Conversely, if u is harmonic in T, there must exist a function v such that u + iv is analytic in T, and v is given by
zf\dun du 7
v = / - —-dx+ --dy J [ dy dx
We will have occasion later in this text to use these rather abstract prop- erties of complex numbers in some practical geophysical applications.
1.4 Problem Set 17 1.4 Problem Set
1. The potential of F is given by (x2 -\- y2)~l. (a) Find F.
(b) Describe the field lines of F.
(c) Describe the equipotential surfaces of F.
(d) Demonstrate by integration around the perimeter of a rectangle in the x, y plane that F is conservative. Let the rectangle extend from x\ to X2 in the x direction and from y\ to ^2 m the y direction, and let x\ > 0.
2. Prove that the intensity of a conservative force field is inversely proportional to the distance between its equipotential surfaces.
3. If all mass lies interior to a closed equipotential surface S on which the potential takes the value C, prove that in all space outside of S the value of the potential is between C and 0.
4. If the lines of force traversing a certain region are parallel, what may be inferred about the intensity of the force within the region?
5. Two distributions of matter lie entirely within a common closed equipotential surface C. Show that all equipotential surfaces outside of C also are common.
6. For what integer values of n is the function {x2-\-y2-\-z2)^ harmonic?
7. You are monitoring the magnetometer aboard an interstellar space- craft and discover that the ship is approaching a magnetic source described by
(a) Remembering Maxwell's equation for B, will you report to Mis- sion Control that the magnetometer is malfunctioning, or is this a possible source?
(b) What if the magnetometer indicates that B is described by
8. The physical properties of a spherical body are homogeneous. De- scribe the temperature at all points of the sphere if the temperature is harmonic throughout the sphere and depends only on the distance from its center.
9. As a crude approximation, the temperature of the interior of the earth depends only on distance from the center of the earth. Based
on the results of the previous exercise, would you expect the temper- ature of the earth to be harmonic everywhere inside? Explain your answer?
10. Assume a spherical coordinate system and let r be a vector directed from the origin to a point P with magnitude equal to the distance from the origin to P. Prove the following relationships:
V - r = 3, Vr = -,
r
V-
(^)=°> ^ o ,
V x r = 0,
r r6
(A-V)r = Ar-.
r
2
Consequences of the Potential
It may be no surprise that human minds can deduce the laws of falling objects because the brain has evolved to devise strategies for dodging them.
(Paul Davies) Only mathematics and mathematical logic can say as little as the physicist means to say.
(Bertrand Russell) In Chapter 1, we learned that a conservative vector field F can be ex- pressed as the gradient of a scalar </>, called the potential of F, and conversely F is conservative if F = V<f>. It was asserted that such po- tentials satisfy Laplace's equation at places free of all sources of F and are said to be harmonic. This led to several important characteristics of the potential. In the same spirit, this chapter investigates a number of additional consequences that follow from Laplace's equation.
2.1 Green's Identities
Three identities can be derived from vector calculus and Laplace's equa- tion, and these lead to several important theorems and additional in- sight into the nature of potential fields. They are referred to as Green's identities.^
f The name Green, appearing repeatedly in this and subsequent chapters, refers to George Green (1793-1841), a British mathematician of Caius College, Cambridge, England. He is perhaps best known for his paper, Essay on the Application of Mathematical Analysis to the Theory of Electricity and Magnetism, and was ap- parently the first to use the term "potential."
19
2.1.1 Green's First Identity
Green's first identity is derived from the divergence theorem (Appendix A). Let U and V be continuous functions with continuous partial deriva- tives of first order throughout a closed, regular region R, and let U have continuous partial derivatives of second order in R. The boundary of R is surface #, and h is the outward normal to S. If A = W E / , then
R R
= Aw-
Using the divergence
R
theorem
•VE/ +
R
yields
VV2U]dS
-1 -l
ss
A-hSv
VVU -ndS
that is,
fw
2Udv+ fvUVVdv= fv^dS. (2.1)
Equation 2.1 is Green's first identity and is true for all functions U and V that satisfy the differentiability requirements stated earlier.
Several very interesting theorems result from Green's first identity if U and V are restricted a bit further. For example, if U is harmonic and continuously differentiate in R, and if V = 1, then V2E/ = 0, VT/ = 0, and equation 2.1 becomes
J On
(2.2)Thus the normal derivative of a harmonic function must average to zero on any closed boundary surrounding a region throughout which the func- tion is harmonic and continuously differentiable (Figure 2.1). It also can be shown (Kellogg [146, p. 227]) that the converse of equation 2.2 is
2.1 Green's Identities 21
F= VU
Fig. 2.1. Region i? subject to force field F. Surface S bounds region R. Unit vector n is outward normal at any point on S.
true; that is, if U and its derivatives of first order are continuous in R, and ^ integrates to zero over its closed boundary, then U must be har- monic throughout R. Hence, equation 2.2 is a necessary and sufficient condition for U to be harmonic throughout the region.
Equation 2.2 provides an important boundary condition for many geo- physical problems. Suppose that vector field F has a potential U which is harmonic throughout some region. Because ^ = F • n on the surface of the region, equation 2.1 can be written as
F-ndS = 0, (2.3)
and applying the divergence theorem (Appendix A) yields V • F dv = 0 .
/ •
R
In words, the normal component of a conservative field must average to zero on the closed boundary of a region in which its potential is harmonic.
Hence, the flux of F into the region exactly equals the flux leaving the region, implying that no sources of F exist in the region. Moreover, the condition that V • F = 0 throughout the region is sufficient to conclude that no sources lie within the region.
Steady-state heat flow, for example, is harmonic (as discussed in Chap- ter 1) in regions without heat sources or sinks and must satisfy equa- tion 2.3. If region R is in thermal equilibrium and contains no heat sources or sinks, the heat entering R must equal the heat leaving R.
Equation 2.3 is often called Gauss's law and will prove useful in subse- quent chapters.
Now let U be harmonic in region R and let V = U. Then, from Green's first identity,
f(VU)
2dv= fu^dS. (2.4)
J J on
R S
Consider equation 2.4 when U = 0 on S. The right-hand side vanishes and, because (VC/)2 is continuous throughout R by hypothesis, (VLQ2 = 0. Therefore, U must be a constant. Moreover, because U = 0 on S and because U is continuous, the constant must be zero. Hence, if U is harmonic and continuously differentiate in R and if U vanishes at all points of S, U also must vanish at all points of R. This result is intuitive from steady-state heat flow. If temperature is zero at all points of a region's boundary and no sources or sinks are situated within the region, then clearly the temperature must vanish throughout the region once equilibrium is achieved.
Green's first identity leads to a statement about uniqueness, some- times referred to as Stokes's theorem. Let U\ and U2 be harmonic in R and have identical boundary conditions, that is,
U1(S) = U2(S).
The function U1 — U2 also must be harmonic in R. But U1 — U2 vanishes on S and the previous theorem states that U\ — U2 also must vanish at every point of R. Therefore, U\ and U2 are identical. Consequently, a function that is harmonic and continuously differentiate in R is uniquely deter- mined by its values on S, and the solution to the Dirichlet boundary- value problem is unique. Stokes's theorem makes intuitive sense when applied to steady-state heat flow. A region will eventually reach thermal equilibrium if heat is allowed to flow in and out of the region. It seems reasonable that, for any prescribed set of boundary temperatures, the region will always attain the same equilibrium temperature distribution throughout the region regardless of the initial temperature distribution.
In other words, the steady-state temperature of the region is uniquely determined by the boundary temperatures.
2.1 Green's Identities 23 The surface integral in equation 2.4 also vanishes if ^ = 0 on S. A similar proof could be developed to show that if U is single-valued, har- monic, and continuously differentiate in R and if Qj£ = 0 on S, then U is a constant throughout R. Again, steady-state heat flow provides some insight. If the boundary of R is thermally insulated, equilibrium tem- peratures inside R must be uniform. Moreover, a single-valued harmonic function is determined throughout R, except for an additive constant, by
the values of its normal derivatives on the boundary.