Vector Calculus ApplicationsŽ

1. Introduction

The divergence and Stokes’ theorems (and their related results) supply fundamental tools which can be used to derive equations which can be used to model a number of physical situations. Essentially, these theorems provide a mathematical language with which to express physical laws such as conservation of mass, momentum and energy. The resulting equations are some of the most fundamental and useful in engineering and applied science.

In the following sections the derivation of some of these equations will be outlined. The goal is to show how vector calculus is used in applications. Generally speaking, the equations are derived by first using a conservation law in integral form, and then converting the integral form to a differential equation form using the divergence theorem, Stokes’ theorem, and vector identities. The differential equation forms tend to be easier to work with, particularly if one is interested in solving such equations, either analytically or numerically.

2. The Heat Equation

Consider a solid material occupying a region of space V . The region has a boundary surface, which we shall designate as S. Suppose the solid has a density  and a heat capacity c. If the temperature of the solid at any point in V is T.r; t/, where r x{ y| zkO is the position vector (so that T depends upon x, y, z and t), then theE total heatE eergyD O containedC O C in the solid is • cT dV : V

Heat energy can get in or out of the region V by flowing across the boundary S, or it can be generated inside V . Let’s suppose that the heat flux vector is called q. This vector measures rate of energy flow past a point per unit area. Thus, if we have a small elementE of surface area dS with outward unit normal vector n, the rate of energy flow outward through this element of surface area is q n dS. Integrating over theO entire surface, the total rate of energy flow (i.e., the flux) out of the regionE
O is therefore — q n dS : E
O This is shown schematically in Fig. 1.

Ž c W. L. Kath, 2004, 2005, 2006, 2010, 2011, 2015. E-mail: [email protected] Version 1.3, February 2015

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z q

n V S

y

x

Figure 1: Schematic diagram indicating the region V , the boundary surface S, the normal to the surface n, and the heat flux vector q. O E

Let’s suppose that the rate at which heat energy per unit volume is being generated is Q. Then, the total rate at which heat is being generated inside V is • Q dV : V

Conservation of energy then requires that the rate at which the energy within the region V changes and the rate at which energy crosses the boundary of region must balance, i.e.,

d • — • cT dV q n dS Q dV : dt V C E
O D V The signs of the terms must appear in this manner; assuming that Q 0 for the moment, if the flux of energy out of the region is positive, then the total energy inside theD region must be decreasing, hence its derivative will be negative. Similarly, if the flux of heat through the boundary is zero, but heat is being generated inside V , then the temperature must be increasing.

To simplify this, we first take the time derivative inside the volume integral:

d • • @ cT dV .cT / dV : dt V D V @t We can do this because the region V over which the integral is being taken is independent of time; V is the same region for all time. Thus, at any fixed time, integration over V just gives a single number, i.e., the energy inside V at that time. The energy will change with time because T (and possibly  and c) changes with time. Note that the time derivative must be turned into

2 Winter 2015 Vector calculus applications Multivariable Calculus a partial derivative when it is moved inside the volume integral, however, because the functions being integrated depend upon both position and time. Similarly, we apply the divergence theorem to the flux integral: — • q n dS q dV : E E
O D V r
E

The result is • @ • • .cT / dV q dV Q dV 0 ; E V @t C V r
E V D or, equivalently, • Ä @  .cT / q Q dV 0 : E V @t C r
E D It would be nice to say at this point that the quantity in brackets is zero; this would allow us to get rid of the integral. This would be true, but the right argument is needed, because an integral can be zero either because the integrand is zero or because it is equally positive and negative, and when integrated there is perfect cancellation.

In the present case, however, there is one more fact: the integral must be zero for arbitrary volumes V . This forces the integrand to be zero everywhere. The proof is by contradiction. First, assume that the integrand is not zero everywhere. Pick a point at which it is positive, and then choose a region V that is entirely within the region where it’s positive. Since we are integrating a quantity that is entirely positive, the resulting integral must be zero. This can’t be true, however, since we know the integral must be zero. Therefore, we must have @ .cT / q Q 0 : @t C rE
E D This is the partial differential equation form of the fundamental principle of energy conservation for heat transfer.

More can be done if one has an explicit result for the heat flux q in terms of the temperature. In many cases, observations show that the heat flux is given by Fick’sE law,

q k T; E D rE i.e., the heat transfer is proportional to the negative of the temperature gradient. Thus, heat tends to flow from hotter regions to colder regions. While this is true in most cases, it is not always true; in particular, if phase transitions are possible it can be violated.

If we use Fick’s law in the equation for conservation of heat energy, it becomes @  Á .cT / k T Q: @t D rE
rE C 3 Winter 2015 Vector calculus applications Multivariable Calculus

Finally, if , c and k are all constant, and Q 0, this equation simplifies to the heat equation D @T  2T; (1) @t D r where  k=c is the thermal diffusivity. D A special case of this equation results if the object is in thermal equilibrium, i.e., the temperature doesn’t change with time:

2T 0 : (2) r D

This is known as Laplace’s equation.

3. Fluid mechanics

3.1. Conservation of mass

Conservation laws are also the basis for the equations of fluid mechanics. Let  be the fluid density and v be the fluid velocity. Then the mass in a region V is given by E •  dV ; V and the rate at which mass leaves the region through its boundary S is — v n dS E
O I see Fig. 2 for a diagram of the situation. Here v is the mass flux vector; this has units of kg=m3 m=sec kg=m2sec, i.e., mass per unit area perE second.  D The rate of change of the mass in V must be balanced by the rate at which mass leaves through S, d • —  dV v n dS 0 : dt V C E
O D Pulling the time derivative inside the V integral, and converting the surface integral into a volume integral using the divergence theorem, we have

• Ä@  v
dV 0 : E V @t C r
E D

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v

n

V S

Figure 2: Schematic diagram indicating the region V , the boundary surface S, the normal to the surface n, the fluid velocity vector field v, and the particle paths (dashed lines). O E

As before, because the region V is arbitrary, we must have the terms between the brackets be identically zero, which gives

@ v
0 : (3) @t C rE
E D This is the partial differential equation form of conservation of mass.

Equation 3 can be simplified somewhat by using the vector identity .v/  v  v rE
E D rE
E C rE
E D v   v, which gives E
rE C rE
E @ v   v 0 : (4) @t CE
rE C rE
E D

The combination @=@t v occurs so much in fluid mechanics that it is given a special notation, CE
rE @ D v : @t CE
rE Á Dt This is called the material or substantial derivative, and physically it represents taking the time derivative ‘following the fluid’, i.e., taking the time derivative of something transported along by the fluid. One can see this by considering .x; t/ not at a fixed position x, but instead .x.t/; t/, where dx=dt v; because the position x movesE at the fluid velocity vE, this is the densityE of a specific fluidE element.DE Taking the time derivativeE using the chain rule, E d dx @ @ dx @ D .x; t/  E E  v  : dt E D rE
dt C @t D @t C dt
rE D @t CE
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In any event, with this notation we have

D  v 0 : (5) Dt C rE
E D

This has a nice physical interpretation: if v > 0, so that the fluid velocity is locally expanding, rE
E then D=Dt < 0, i.e., the local density is decreasing. Similarly, if v < 0, so the fluid velocity E is locally compressing, then D=Dt > 0 and the density is increasing.r
E

We can also consider the case of an incompressible fluid: if  is constant, then D=Dt 0 and therefore D

v 0 : (6) rE
E D Water is an example of a fluid that is almost incompressible.

3.2. Conservation of momentum

The next main conservation law is momentum. Actually, of course, the law is that the time rate of change of momentum is equal to force, d(momentum)/dt = force. To apply it to a fluid, we need to identify all of the sources of momentum and all of the forces involved.

First, the momentum density (per unit volume) of the fluid is v. Therefore, the momentum in a volume V of fluid is E • v dV : V E Of course, what we really need is the time rate of change of this momentum, d • v dV : dt V E Another way that the momentum in V can change is if the fluid transports momentum out of the region across the boundary S. The flux of momentum out of V is then — v .v n/ dS : E E
O This is similar to the previous case of conservation of mass, where we had mass density times the normal component of the velocity, giving mass flux. In this case, we have momentum density times the normal component of the velocity, giving momentum flux.

We also need the forces acting on V . First of all, we have the net pressure force, which is — p n dS ; O

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since the pressure acts normally to each element of the surface (with an inward force when the pressure is positive, hence the minus sign). Similarly, if we have any body forces, such as gravity, we have to include them. It’s traditional to use fE as the body force per unit mass, so that • fE dV V is the total body force.

Putting everything together, we have conservation of momentum:

d • — — • v dV v .v n/ dS p n dS fE dV : (7) dt V E C E E
O D O C V

Note that other forces have been neglected, such as forces due to viscosity, where a bit of fluid exerts a drag force on nearby fluid elements.

The next step should be to convert the surface integrals to volume integrals using the divergence theorem. Unfortunately, the form of the surface integrals is not right for the divergence theorem, because the integrands of the surface integrals are vectors, not scalars. Thus, we need to somehow modify the divergence theorem.

Let’s look first at the force due to pressure. Here, we are missing a vector. We can fix this by taking the dot product of the integral (neglecting the sign for the moment) with a constant vector a: E — — — a p n dS p a n dS .p a/ n dS : E
O D E
O D E
O Now we can apply the divergence theorem: — • .p a/ n dS .p a/ dV E E
O D V r
E

Now we use the vector identity .p a/ p a p a. This is equal to p a, since a 0, E E E E E because a is a constant vector. Thus,r
E D r
E C r
E r
E r
E D E — — • • • a p n dS .p a/ n dS .p a/ dV p a/ dV a p dV : E E E E
O D E
O D V r
E D V r
E DE
V r Thus, we have Ä — •  a p n dS p dV 0 : E E
O V r D Because a is an arbitrary vector, this means that the terms in brackets must be zero. Therefore E

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— • p n dS p dV : (8) E O D V r

This is an extension of the divergence theorem.

Next, consider the integral for the flux of momentum, — v .v n/ dS : E E
O

We can’t apply the divergence theorem because of the extra v. It’s also not so easy to do something like what we did for the pressure force, so instead let’s expandE the extra v into components, v P3 E E D n 1 invn: D O 3 — X — v .v n/ dS in vn .v n/ dS : E E
O D O E
O n 1 D Now we can apply the divergence theorem to the integral,

3 — 3 • X X
in vn .v n/ dS in vnv dV O E
O D O rE
E n 1 n 1 V D D Applying vector identities, we have




vnv vn v vn v vn v v vn vn v rE
E D rE
E D rE
E C rE
E D E
rE C rE
E The result of the divergence theorem then simplifies to

3 • • X  Á
 Á
in  v vn vn v dV  v v v v dV O E
rE C rE
E D E
rE E CE rE
E n 1 V V D

Collecting all of the converted terms in Eq. (7) and dropping the integrals because the region is arbitrary, we then have @  Á .v/  v v v .v / p f : @t E C E
rE E CE rE
E D rE C E This can be further simplified since

@ @ @v @v .v/ v  E v .v /  E ; @t E DE @t C @t D E rE
E C @t where for this last part we have used conservation of mass, Eq. (5), to eliminate @=@t. Cancelling the terms, this then gives

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Ä@v  Á   E v v p f ; (9) @t C E
rE E D rE C E

or, alternatively

Dv  E p f : (10) dt D rE C E

This is the equation for conservation of momentum, the fluid-mechanical equivalent of Newton’s law ma F . Together with the equation for conservation of mass, they are called the Euler E D E equations . Note that to complete the equations, the force fE and an equation for the pressure p need to be specified. Sometimes the pressure can be given as a function of the density; this is called the equation of state. A fully complete set of equations, however, also has to include one additional equation, that for conservation of energy.

3.3. Irrotational flows and Bernoulli’s equation

First, use the last vector identity in the Appendix with F v and G v to write E DE E DE  Á 1 v v .v v/ v . v/ : (11) E
rE E D 2rE E
E E  rE  E

Let’s also write ! v; here ! is called the fluid vorticity, and measures the local rotation rate E of the flow. Then,E D conservationr  ofE momentum, Eq. (9), becomes

@v 1 p E .v v/ v ! rE f : (12) @t C 2rE E
E E  E D  C E Now, take the curl of this equation. We need another vector identity,  Á  Á .v !/ . !/ v . v/ ! ! v v !: rE  E  E D rE
E E rE
E E C E
rE E E
rE E

Of course, ! 0. We then have for the curl of the equation (remember the curl of a gradient E is zero) r
E D ! @!  Á  Á p E v ! ! v . v/ ! rE f : @t C E
rE E D E
rE E rE
E E rE   C rE  E This can also be written D!  Á 1 E ! v . v/ !  p f : Dt D E
rE E rE
E E C 2 rE  rE C rE  E

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If  p 0, which will happen if  is a constant or if p is a function of , and if f 0, rE  rE D rE  E D which will happen if fE is a conservative force field, then D!  Á E ! v . v/ ! Dt D E
rE E rE
E E The interesting thing about this equation (called the vorticity equation), is that if ! is zero initially, then it shows that D!=Dt 0, so that the vorticity won’t change. Thus, ! willE remain zero for all time (or until someE sourceD of vorticity is introduced into the fluid). E

If the vorticity ! 0, this means that v 0. Such a fluid flow is said to be irrotational. E D rE  E D In this case, we know that there is a scalar potential ' such that v '. The advantage of this is E that it reduces three unknowns (the components of v) to one (theE potentialD r '). E In the case when the density  is constant, conservation of mass becomes v 0. As described E earlier, this flow is said to be incompressible. If, in addition, the flow is alsor
EirrotationalD , so that v ', then the equation of conservation of mass becomes Laplace’s equation, E D rE 2' 0 : r D This equation is a lot simpler than the original version of conservation of mass.

Another interesting result is that happens to conservation of momentum in this case. Substituting v ' in Eq. (12), we have E D rE @ 1  Á Âp à . '/ ' 2 U 0 ; @t rE C 2rE jrE j C rE  C rE D

where we have assumed f U . This can then be written E D rE Ä@' 1 p  ' 2 U 0 : rE @t C 2jrE j C  C D For the gradient of a function to be zero it must be constant. Therefore,

@' 1 p ' 2 U constant : (13) @t C 2jrE j C  C D

This is Bernoulli’s equation. It can be thought of as an equation similar to conservation of energy; note that in the case with no body force (U 0), where the fluid speed (i.e., ' ) is small, the E pressure will be large, and where the fluid speedD is large, the pressure will be small.jr j

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4. Maxwell’s equations

The equations describing the behavior of electric and magnetic fields can also be derived in a manner similar to that done for fluids. In each case, the starting point is a principle based upon observation. Note that fundamentally, conservation of mass, momentum and energy are princi- ples that are based upon observation. The situation is much the same for electromagnetics; the principles are just a little different.

Historically, one of the first results to be obtained was that for the force between charged objects. In particular, the force between two metal spheres with charges Q1 and Q2 was observed to be

Q1Q2 r F O ; E D 4 r 2 where r is the distance between the centers of the two spheres, r is a unit vector pointing from O Q1 to Q2 when we want the force on Q2 due to the charge on Q1, and  is a constant (called the permittivity, or the dielectric constant).

If the second object is replaced with a test charge q, then the force on the test charge due to the charge Q1 is Q1 r F qE; where E O : E D E E D 4 r 2 Here E is the electric field produced by the charge Q ; in this case r is a unit vector that points E 1 away from its position. O

In the case where more than one charge is present, or there is a distribution of charge with density .r; t/, experimentally one can verify Gauss’s law: E — E n dS Q; (14) E
O D

i.e., the total flux of the electric field out through the surface S is proportional to the total charge Q. Furthermore, if one has a distribution of charge with density .r; t/, then the above becomes E — • E n dS  dV : (15) E
O D V

Applying the divergence theorem to the flux integral, and noting that the volume V is arbitrary, this becomes  Á E  : (16) rE
E D

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The vector D E is called the electric displacement, or the electric flux density (E is still the E E E electric field). ThisD is the first of Maxwell’s equations.

A similar result holds for the magnetic flux density, B, except there are no free magnetic charges. E This means that we have — B n dS 0 ; (17) E
O D

or equivalently,

B 0 : (18) rE
E D This is the second of Maxwell’s equations.

The last two of Maxwell’s equations don’t follow from the divergence theorem, but rather from Stokes’ theorem. First of all, we have Faraday’s law:

I @ “ E dr B n dS : (19) E E C
E D @t S
O

The line integral of E is called the electromotive force. Essentially, it measures the increase or E decrease in the electric potential (i.e., voltage) when going around the contour, since E is the gra- E dient of the potential. Faraday’s law says that the voltage drop around the contour is proportional to the time rate of change of magnetic flux passing through a surface spanning the contour. This was discovered experimentally by measuring electric fields (or rather, voltage across a gap in a circular wire) in the presence of time varying magnetic fields.

To eliminate the integrals in the above, we apply Stokes’ theorem: I “ E dr n E dS : E E E C
E D S O
r  We therefore have " # “ @B n E E dS 0 : E E S O
r  C @t D Since the contour C is arbitrary, this means the surface S must be arbitrary. By the same argument used for the divergence theorem, the only way we can get zero is for the terms in the brackets to be zero, i.e., @B E E : rE  E D @t This is the differential equation form of Faraday’s law, and is the third of Maxwell’s equations.

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To derive the last of Maxwell’s equations, we start from Ampere’s law, which says that

I B “ E dr J n dS : (20) E C 
E D S
O

Here  is the magnetic permeability, J is the current density, C is an arbitrary closed contour and E S is an open surface that has C as its boundary. The basic form of this law was discovered from experiments. Basically, this is the mathematical statement of the well-known fact that currents generate magnetic fields. It is traditional to define H B= to be the magnetic field intensity. E D E To eliminate the integrals in the above, we apply Stokes’ theorem: I “ H dr n H dS : E E E C
E D S O
r  We therefore have “ h i n H J dS 0 : E E E S O
r  D Since the contour C is arbitrary, this means the surface S must be arbitrary. By the same argument used for the divergence theorem, the only way we can get zero is for the terms in the brackets to be zero, i.e., H J: rE  E D E Here, the current density J is generated by movement of charges. E There is a problem with this equation, however. If we take the divergence of both sides, we get J 0, since . H/ 0. This doesn’t make sense, though. If we think about conservation E E E E E ofr
chargesD like conservationr
r  ofD heat or fluid mass, we can write down a conservation law for charge d • “  dV J n dS 0 ; E dt V C S
O D which when applying the divergence theorem and a similar argument as before gives @ J 0 : @t C rE
E D Thus, there is an inconsistency with the current differential form of Ampere’s law (i.e., J 0). rE
E D Maxwell’s tremendous insight was to realize that if one differentiates with respect to time the equation for the divergence of the electric displacement field, Eq. (16), one gets ! @D @ E ; rE
@t D @t

13 Winter 2015 Vector calculus applications Multivariable Calculus and thus when put in the equation of conservation of charge we get ! @D J E 0 : rE
E C @t D

He then realized that the time rate of change of the electric displacement field is essentially a current, called the displacement current, and that Ampere’s law should be generalized to be

@D H J E : (21) rE  E D E C @t

This is the differential equation form of Ampere’s law, and is the third of Maxwell’s equations.

Note that if one now takes the divergence of the above equation, one gets

@ @ 0 J D J ; D rE
E C @t rE
E D rE
E C @t bringing back directly the equation for conservation of charge.

In summary, we have the full set of Maxwell’s equations:

@B D  ; E E ; rE
E D rE  E D @t (22) @D B 0 ; H J E ; rE
E D rE  E D E C @t where D E and B H. One more thing: in the case of a resistive material, J E. E D E E D E E D E As a last application, let’s consider Maxwell’s equations in free space, with no free charges ( 0 and J 0). In addition, in this case  and  are constant. Then Maxwell’s equations E becomeD D

@B E 0 ; E E ; rE
E D rE  E D @t @E B 0 ; B  E : rE
E D rE  E D @t

Let’s eliminate the magnetic field B. We do this by taking the curl of Faraday’s law: E @  Á 1 @2E . E/ B E ; rE  rE  E D @t rE  E D c2 @t 2

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where we used Ampere’s law for B, and have written  1=c2. The constant c turns out to rE  E D be the speed of light. We can also use a vector identity to simplify . E/, i.e., rE  rE  E . E/ . E/ 2E: rE  rE  E D rE rE
E r E Thus, we get 1 @2E 2E E ; r E D c2 @t 2 because E 0. This is known as the wave equation. rE
E D To understand the behavior of the wave equation’s solutions, consider the very special case when

E E1.x; t/i. Then the equation becomes E D O @2E 1 @2E 1 1 : @x2 D c2 @t 2 It is relatively easy to verify that a solution of this equation is

E1.x; t/ f .x ct/ ; D where f . / is an arbitrary function. The solution is a wave propagating in the positive x direction with velocity
c. To see this, let x ct  and note that if f ./ has its maximum at  0, then the position of this maximum will be atDx ct 0, or x ct. Thus, the maximum willD move in the positive x direction with speed c. D D

Another result follows from Maxwell’s curl equations for the electric field E the the magnetic E field B, E @B 1 @E E E and B 0 E : rE  E D @t 0 rE  E D @t

Assuming 0 and 0 are constant, using the vector identity  Á  Á  Á F G G F F G rE
E  E D E
rE  E E
rE  E with F E and G B=0, we have E D E E D E 1 1  Á 1  Á .E B/ B E E B 0 rE
E  E D 0 E
rE  E 0 E
rE  E Now we use Maxwell’s curl equations. This gives ! ! 1 1 @B @E .E B/ B E E 0 E 0 rE
E  E D 0 E
@t E
@t

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Finally, we use ! @B @ Â1 Ã @ Â1 ˇ ˇ2Ã B E B B ˇBˇ E
@t D @t 2 E
E D @t 2 ˇ Eˇ

and a similar result for E. Therefore E 1 @ Ä 1   .E B/ B 2 0 E 2 : 0 rE
E  E D @t 20 j Ej C 2 j Ej or 1 @ Ä 1   .E B/ B 2 0 E 2 0 : 0 rE
E  E C @t 20 j Ej C 2 j Ej D

The quantity .E B/=0 is the electromagnetic energy flux, also called Poynting’s vector, and E  E 1  B 2 0 E 2 20 j Ej C 2 j Ej is the electromagnetic energy density. If we integrate the above equation over an arbitrary volume,

@ • Ä 1   • 1 B 2 0 E 2 dV .E B/ dV 0 ; E E E E E @t V 20 j j C 2 j j C V 0 r
 D and then convert the last volume integral to a surface integral using the divergence theorem, we get

@ • Ä 1   — 1 B 2 0 E 2 dV .E B/ n dS 0 : E E E E @t V 20 j j C 2 j j C 0 
O D This result is a conservation law for electromagnetic field energy. It says that the time rate of change of the electromagnetic energy inside any volume V must be balanced by the flux of energy out of that region, where energy flux is defined in terms of Poynting’s vector.

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A. Appendix: the main theorems and formulas

Divergence Theorem: — • F n dS F dV E E E
O D V r

Stokes’ Theorem: I “ F dr n F dS E E E C
E D S O
r 

Vector Operators & Identities:  Á  Á 'F ' F ' F; 'F ' F ' F; rE
E D rE
E C rE
E rE  E D rE  E C rE  E  Á  Á div curl F F 0 ; curl grad ' ' 0 ; E D rE
rE  E D D rE  rE D  Á  Á F F 2F; rE  rE  E D rE rE
E r E  Á  Á  Á F G G F F G ; rE
E  E D E
rE  E E
rE  E  Á  Á  Á  Á  Á F G G F F G G F F G; rE  E  E D rE
E E rE
E E C E
rE E E
rE E  Á  Á  Á  Á  Á F G F G G F F G G F: rE E
E D E  rE  E C E  rE  E C E
rE E C E
rE E

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