether it is

conservative and, if so, what is its potential, is more involved. In order to solve it we need to discuss

some more the properties of conservative fields.

It is fairly straightforward to say if a field is not conservative. In fact, suppose thatF(x, y) =F1(x, y)i +F2(x, y)j is a two dimensional conservative vector field with potential:

F = =

F1= x

,

F2= y

,=

F1y

= 2

yx,

F2x

= 2

xy.

The mixed derivatives of are identical,

2

yx=

2

xy = F1

y =

F2x

. (3)

Therefore, ifF(x, y)is a conservative two dimensional vector field then (3)must hold. The inverseof this statement tells us that a field F does not satisfy relation (3)then it cannot be conservative.


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3.3 Potentials of conservative vector fields

Suppose that the fieldF(x, y) = F1(x, y)i+ F2(x, y)j is conservative. There are two methods tofind its potential, i.e. a function (x, y)such that = F.

3.3.1 Integral method

We can use the theorem that say that that the line integral of a conservative vector field on any path cjoining two pointsP0= (x0, y0)andP1= (x1, y1)is equal to the difference of the potentials betweenthe two points,

c

F ds= (x0, y0) (x1, y1),

to define the potential function. Choose a reference pointP0 = (x0, y0)and assign the value of thepotential at this point,

(x0, y0) =C.

The value of the potential at any point P1= (x1, y1)is

(x1, y1) =c

F ds +(x0, y0) =c

F ds +C, (6)

where the line integral is evaluated on any path c that joins P1 with the reference point P0. Equa-tion (6) defines the function potential at every point (x, y): this function is, by definition, thepotential of the vector field F(x, y).

Example - Find the potential of the conservative vector field F(x, y) =yi +xj.We choose the origin as the reference point and we set the potential to be equal to an arbitrary

constantC there:

(0, 0) =C.

To evaluate the potential at a generic point P1 = (x1, y1)we choose as path a straight line r(t)fromthe origin to the point P1parametrised by

r(t) =tx1i +ty1j, 0 t 1.The potential is given by

(x1, y1) =

r

F ds + C=

1

0

F[r(t)] drdt

dt+C=

1

0

(ty1i +tx1j) (x1i +y1j) dt+C=x1y1+C.

3.3.2 Differential method

The potential(x, y)of the conservative fieldF(x, y) =F1(x, y)i+F2(x, y)j must satisfy the twodifferential equations

x= F1(x, y), (7)

y = F2(x, y). (8)


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We can integrate (7)with respect tox, consideringy as a parameter. Its solution is

(x, y) =

F1(x, y) dx+(y), (9)

where(y)is the integration constant: notice that it is constant with respect to x, but may still be afunction of the other variabley . We can now substitute (9)into (8)to obtain an ordinary differential

equation for(y):

d

dy = F2(x, y) +

y

F1(x, y) dx.

Example - Find the potential of the conservative field F =yi +xj.We are looking for a function (x, y)such that

x = y,

y = x.

The solution of the first equation, obtained by integrating overx while consideringy as a constant is

(x, y) = xy+(y).

If we substitute this expression for into the second equation we obtain an ordinary differentialequation for(y):

y = x+ d

dy = x = d

dy = 0 = (y) =C,

where C is an arbitrary real constant. Therefore the potential of the conservative vector field F =yi +xj is

(x, y) =

xy+C,

the same result obtained using the previous method.

4 Stokes theorem

4.1 Introduction

In this section we will discuss how to find a simple condition to determine whether or not a given

vector field is conservative. For example, we know that if a two dimensional vector field, F(x, y) =F1(x, y)i +F2(x, y)j, is conservative then

xF2 = yF1. (10)

However, we also know that this condition is not sufficient, per se. A field can satisfy it and yet not

be conservative. Is it possible to generalise Equation (10), so that it is not only necessary, but also

sufficient? The answer is yes and Stokes theorem tells us what we need to do. However, before being

able to answer this question we need to introduce a first derivative of a vector field, the curl of a

vector field.


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4.2 The curl of a vector field

All the derivatives of vector fields are based on the operator ,nablaor grad. This symbol can be

considered as a vector operator

=

xi +

yj+

zk,

that acts on functions of(x,y,z). The gradient of a scalar function, for example, is the result ofapplying the operatorgradto the function itself:

f=

xi +

yj+

zk

f(x,y,z) =

f

xi +

f

yj+

f

zk.

This same approach can be used to introduce new derivatives, that instead of differentiating

scalar functions, likef(x,y,z), differentiate vector fields. One of these derivatives is the curl of avector field,F(x,y,z).Definition- Thecurlof a differentiable vector field Fis the vector field F. Equivalent ways ofevaluating this derivative are:

F =

xi +

yj+

zk

(F1i +F2j+F3k)

=

i j k

x y zF1 F2 F3

= i

F3y

F2z

+j

F1z

F3x

+ k

F2x

F1y

The curl of a vector field is related to rotation, it measures how the field rotates, swirls at

different points of space. Consider, for example, the rotation of a rigid body with constant angular

velocity, = 0k. The velocity at a point(x,y ,z)is given by

v(x,y,z) = r= y0i +x0j.

The curl of this vector field is

v=

i j k

x y zy0 x0 0

=k0[xx y(y)] = 20k.

The curl of the velocity field is twice the angular velocity of the rigid body.


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F = 0

x

y

Figure 5: A centrally symmetric field has zero curl.

Properties of the curl

1. (gF) = g( F) = g F, whereg(x,y,z) is a scalar function and F(x,y,z) avector field.

2. The curl of a conservative vector field is zero: () = 0.This property is the three dimensional version of equation ( 10). It says that ifFis a conservative

vector field, i.e. F = , then its curl is zero:

Fconservative = F = 0.

Stokes theorem allows us to invert the direction of the arrow, provided that F satisfies some

other properties.

3. The curl of a centrally symmetric field is zero:

F =f(r)r = F = 0.

This results is in line with the interpretation of the curl as a measure of the rotation of a vector

field. A centrally symmetric field does not rotate and its curl is zero (see Figure 5).

4.3 Stokes theorem

We have defined a derivative of a vector field, the curl. Before stating Stokes theorem we need tospecify the surfaces we wish to work with. The first requirement is that the surface is orientable, so

that we can evaluate flux integrals across it. The second requirement is that the surface must have a

boundary, i.e. it must be delimited by a piecewise smooth curve, eventually composed of many bits

and pieces (see Figure6).

Finally we need to define a positive orientation for the boundary and the vector normal to the

surface. The orientation of the boundary ispositiveif walking along it the surface is on our left. The


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x

y

zz

x

y

z

x

y

nn

Figure 6: A sphere is a surface with no boundary: it is not delimited by a curve. The other two surfaceshave boundaries formed by one piece (centre) or two (right). The inner boundary is traversed in the opposite

direction of the outer boundary.

vector normal to the surface is defined to point upwards. This rules implies that the borders of holes

in the surface are traversed in opposite direction to the outer rim, Figure 6, or that the bases of a

cylinder are traversed in opposite ways, Figure7. We are now ready to state Stokes theorem and its

two dimension version, Greens theorem.

Theorem (Stokes) Let S be a piecewise smooth, oriented surface in three dimensions, having unit

normal n and boundary cconsisting of one or more piecewise smooth, closed curves with positiveorientation. Then

c

F ds=S F dS

Theorem(Green) LetRbe a closed region in thexy-plane whose boundaryc consists of one or morepiecewise smooth non self-intersecting closed curves that are positively oriented. IfF =F1i +F2jis a differentiable vector field onR then

c

F ds=R

F2x

F1y

dA.

Example - EvaluateS F dSover the hemispherex2 +y2 +z2 =a2,z 0, whereF(r) =

yi +xzj+yk.

The boundary of the surface is the circumference x2 +y2 =a2,z = 0, (see Figure8)which canbe parametrised by the curve

r(t) =a cos(t)i +a sin(t)j 0 t 2 .Stokes theorem states that

S F dS=

c

F ds=

2

0

dt F[r(t)] drdt

=

2

0

dt [a sin(t)i +a sin(t)k] [a sin(t)i +a cos(t)j]dt= a2

Remark - A consequence of Stokes theorem is that the flux of the curl across all the surfaces that

have the curve c(t) as border is the same. Therefore, instead of evaluating the flux of F overthe hemisphereS, we can evaluate the flux across the baseB (a much simpler integral) and obtain thesame result:

S F dS=

B F dS= a2


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n

B

S

r

x

y

z

Figure 7: The bases of a cylinder are traversedin opposite directions. The vector normal to the

surface points outwards.

Figure 8: The flux of Facross Sis the sameas that acrossB and as the line integral along r.

4.4 Stokes theorem and conservative fields.When studying conservative fields we have established that a field on a connected domain D is con-

servative if and only if

c

F ds= 0

for all closed paths c in D. This theorem while giving a complete characterisation of a conservativevector field is rather cumbersome to use: there are infinite closed paths. Stokes theorem allows us to

get around this problem.

IfDis a simply connected domain then any closed path c is the border of an orientable surface S.To each of these surfaces is possible to apply Stokes theorem,

c

F ds=S F dS.

If F = 0everywhere in a simply connected domainD thenS F dS= 0, for all surfaces =

c

F ds= 0, for all closed path = Fis conservative.

On the other hand we have already stated that if F is conservative then its curl is zero. We

therefore a simple criterion to establish whether a field is conservative:

A vector fieldFdefined on a simply connected domainD is conservative ifand only if

F = 0 inD.


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For example, the Biot-Savart field,

F =yi +xj

x2 +y2 , (x, y) = (0, 0),

is not conservative even though F = 0, because its domain of definition, thexy-plane minus theorigin, is not simply connected.

5 Divergence and Divergence theorem

There are two types of derivatives of a vector field. We have seen the curl, Fobtained by takingthe cross product of the grad operator, with a vector fieldF. By taking the dot product instead, we

obtain a second type of derivative, the divergence: F.Definition- TheDivergenceof a differentiable vector fieldF(x,y,z)is the scalar function F:

F = =

xi +

yj+

zk

(F1i +F2j+F3k)

= F1

x +

F2

y +

F3

x.

While the curl F is a vector, the divergence is a scalar function.The key to the physical meaning of the divergence of a vector field is given by the divergence

theorem. The divergence theorem relates surface integrals of vector fields to volume integrals of their

divergence. It allows

To get a clearer idea of the physical meaning of divergence. To express physical laws as relations between derivatives of vector fields. To find the value of a surface integral by evaluating a volume integral (the latter is usually

simpler than the former).

Theorem - LetF(r) be a a vector field defined in a volume V. LetSbe the orientable surface thatencloses the volumeV. IfF(r)is differentiable inV then

V FdV =

S

F dS.

In words, the volume integral of the divergence is equal to the flux of the vector field.

Example- Evaluate the flux of the vector field F(x,y,z) =yi across the sphere of radius one centredat the origin and compare the result with the volume integral of the divergence ofF.

The sphere is represented by the vector function

r(, ) = cos()sin()i + sin()sin()j+ cos()k, 0 2, 0 .The surface element is

dS= [cos()sin()i + sin()sin()j+ cos()k]sin()dd.


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The flux ofF =yiis

S

F dS=S

F[r(, )] dS=

2

0

d

0

d [sin()sin()i] [cos()sin()i + sin()sin()j+ cos()k]sin() =

20

d 0

dcos()sin()sin3() = 0.

On the other hand the divergence ofF =F1i +F2j+F3kis

F = F1x

+ F2

y +

F3z

= y

x= 0,

so that the volume integral of the divergence is

V FdV =

V

0 dV = 0

as stated by the divergence theorem.

6 Physical Applications of the divergence theorem

6.1 Divergence and the sources of vector fields

The divergence of a field is associated with the sources and the sinks of the field (see Figure 9). A

sourceis a region in space from which field lines flow outward (for example, the neighbourhood of a

positive charge or a source of water). Asinkis a region of space where the field lines converge to

(for example, the neighbourhood of a negative charge or a hole where water disappears).

Consider a small region in a vector field . The volume integral of the divergence is approximately:

V FdV FV ,

where V is the volume of the small region. The divergence theorem states that the value of thedivergence is related to the flux across the surface of the region:

V FdV =

S

F dS = FVS

F dS.

The field flows outward form a source, its flux is positive and so is its divergence. The field flows

inward towards a sink, its flux is negative and so is its divergence. A positive value of the divergence

is associated to the sources of the field, a negative one to its sinks. In other words, the divergence of a

vector field is a measure of the inflow or outflow of the field from a small region of space.

6.2 Divergence and Conservation laws

The divergence of a vector field is an extremely useful tool to express conservation laws. The total

mass of fluid contained in a volume Vcan change only if there is a mass outflow or inflow across the


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Source Sink

Positive Divergence Negative Divergence

Figure 9: Field lines and equipotential for two equal and opposite point charges. The divergence is negativearound the negative charge (sink), positive around the positive one (source).

surfaceSthat delimits the volume. This statement can be expressed in integral form as:

d

dt(Mass insideV)= (Mass outflowing throughS) =

V

t +

S(v)

dS= 0 ,

where (r, t) is the mass density of the fluid and v(r, t) is the speed of a fluid particle at the pointr = xi+yj+ zk. This relation involves integrals. It can be put in a differential form (an equationthat contains only derivatives of the field), by making use of the divergence theorem:

S(v) dS=

V (v)dV .

The mass conservation law becomes:V

t + (v)

dV = 0

This relation must hold for all volumesV. This is possible only if the integrand itself is zero:

t+ (v) = 0 .

This equation is the mass conservation law expressed in differential form. It shows that the divergence

of a field is usually associated with a conserved quantity. For example, an equation exactly analogous

to this expresses the charge conservation law in electromagnetism.


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6.3 Divergence and electrostatic

Gauss law states that the flux of the electric field across a closed surface S is equal to the chargecontained in the volumeVenclosed by the surface:

1

0

V

(r)dV =

S

E dS,

where(r)is the charge density and E(r)is the electric field. This equation can be put in differentialform by applying the same procedure as in the previous example:

S

E dS=V EdV =

V

(r)

0 E

dV = 0

This relation must hold for all volumes. This is possible only if the integrand is zero:

E= 0

This is the first of Maxwells equations. This equation can be written in a simpler form: since the

electrostatic fieldE is conservative, E =, we can rewrite Maxwells equation as an equationfor the potential, . The advantage of doing so is that only scalar functions appear in it, instead ofvectors.

() = 0

2= 0

Poissons equation

where the symbol 2(calledLaplacian) means

2= 2

x2+

2

y2 +

2

z2.

The equation for the potential in absence of charges, = 0, is

2= 0 Laplace equation

These last two equations appear in many branches of physics, from the theory of elasticity, to electro-

magnetism, to quantum mechanics.

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Divergence Curl Vector Feilds