Exercise 2.2 Prove the previous theorem
2.2 Helmholtz Theorem
We said in Chapter 1 that a vector field F is conservative if the work required to move a particle through the field is independent of the path of the particle, in which case F can be represented as the gradient of a scalar 0,
called the potential of F. Conversely, if F has a scalar potential, then F is conservative. These concepts are a subset of the Helmholtz theorem (Duff and Nay lor [81]) which states that any vector field F that is continuous
2.2 Helmholtz Theorem 29 and zero at infinity can be expressed as the gradient of a scalar and the curl of a vector, that is,
F = V0 + V x A , (2.12) where V0 and V x A are orthogonal in the integral norm. The quantity
<fi is the scalar potential of F, and A is the vector potential.
2.2.1 Proof of the Helmholtz Theorem
Given that F is continuous and vanishes at infinity, we can construct the integral
[ d v , (2.13)
4TT J r
where Q is the point of integration, r is the distance between P and Q, and the integral is taken over all space. Each of the three cartesian components of W has a form like
l (2J4)
At this point, we borrow a result from Chapter 3: Equation 2.14 is a solution to a very important differential equation, Poisson's equation:
V2WX = -FX. (2.15)
The relationship between equations 2.14 and 2.15 follows from Green's third identity because the integration in equation 2.14 is over all space, and we have stipulated that F and, therefore, the three components of F vanish at infinity.
Exercise 2.4 Show that equations 2.14 and 2.15 are consistent with Green's third identity.
With W defined as in equation 2.13, the relationship between equa- tions 2.14 and 2.15 suggests that
V2W = - F , (2.16)
where each component of F leads to an example of Poisson's equation.
A vector identity (Appendix A) shows that V2W can be represented by a gradient plus a curl, that is,
- V2W = - V ( V • W) + V x (V x W ) , (2.17) and hence F is represented as the gradient of a scalar (V • W) plus the
curl of a vector (V x W). We define (ft = —V • W and A = V x W and substitute these definitions along with equation 2.16 into equation 2.17 to get the Helmholtz theorem,
F = V0 + V x A.
E x e r c i s e 2 . 5 P r o v e t h e vector identity V ( V - W ) - V x V x W = V2W . Hence, the Helmholtz theorem is proven: If F is continuous and vanishes at infinity, it can be represented as the gradient of a scalar potential plus the curl of a vector potential.
The Helmholtz theorem is useful, however, only if the scalar and vec- tor potentials can be derived directly from F. This should be possible because of the way (ft and A were defined, and the relationships can be seen by taking the divergence and curl of both sides of equation 2.12.
The divergence yields
V2(ft = V - F ,
which, comparing with equations 2.14 and 2.15, has the solution
<£ = - T - f^-^-dv. (2.18)
4TT J r The curl of equation 2.12 provides
V2A = V(V- A ) - V x F .
For convenience, we define A so that it has no divergence, and conse- quently
V2A = - V x F .
Comparing this result with equations 2.13 and 2.16 leads to
V X— dv. (2.19)
= - / •
4W
Exercise 2.6 Show that equation 2.19 implies that V • A = 0.
Consequently, the scalar potential (ft and vector potential A can be de- rived from integral equations taken over all space and involving the di- vergence and curl, respectively, of F itself.
Exercise 2.7 Prove the last statement of the Helmholtz theorem; that is, show that V</> and V x A, both vanishing at infinity, are orthogonal under integration over three-dimensional space.
2.2 Helmholtz Theorem 31 2.2.2 Consequences of the Helmholtz Theorem
The Helmholtz theorem shows that a vector field vanishing at infinity is completely specified by its divergence and its curl if they are known throughout space. If both the divergence and curl vanish at all points, then the field itself must vanish or be constant everywhere.
In addition to this statement, the following important observations follow directly from the Helmholtz theorem and from the integral repre- sentation for scalar and vector potentials.
Irrotational Fields
A vector field is irrotational in a region if its curl vanishes at each point of the region; that is, F is irrotational in a region if V x F = 0 throughout the region. Such fields have no vorticity or "eddies." For example, if the flow of a fluid can be represented as an irrotational field, then a small paddlewheel placed within the fluid will not rotate. Examples of irrotational fields are common and include gravitational attraction, of considerable importance to future chapters.
Consider any surface S entirely within a region where V x F = 0.
Integration of the curl over the surface provides (V x F) -hdS = 0, s
and applying Stokes's theorem (Appendix A) provides F • ds - 0 ,
/ •
where the closed line integral is taken around the perimeter of S. Because the integral holds for any closed surface within the region, no net work is done in moving around any closed loop that lies within an irrotational field, that is, work is independent of path, and F is conservative, a sufficient condition for the existence of a scalar potential such that F = V</>. Hence, the condition that V x F = 0 at each point of a region is sufficient to say that F = V</>. Furthermore, a field that has a scalar potential has no curl because V x F = V x V</> vanishes identically (Appendix A). Hence, the property that V x F = 0 at every point of a region is a necessary and sufficient condition for the existence of a scalar potential such that F = V0.
Solenoidal Fields
A vector field F is said to be solenoidal in a region if its divergence vanishes at each point of the region. A physical meaning for solenoidal fields can be had by integrating the divergence of F over any volume V within the region,
/ •
V • F dv = 0 , v
and applying the divergence theorem (Appendix A) to get
F-ndS = 0, (2.20) s
where S is the closed boundary of V. Hence, if the divergence of F vanishes in a region, the normal component of the field vanishes when integrated over any closed surface within the region. Or put another way, the "number" of field lines entering a region equals the number that exit the region, and sources or sinks of F do not exist in the region. For example, gravitational attraction is solenoidal in regions not occupied by mass.
It was stated in Section 2.1.1 that if a function </> can be found such that F = V</>, then the condition expressed by equation 2.20 is necessary and sufficient to say that <fi is harmonic throughout the region. From the Helmholtz theorem, V • F = V20 + V • V x A. The last term of this equation vanishes identically (Appendix A), and V • F = V20. Hence, if the divergence of a conservative field vanishes in a region, the potential of the field is harmonic in the region.
Note that if F = V x A, then V • F = 0; that is, the divergence of a vector field vanishes if the vector can be expressed purely as the curl of another vector. Furthermore, the converse can be shown to be true by taking the curl of both sides of equation 2.19. Hence, the property that V • F = 0 is a necessary and sufficient condition for F = V x A.
2.2.3 Example
Equation 2.18 is important to the geophysical interpretation of gravity and magnetic anomalies caused by crustal masses and magnetic sources, respectively. To see this, we use the magnetic field as an example and anticipate the results of future chapters.
2.2 Helmholtz Theorem 33 A set of differential equations, called Maxwell's equations, describes the spatial and temporal relationships of electromagnetic fields and their sources. One of Maxwell's equations relates magnetic induction B and magnetization M in the absence of macroscopic currents:
V x B = /i0V x M ,
where /io is the permeability of free space. Magnetic field intensity H is related to magnetic induction and magnetization by the equation
B = /io(H + M ) . (2.21)
Hence, in the absence of macroscopic currents, V x H = 0,
and it follows from the Helmholtz theorem that magnetic field intensity is irrotational and can be expressed in terms of a scalar potential, that is, H = — VV, where the minus sign is a matter of convention as discussed in Chapter 1. Moreover, equation 2.18 provides an expression for that scalar potential:
V=
T- / ^ V (
2-
22)
4TT J r
where it is understood that the integration is over all space. Another of Maxwell's equations states that magnetic induction has no divergence, that is, V • B = 0. This fact plus equation 2.21 yields
V H = - V M , (2.23) and substituting into equation 2.22 provides
™*dv. (2.24) r
Again integration is over all space. Equation 2.24 provides a way to cal- culate magnetic potential and magnetic field from a known (or assumed) spatial distribution of magnetic sources. This is called the forward prob- lem when applied to the geophysical interpretation of measured magnetic fields. Equation 2.24 also is a suitable starting point for discussions of the inverse problem: the direct calculation of the distribution of magne- tization from observations of the magnetic field. We will return to this equation in Chapter 5 and subsequent chapters.