physical applications of vector calculus

8/7/2019 Physical Applications of Vector Calculus

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SOME PHYSICAL APPLICATIONS OF VECTOR CALCULUS

1. Fundamental Theorems of Vector Calculus

Let us first recall the fundamental theorems of vector calculus. They will beused many times in what follows.

Theorem 1.1. Let γ be an oriented curve in R3 with initial and final points p0

and p1, respectively. Let h(x,y,z) be a scalar function. Then

(1.1)

γ

∇h · dr = h(p1)− h(p0)

Theorem 1.2. Let M be a oriented surface in R3 with boundary given by the closed

curve γ , with orientation induced from that of M . Let F(x,y,z) be a vector field.

Then

(1.2)

M

(∇×F) · ndS =

γ

F · dr

Theorem 1.3. Let E be a bounded solid region in R3 with boundary given by the

closed surface M , with the outward pointing orientation. Let F(x,y,z) be a vector

field. Then

(1.3)

E

(∇ · F) dV =

M

F · ndS

We also have the following two theorems which characterize conservative andsolendoidal fields, respectively:

Theorem 1.4. A vector field F in R3 is said to be conservative or irrotational if

any of the following equivalent conditions hold:

∇×F = 0 at every point.

γ

F

·dr is independent of the path joining the same two endpoints.

γ

F · dr = 0 for any closed path γ .

F = ∇h for some scalar potential h.

In fact this theorem is true for vector fields defined in any region where all closedpaths can be shrunk to a point without leaving the region.

1



2 SOME PHYSICAL APPLICATIONS OF VECTOR CALCULUS

Theorem 1.5. A vector field F in R3 is said to be solenoidal or incompressible if

any of the following equivalent conditions hold:

∇ · F = 0 at every point. M

F · ndS is independent of the surface M having the same boundary curve.

M

F · ndS = 0 for any closed surface M .

F = ∇×A for some vector potential A.

Similarly, this theorem is actually true for vector fields defined in any regionwhere all closed surfaces can be shrunk to a point without leaving the region.

The above two theorems should look very similar. Everything is shifted up byone dimension and the curl is replaced by the divergence, but the theorems areidentical in form.

2. Fluid Dynamics

Let v(x,y,z,t) be a time dependent vector field whose value at any point givesthe velocity of a fluid at that point in space and time. Note that since fluids (liquidsand gases) are not rigid like solids, different parts of the fluid can be moving atdifferent velocities. Similarly let ρ(x,y,z,t) denote the density of the fluid, a scalarquantity. We can compute the total mass m(t) of a three dimensional boundedsolid region R of the fluid by integrating the density over R:

m(t) =

R

ρ(x,y,z)dxdydz

The rate of change of the mass of the fluid in the region R is given by the timederivative of this expression:

dmdt

= ddt

R

ρdxdydz =

R∂ρ∂t

dxdydz

where we can differentiate under the integral sign since we are assuming that theregion R in question is not changing with time. Now the mass of the fluid inthe region R can change only because of fluid entering or leaving R through itsboundary surface M . The rate of flow of fluid out through the surface M is givenby the flux integral of ρv over M . Note that the flux of v gives the rate of volumeflow, and we need to multiply by the density at each point to get a rate of massflow. Now the mass m(t) will decrease if fluid is flowing outward , so we need aminus sign:

dm

dt= −

M

ρv · ndS

Setting equal the two expressions for the rate of change of mass flow, and using the

divergence theorem 1.3, we obtain:

(2.1)

R

∂ρ

∂tdxdydz +

M

ρv · ndS =

R

∂ρ

∂t+∇ · (ρv)

dxdydz = 0

Since this must hold for any region R, the integrand must be identically zero.This yields the equation of continuity :

(2.2)∂ρ

∂t+∇ · (ρv) = 0



SOME PHYSICAL APPLICATIONS OF VECTOR CALCULUS 3

This is the mathematical formulation of conservation of mass.If the fluid is incompressible, then the density ρ is a constant, independent of po-

sition and time, and equation 2.2 reduces to∇·

v = 0, which is the historical reasonfor calling such vector fields incompressible. In this case, applying the divergencetheorem to ∇ · v = 0 tells us that the flux through any closed surface M is zero.This makes sense physically because since the fluid is incompressible, it cannot bepiling up inside the region, so whatever volume of fluid goes in must come out andhence the total flux must be zero.

We can also use Stokes’ Theorem 1.2 to calculate the circulation of the fluidabout a closed curve γ . This is just the line integral of v over γ , which we canrewrite as

M

(∇× v) · ndS for any surface M which has γ as boundary. Thisis a measure of the fluid’s tendency to circulate around this path. If the fluid isirrotational, ∇× v = 0, and the circulation is zero. Hence we see here the reasonfor calling such fields irrotational.

We can also use vector calculus to determine the equation of motion for the

fluid, which is governed by Newton’s second law. The time rate of change of thetotal momentum of the fluid must equal the total force acting on the fluid. Themomentum in a solid bounded region can change due to flow of the fluid out of the region, due to the pressure exerted on the fluid inside by the rest of the fluidexterior to the region, and by various external forces such as gravity or electricityand magnetism, in the case of charged fluids. This situation occurs in the interiorsof stars, for example. An analysis that is similar to that which led to equation 2.2but somewhat more involved yields the classical equation of motion for fluids:

(2.3) ρ∂ v

∂t+ ρ (v · ∇) v = −∇P + ρF

where P is the pressure of the fluid and F is the external force density. This isthe Navier-Stokes equation . This is an example of a nonlinear differential equation.Even today, we know very little about the behaviour of solutions to this equation,which is part of the reason why the weather is so difficult to predict. Phenomenalike turbulence, tornadoes, and whirlpools are mathematical consequences of thenonlinearity of this equation.

3. Electricity and Magnetism

We begin by stating Maxwell’s equations of electromagnetism. We will not con-cern ourselves here with the physical derivations of these equations and insteadexamine their mathematical consequences. Let E(x,y,z,t) and B(x,y,z,t) denotethe electric and magnetic fields in space, respectively. These depend on both po-sition and time, in general. Further, we denote by ρ(x,y,z,t) the charge density

and J(x,y,z,t) the current density in space. Note the current density is a vectorfield, since a current is given by both a magnitude and a direction. Here are theequations:




∇ ·E =

ρ

ǫ0

Gauss’s Law(3.1)

∇×E = −∂ B

∂tFaraday’s Law(3.2)

∇ ·B = 0(3.3)

∇×B = µ0J + µoǫ0

∂ E

∂tAmpere-Maxwell Law(3.4)

Here ǫ0 = 8.85×10−12 C 2

Nm2 is the permittivity of free space and µ0 = 4π×10−7 Ns2

C 2

is the permeability of free space. These are just constants. In different systemsof units they would have different values, and one can choose units in which theydon’t appear in the equations at all. So we see in particular that the magnetic fieldB is always solenoidal, and can hence be written as the curl of a vector potential

B = ∇×A. Hence we know from theorem 1.5 that the magnetic flux through any

closed surface is always zero.The first thing we can do is check that these equations are consistent. Remem-

ber that the divergence of any curl is always zero. We compute explicitly usingMaxwell’s equations:

∇ · (∇×E) = ∇ ·−∂ B

∂t

= − ∂

∂t(∇ ·B) = 0

For the magnetic field we get:

∇ · (∇×B) = ∇ ·

µ0J + µoǫ0

∂ E

∂t

= µ0∇ · J + µ0ǫ0

∂

∂t(∇ ·E)

= µ0

∇ · J + ∂ρ

∂t

Hence we see that to force consistency we require that ∇ · J + ∂ρ∂t

= 0. But this isnothing more than the statement of conservation of charge. In fact, if you followthe development that led to equation 2.2 and replace all references to mass withcharge, we arrive at exactly this equation. Note that J is a charge flux density,which plays the analogous role to the mass flux density ρv of fluid dynamics.

For now let us consider the case of electrostatics and magnetostatics. This meansthe two fields E and B are constant in time, so the two time derivatives drop outof Maxwell’s equations.

In this situation, the curl of the electric field is zero, so we can write E =

−∇Φ for some scalar potential function Φ(x,y,z). The minus sign is chosen for

convenience, because with this choice, positive charges tend to flow from points of higher potential to points of lower potential. Note in this case theorem 1.4 tells usthat the circulation of the electric field over any closed path is zero. Now we have:

(3.5) ∇ ·E = −∇·∇Φ = −∇2Φ =ρ

ǫ0

The operator ∇2 = ∇ · ∇ = ∂ 2

∂x2+ ∂ 2

∂y2+ ∂ 2

∂z2is called the Laplacian and is of the

utmost importance in physics. It arises in many different situations, including heat



SOME PHYSICAL APPLICATIONS OF VECTOR CALCULUS 5

diffusion, wave motion, and quantum mechanics. The equation ∇2Φ = − ρǫ0

is calledPoisson’s equation and it is studied in courses on partial differential equations.

We can use Gauss’s Law and Ampere’s Law to calculate electric and magneticfields in cases where there is a high degree of symmetry. Suppose we had a uniformlycharged solid sphere of some radius R. Since there is no preferred direction in thiscase, symmetry tells us that outside the charged sphere the electric field must bein the radial direction and depend only on the distance r from the origin. SoE ·n = E (r) because the electric field is parallel to the normal vector. Now we canintegrate both side of Gauss’s Law over a solid sphere Br of some constant radiusr > R and use the divergence theorem:

Br

(∇ ·E) dV =

Br

ρ

ǫ0

dV =Q

ǫ0

M

E · ndS =

M

E (r)dS = 4πr2E (r)

where Q is the total charge of the sphere, since ρ is constant in the charged sphereand zero outside it, and E (r) is a constant on the sphere of radius r. Thus we see

E (r) = Q4πǫ0r2

which is the same as the electric field due to a point charge Q at theorigin. Inside the charged sphere the field is slightly more complicated.

We can do an analogous calculation for magnetic fields. Suppose we have aninfinitely long thick wire (an infinitely long cylinder) of some radius R. Current isflowing through this cylinder with some uniform current density J. Now becausethe force on a moving charge due to a magnetic field is perpendicular to both thedirection of motion of the charge and the direction of the field, symmetry tellsus that the magnetic field due to this infinite wire must be tangential to circlesperpendicular to and centred on the wire. That is, if we point the thumb of ourright hand in the direction of the current, the field lines go around the wire in thedirection of our fingers. By symmetry, the magnitude of the magnetic field depends

only on the perpendicular distance r from the wire. Now we integrate both side of Ampere’s Law over a solid disc Dr of some constant radius r > R and use Stokes’theorem:

Dr

(∇×B) dS =

Dr

µ0JdS = µ0I

γ

B · dr =

γ

B(r)dr = 2πrB (r)

where I is the total current through the wire, since J is constant in the wire and zerooutside it, and B(r) is a constant on the circle of radius r. Thus we see B(r) = µ0I

2πr

which is the same at the magnetic field due to an infinitely thin wire with currentI . Inside the wire the field is slightly more complicated.

As a final illustration of the use of vector calculus to study electromagnetic

theory, let us consider the case where the fields are time varying, but we are infree space where the charge and current densities are both zero. We will need tomake use of the following identity for a vector field F, which can be easily provedby writing down the definitions and checking each component:

∇× (∇×F) = ∇ (∇ · F)−∇2F

We apply this identity to both the electric and magnetic fields, and use all of Maxwell’s equations to simplify the results, remembering that both ρ and J are




assumed to be zero:

∇× (∇×E) = ∇ (∇ ·E)−∇2E = −∇2E

= ∇×−∂ B

∂t

= − ∂

∂t(∇×B) = −µ0ǫ0

∂ 2E

∂t2

and similarly:

∇× (∇×B) = ∇ (∇ ·B)−∇2B = −∇2B

= ∇×

µ0ǫ0

∂ E

∂t

= µ0ǫ0

∂

∂t(∇×E) = −µ0ǫ0

∂ 2B

∂t2

Thus we see that each of the three components of both the electric and magneticfields satisfy the differential equation

(3.6)∂ 2f

∂t2= c2∇2f

for c = 1

√ µ0ǫ0. This equation represents the motion of a wave with speed c. Hence

we see that in free space the electric and magnetic fields propagate as waves withspeed

1√µ0ǫ0

=1

4π × 10−7 Ns2

C 2

8.85× 10−12 C 2

Nm2

= 2.99863× 108 m

s

which is exactly the speed of light! Maxwell did precisely this same calculationaround 1880 and since the speed of light had been measured by then, he was ableto deduce that light is an electromagnetic wave. There are many different kinds of electromagnetic waves: gamma rays, X-rays, ultraviolet rays, light, infrared rays,microwaves, radio waves. They are all propagating electric and magnetic fields, theonly difference being the frequency of the wave. All travel at the same velocity.They are listed above in decreasing order from highest to lowest frequency. The

energy of the wave is proportional to the frequency, which is why X-rays are farmore harmful to us than radio waves, for example.

Hopefully the above discussions have sufficiently aroused your curiousity, andyou will take more mathematics and physics courses in the future. Vector calculusand electromagnetic theory are only the tip of a giant iceberg.

physical applications of vector calculus

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