12
Vector calculus
In Chapter 8 we introduced the idea of a vector as a quantity with both magnitude and direction and we discussed vector algebra, particularly as applied to analytical geometry, and the differentiation and integration of vectors with respect to a scalar parameter. In this chapter we extend our discussion to include directional derivatives and integration over variables that are themselves vectors. This topic is called vector calculus or vector analysis. It plays a central role in many areas of physics, including fluid mechanics, electromagnetism and potential theory.
12.1 Scalar and vector fields
If scalars and vectors can be defined as continuous functions of position throughout a region of space, they are referred to as fields and the region of space in which they are defined is called a domain. An example of a scalar field would be the distribution of temperature T within a fluid. At each point the temperature is represented by a scalar field T(r) whose value depends on the position r at which it is measured. A useful concept when discussing scalar fields is that of an equipotential surface, that is, a surface joining points of equal value. This is somewhat analogous to the contour lines on a two-dimensional map, which join points of equal height. An example of a vector field is the distribution of velocity v(r) in a fluid. At every point r, the velocity is represented by a vector of definite magnitude and direction, both of which can change continuously throughout the domain. In this case, we can define flow lines such that the tangent to a flow line at any point gives the direction of the vector at that point. Flow lines cannot intersect. This is illustrated in Figure 12.1.
Figure 12.1 The motion of a fluid around a smooth solid. The coloured lines are the flow lines and the arrows show the direction of the vector field, in this case the velocity v(r).
In the rest of this section, we shall extend our discussion of differentiation to embrace scalar and vector fields. Since we are primarily interested in applications where these fields are physical quantities, we shall assume throughout this chapter that they and their first derivatives are single-valued, continuous and differentiable.
12.1.1 Gradient of a scalar field
If we consider the rate of change of a scalar field ψ(r) as r varies, this leads to a vector field called the gradient of ψ(r), written as 
. We will define and derive an expression for this by reference to Figure 12.2.
Figure 12.2 Diagram for the derivation of the gradient of a scalar field ψ(r). PR is normal to the equipotential surface ψR at the point of intersection P.
Consider the point P on an equipotential surface ψ = ψP of the scalar field ψ(r). Let R be a point on the normal to the surface through P that also lies on an equipotential surface ψR > ψP. If we take the two surfaces to be close, then they will be approximately parallel to each other and grad ψ(r) at P is defined as a vector in the direction PR of magnitude
This definition is similar to the definition of a derivative; hence the name gradient. Now let PQ be the signed distance from P to the surface ψR measured in the positive x-direction, and let α be the angle between PR and the x-direction, as shown in Figure 12.2. Then as PR → 0, 
, and thus the component of 
in the x-direction is
since the point Q is on the surface ψ = ψR. The right-hand side of (12.1) is 
. Similarly, the components of grad ψ(r) in the y and z directions are 
and 
, and hence
Further, if we make a small displacement δr = (δx, δy, δz) from P in any direction, we have
(12.3) 
which using (12.2) is
Similarly the corresponding differential [cf. (7.10), (7.11)] is given by
It follows from (12.4a) and (12.4b) that the rate of change of the field with respect to an infinitesimal displacement depends on the direction of travel. For this reason, it is called the directional derivative. To find it, consider moving a distance ds in the direction specified by a unit vector 
, so that 
. Substituting this expression into (12.4b) and dividing by ds then gives
as the directional derivative of ψ in the direction 
.
Another way of writing (12.2) and (12.5) is in terms of an object 
, called del, and defined in the Cartesian system by
(12.6) 
so that (12.2) becomes
Del is called a vector operator, meaning that it acts on (i.e. operates on) a scalar field ψ to give a vector field 
. It is also an example of a differential operator in that it involves derivatives and, like all operators,1 it acts only on objects to its right.
The directional derivative (12.5) can also be rewritten using (12.7) and the definition of 
, when it becomes
Since 
is, by definition, normal to the equipotential surface at P(r), (12.8) satisfies the requirement that 
if 
is along the direction of a tangent to the equipotential surface at P and it attains its maximum value of 
when 
and 
are in the same direction. The relation between these quantities is shown in Figure 12.3.
Figure 12.3 Gradient and directional derivative, where the vector u indicates the direction of 
defined in the text.
12.1.2 Div, grad and curl
Because del is a vector operator, it can form both scalar and vector products with vector fields. Thus if V is the vector field
then the scalar product with del is
This is called the divergence of V and is written 
. Therefore 
. The vector product of del with V is
and is called the curl of V. Thus 
. The origin of these names will emerge later in the chapter. Note that in both cases del is to the left of V because it is an operator. Thus, for example, 
and 
do not have the same meaning, even though they are both scalar products of the same quantities. The former is the simple scalar given in (12.9), the latter is the scalar differential operator
Various combinations of div, grad and curl can also be formed. For example, if ψ is a scalar field, then 
is a vector field. Hence, if we choose 
, we can take its divergence to give
where the scalar operator
(12.12) 
is called the Laplacian operator and 
is called the Laplacian of ψ. The Laplacian is an important operator in physical science and occurs very frequently, for example, in the wave equation
where 
is the wave velocity. Similarly we note that, since the divergence of a vector field V is itself a scalar field, we can take its gradient to give
From this we see that grad div acts on a vector field V to give a vector field (12.11b) and is quite different from div grad, which acts on a scalar field ψ to give a scalar field (12.11a). This illustrates again that care is required with the order of factors when operators are involved. The Laplacian can also operate on a vector field V to give another vector field 
defined by
(12.13) 
The combination 
and grad div are two of only five valid combinations of pairs of div, grad and curl. The other three, together with important identities which they satisfy, are
(12.14b) 
For example, from (12.2) and (12.10) we have
and the other two identities also follow from the definitions of div, grad and curl.
In addition, there are many other identities involving del and two or more scalar or vector fields. They can all be verified by using the previous formulas, taking 
to be a differential vector operator. Some useful identities involving two fields are given in Table 12.1, where a, b are arbitrary vector fields, and ψ and φ are arbitrary scalar fields.
Table 12.1 Some useful identities involving del
 |
12.1.3 Orthogonal curvilinear co-ordinates
So far, we have defined div, grad and curl in Cartesian co-ordinates. However, in problems with spherical or cylindrical symmetry, it is much easier to work in spherical or cylindrical polar co-ordinates, which reflect the symmetry of the problem. As we saw in Section 11.3, these two co-ordinate systems are examples of orthogonal curvilinear co-ordinates ui(i = 1, 2, 3), which are such that distances dr are obtained from formulas of the type
(12.15) 
where the unit vectors ei are orthogonal. The scale factors hi are given by (11.30b), which for the special case of polar co-ordinates are [cf. (11.36) and (11.42)]
Here we shall first give the forms of 
etc. in orthogonal curvilinear co-ordinates, and then obtain the corresponding expressions from them for the cases of spherical and cylindrical polar co-ordinates.
Consider firstly the gradient of a scalar ψ, that is, 
. Returning to Figure 12.2, we let PQ be in the direction of u1, rather than x as before. Then the component of 
in the direction of u1 (with u2 and u3 held fixed) is the directional derivative 
, where ds = h1 du1, that is, the component of 
in the direction e1, is
and similarly for the other directions. Thus,
For example, for spherical polar co-ordinates
and using (12.16), gives
The derivations of the corresponding results for 
and 
using the technique above are more difficult. The derivations are much easier using results we will obtain in Sections 12.3 and 12.4, and will be given there. For the present we will just quote the results
and
The expressions for 
given earlier for the special case of Cartesian co-ordinates are easily regained by setting (u1, u2, u3) = (x, y, z) and hx = hy = hz = 1 in (12.17) to (12.20), respectively. The corresponding results for spherical polar and cylindrical polar co-ordinates are similarly obtained using (12.16) for the scale factors and are shown in Table 12.2.
Table 12.2 Grad, div, curl and the Laplacian in polar co-ordinates
12.2 Line, surface, and volume integrals
In this section, we shall extend the discussion of line integrals given in Chapter 11 to embrace integrals of a vector field and use vector methods to define integrals of a vector field over a curved surface.
12.2.1 Line integrals
In Section 8.3.1, we saw that the running vector
(12.21) 
where a and b are fixed vectors, described a straight line passing through the point r = a in the direction 
as the scalar parameter s varied in the range − ∞ < s < ∞. More generally, any running vector r(s), where r is a differentiable function of s, will describe a curve in space and for any given s the differential
is an infinitesimal vector directed along the tangent to the curve, as shown in Figure 12.4 for the point P corresponding to s = s0. For example, in Cartesian co-ordinates
is the vector equation of the parabola y2 = 4ax lying in the plane z = 0, and the corresponding differential
is an infinitesimal vector directed along the tangent to the curve, as shown in Figure 12.4 for the point P corresponding to s = s0.
Figure 12.4 Definition of a space curve for the line integral of a scalar product. The path is defined in terms of a parameter s, so that r = r(s).
Now suppose we have a vector field V(r). Then we can define two line integrals
(12.23) 
where C as usual denotes the path, or contour, of integration. The first of these is by far the most important in physics, and is the only one we shall discuss. One important example is the work done by a force field F(r). If F(r) is the force acting at the position r, then F(r) · dr is the work done by the force in moving from r to r + dr, and the integral
(12.24) 
is the work done in moving from the initial position ri to the final position rf along the path C. If the force returns to ri, then ri = rf and the integral is denoted
where the circle on the integral sign emphasises that the path is a closed loop.
So far, we have not used any co-ordinates to define the integrals. In general, if we use a set of co-ordinates u1, u2, u3, such that
and
then,
In Cartesian co-ordinates
so that (12.25) becomes
Here x, y, z (and in general u1, u2, u3) are not independent variables along the path C, but are specified by a single parameter s, so that C = r(s) and (12.26a) becomes
(12.26b) 
using (12.22). In particular, s may be chosen to be one of the co-ordinates themselves, for example x, when (12.26a) may be used directly together with the relations y = y(x), z = z(x) along the contour C.
At this point we note that (12.26) is identical with the line integral (11.13) discussed in Section 11.1.3, if the functions Q, R, P are replaced by functions Vx, Vy, Vz. Hence the methods and results discussed in Section 11.1.3 can be carried over, with a trivial relabeling, to the line integrals (12.26) as we shall illustrate in the next section.
12.2.2 Conservative fields and potentials
The result of a line integral of a vector between any two points will in general depend on the path taken between them. If, however, the line integral is independent of the path for any choice of end points within the field, the vector field V is said to be conservative. Conservative fields play an important role in physics, as we shall now see.
Suppose that
Then, using the expression (12.10) for the curl in Cartesian co-ordinates, we see that (12.27) implies that
which is precisely the condition [cf. (11.23b) and (7.19b)] that
is an exact, or perfect, differential. From (12.28) we immediately see that
i.e.
and that
where ψA and ψB are the values of ψ at the points A and B. Hence V is a conservative field and can be derived from a scalar field ψ, called a potential field, or just a potential. We note that ψ(r) is only defined up to a constant by (12.29) and (12.30). This is usually chosen by requiring that ψ has a given value ψ0 at a reference point r0, or sometimes that ψ(r) → 0 as |r | → ∞.
The above argument shows that (12.27) is a sufficient condition for V to be a conservative field. That it is also a necessary condition is seen by reversing the argument. If V(r) is a conservative field, then we can define a potential by
since the integral is independent of the chosen path between the reference point r0 and the point r. This implies that
so that 
and 
by (12.14a). Hence 
is not only a sufficient condition, but also a necessary condition for V to be a conservative field.
An important example of a conservative field is the gravitational field. In general, if F(r) is the force acting on a particle at a position r, it is usual to introduce the potential energy due to gravity such that
that is, the force acts in the direction of maximally decreasing potential energy. The work W done when F moves a particle from A to B is
so that the work done by the force equals the loss of potential energy.
Of course not all forces are conservative. If dissipative forces such as friction are involved, then energy will be lost in moving from A to B in a way that depends on the path and a potential cannot be defined.
12.2.3 Surface integrals
In three dimensions, a surface is defined by an equation of the form
where f(x, y, z) is a given function and d is a constant.2 Simple examples are the equation of a plane [cf (8.45a)],
(12.34) 
which is an example of an open surface; and the equation of a sphere
(12.35) 
which is an example of a closed surface. In addition, since (12.33) defines an equipotential surface for the scalar field f, it follows from the discussion of 
in Section 12.1.1 (cf. Figure 12.2) that at any point on the surface
(12.36) 
are the two vectors normal to the surface.
We now introduce integrals of a vector field V(r) over a surface S as follows. Given a small surface element ds, we form a vector surface element
where 
is a unit vector normal to the surface at the position of ds, so that the direction of 
varies continuously over S. Surface integrals can now be defined of the form
(12.38) 
In each case, the integral is a double integral over a surface S, which may be open or closed. If the surface is closed, 
is chosen to point outwards from the closed region. If the surface is open it must be two-sided, that is, it is only possible to get from one side to the other by crossing the curve bounding the surface. Figure 12.6a shows a two-sided surface, whereas Figure 12.6b shows a so-called Mobius strip, which is one-sided.
Figure 12.6 Examples of open surfaces that are (a) two-sided (b) one-sided and (c) the use of a projection of a two-sided surface onto a plane to define the direction of 
.
For open surfaces, one must choose the direction for 
. However, if a direction is associated with a boundary curve that surrounds the surface, as it is in some very important applications, then 
is chosen to be ‘right-handed’. To see what this means, let us suppose that the surface and its boundary curve were to be projected onto a plane. Then, as shown in Figure 12.6c, 
is chosen so that the direction of integration around the contour of integration corresponds to that of a right-hand screw pointing in the direction of 
.
Of the two integrals (12.37), the scalar integrals are by far the more important. Their evaluation is often facilitated by choosing an appropriate co-ordinate system. For example, if one is integrating over a planar surface that lies in the x–y plane, then ds = dx dy k and
On the other hand, suppose S lies on surface of a sphere of radius a. Then if we take the origin to be at the centre, the equation of the sphere in spherical co-ordinates is r = a, and one sees from Figure 11.14 that
(12.39a) 
Similarly, θ = θ0 is the equation of a cone with its axis along the z-direction and, from Figure 11.14, one sees that in this case
(12.39b) 
More generally, given any set of orthogonal curvilinear co-ordinates (u1, u2, u3), keeping any one of them constant defines a surface with, for example,
(12.40) 
if u1 is constant. Hence if
an integral over a surface on which u1 is constant reduces to
with similar expressions if either u2 or u3 is constant.3 These are straightforward double integrals, which can be evaluated using the methods discussed in Section 11.2.
If the surface does not correspond to a constant value of a suitably chosen orthogonal curvilinear co-ordinate, the integral can be evaluated using the projection method. In this method, the surface is projected onto a plane and the integral evaluated using Cartesian co-ordinates. This is illustrated in Figure 12.7, which shows an element of surface ds projected onto an element dA in the xy plane. From this figure,
Figure 12.7 Diagram used to illustrate the evaluation of surface integrals by the projection method.
If the surface S is given by f(x, y, z) = d, then 
evaluated at the point on the surface, and so
This general formula can be used to convert an integral over a curved surface S to an integral over A in the xy plane, as illustrated in Example 12.11 below.
12.2.4 Volume integrals: moments of inertia
In Section 11.4, we considered volume integrals of the form
in Cartesian co-ordinates, where the integral extends over the region Ω, and the abbreviated notation on the left-hand side is sometimes used for convenience in what follows. We can now also define similar integrals over a vector field, i.e.
(12.43) 
whose evaluation essentially involves evaluating three integrals of the form (12.42).
To illustrate this, let us consider a solid body with variable density ρ occupying a region of space Ω. Then since the mass of a volume element is 
, the total mass of the body is given by
(12.44) 
while the formula
for the centre-of-mass 
of a system of point particles of masses mi at positions r, becomes
for an arbitrary solid body. Similarly, the formula
for the moment of inertia I, where roi is the perpendicular distance from the mass mi to the axis of rotation, becomes
12.3 The divergence theorem
The divergence theorem4 states that, for any vector field V,
for any surface S enclosing a region Ω. The quantity V · ds is called the flux of V through ds and the circle on the double integral is to emphasise that S is a closed surface, by analogy with closed paths in line integrals. This circle is sometimes omitted and (12.49) is also sometimes written in the abbreviated form
already used in (11.49) for volume integrals. However, in whatever form it is written, the theorem states that the volume integral of the divergence of V is equal to the total flux out of the bounding surface S, since 
points out of a closed surface.
We shall derive the divergence theorem and two well-known identities resulting from it in Section 12.3.1 below. Before that, we point out that the divergence theorem is central to the physical interpretation of divergence. To see this, we apply (12.49) to the case when S encloses a small volume element that shrinks to a point as 
. In this limit, the variation of V in 
can be neglected, so the left-hand side of (12.49) becomes 
, implying
In other words, 
at a point r is the flux per unit volume out of an infinitesimal volume 
surrounding r. For example, if V = ρ v, where ρ is the density and v is the velocity field of a fluid, the flux V · ds is the rate of flow of mass through the surface. Hence if 
is greater than zero, there is a net flow of mass away from r, so that either the density is decreasing at the point, or a source (i.e. a point where fluid is entering the system) is present. On the other hand, if there is no source or sink (i.e. a point where fluid is leaving the system) at r, and the density is constant, which is normally a good approximation for a liquid, then 
. In this latter case V is called a solenoidal field. Although we have chosen the example of a fluid, the same ideas may be applied to other situations, including the flow of electric current.
12.3.1 Proof of the divergence theorem and Green's identities
To derive the divergence theorem, we consider a segment through the region Ω lying parallel to the x-axis and with constant infinitesimal cross section dy dz, as shown in Figure 12.85. Further, let the unit vectors 
and 
be the outward normals on the surface elements 1 and 2, respectively, where the segment intersects the surface of the region Ω, so that 
and
Figure 12.8 A segment through the region Ω lying parallel to the x-axis and with constant infinitesimal cross section dy dz, used in the derivation of the divergence theorem.
Then, since
at fixed y, z, we have
where the right-hand side is the net flux through the surface elements 1 and 2 from the x-component Vxi.
All that remains now is to add together the contributions from enough segments to cover the whole region Ω, so that (12.51) becomes
The contributions from the y and z components of V can be calculated in a similar way, and adding all three components we obtain
which is the divergence theorem.6
Finally, we use the divergence theorem to derive two other useful results as follows. Let φ and ψ be two scalar fields continuous and differentiable in some region Ω bounded by a closed surface S. Applying the divergence theorem to 
gives
(12.52a) 
This is known as Green's first identity. Similarly, interchanging φ and ψ gives
Subtracting these two equations gives
(12.52b) 
which is Green's second identity.
*12.3.2 Divergence in orthogonal curvilinear co-ordinates
Having derived the divergence theorem (12.49) using Cartesian co-ordinates, then the corollary (12.50) follows, and can be regarded as an alternative definition of the divergence, independent of the co-ordinate system. In particular, it can be used to find the general expression (12.18) for the divergence in an arbitrary set of orthogonal curvilinear co-ordinates (u1, u2, u3).
To do this, we consider the region bounded by surfaces of constant ui and constant ui + δui as shown in Figure 12.9. The edges AB, AD and AA' are along the orthogonal co-ordinate axes, and so are of approximate length h1δu1, h2δu2 and h3δu3, where hi are the coefficients defined in (11.30b). We first calculate the contribution to the integral
from the faces ABCD and A'B'C'D'. If V1, V2, V3 are the components of V along u1, u2, u3, then the contribution from the face ABCD is approximately
evaluated at u3, while the contribution from A'B'C'D' is approximately
evaluated at u3 + δu3, where terms of third order in δui have been neglected. Applying the Taylor series to h1h2V3 at fixed u3 and neglecting terms of order (δu3)2 gives
so that the net contribution from these two faces is
Figure 12.9 Construction to derive the divergence in orthogonal curvilinear co-ordinates.
Now the volume element is
to the same order and so from (12.50) the contribution to the divergence is
on taking the limit 
. Contributions from other pairs of faces may be found in a similar way and putting these together yields
which is the required result (12.18). We leave it as an exercise for the reader to show that the corresponding result (12.19) for the Laplacian follows from combining this result with (12.17) for the gradient, and that the corresponding results for cylindrical and spherical spherical co-ordinates given in Table 12.2 follow on substituting the appropriate values for h1, h2 and h3.
*12.3.3 Poisson's equation and Gauss' theorem
The electrostatic field E obeys the fundamental equation
where ρ is the electric charge density and the
constant ϵ0 is the electric permittivity of free space. This equation is called Poisson's equation. Since 
is the flux of E per unit volume away from the point at which it is evaluated, the interpretation of Poisson's equation is that the electric charge is the source of the electrostatic field.
If we now apply the divergence theorem (12.49) to the field E, and use (12.53), we immediately obtain
(12.54) 
This relation is called Gauss' theorem. It says that the electric flux through a closed surface S is equal to ϵ− 10 times the total charge enclosed by the surface.
Gauss' theorem is useful in that it allows the field due to a given charge distribution ρ(r) to be evaluated relatively easily in cases where there is a high degree of symmetry. For example, let us suppose that we have a charged sphere centred at the origin with radius R and total charge Q, and that the charge density within the sphere is also spherically symmetric, that is, ρ(r) = ρ(r). Then the resulting field must also be spherically symmetric, that is, it must be of the form
(12.55) 
so that E(r) points away from (or towards) the origin and its magnitude is the same in all directions. Hence if we choose the surface S to be a sphere of radius r > R, as shown in Figure 12.10, then E(r) is perpendicular to S and
by Gauss' theorem. Consequently,
which reduces to Coulomb's law
for a point charge at the origin if we allow R → 0 at fixed Q.
Figure 12.10 The spherical surface S used to calculate the electric field due to a spherical charge distribution of radius R for r > R.
This analysis is easily generalised to other inverse square law forces. In particular, if g is the gravitational field, so that the force on a point particle of mass is F = mg, then g obeys the Poisson equation
(12.58) 
where ρ is the mass density and G is the gravitational constant. The result corresponding to (12.56) for a spherically symmetric sphere of total mass M is
which reduces to
when R → 0 at fixed M. This is basis of the approximation that the Earth may be treated as if all its mass were concentrated at its centre when calculating its gravitational field for r > R. However, the approximation is not exact, because the earth is flattened at the poles.
Finally, we note that both the electrostatic and gravitational fields are conservative, satisfying
so that we can introduce scalar potentials φ and ψ by
in accordance with the discussion of Section 12.2.2. For the electrostatic case, substituting (12.59) into (12.53) gives
(12.60) 
which is Poisson's equation for the electrostatic potential; and if one requires φ → 0 as r → ∞, one easily shows that the potential corresponding to (12.57) is the familiar Coulomb potential
(12.61) 
Figure 12.11 The infinitesimal cylinder used in Example 12.14, where S is the surface of the conductor and E = 0 within the conductor.
*12.3.4 The continuity equation
Let us consider a fluid of density ρ(r, t) with a velocity field v(r, t) at time t. Then if we consider a surface element ds, which may lie within the body of the fluid, the mass of liquid passing through ds in unit time is j · ds, where j = ρv is the current vector. If mass is conserved, the rate of change of the mass
contained in a given region Ω must be balanced by the rate at which mass flows out through the surface S bounding Ω, i.e.
Equation (12.62) is the statement of mass conservation in integral form. However, it is often more convenient to express it in differential, or local, form, that is, one that refers only to quantities at a single point in space. This can be achieved by using the divergence theorem on the right-hand side of (12.62) and taking the derivative inside the integral on the left-hand side to give
Since this must hold for any region Ω, we must have
at any point in space.
Equation (12.63) is called the equation of continuity and is the statement of mass conservation in differential, or local, form. Furthermore, any ρ(r, t) that satisfies a relation of the form (12.63), whatever the relation between the density ρ and the current j, is the density of a conserved quantity. This is because the argument can be reversed, that is, (12.62) follows from (12.63) using the divergence theorem. Then, if we let the surface S recede to infinity, we obtain
(12.64) 
where the integral extends over all space, provided ρ, j → 0 sufficiently rapidly at infinity, as they usually do. Many examples of conserved quantities occur in physics, including electric charge and energy. However, the relation between the density ρ and the current j is not always as simple as j = ρv, as shown in Example 12.15.
12.4 Stokes' theorem
Given a closed contour C, spanned by a surface S, and a vector field V defined on S, then Stokes' theorem states that
where the sense of the vector element ds is given by a right-handed screw rule with respect to the direction of integration around C.
The line integral on the right-hand side of (12.68) is called the circulation of V around the loop C. Thus the theorem states that the surface integral of 
is equal to the circulation of V around the bounding curve C. This is closely related to the interpretation of curl. To see this, we apply (12.68) to a loop C that encloses a small surface element 
, which shrinks to a point when ds → 0. In this limit, the variation of V and 
can be neglected on ds, so that the left-hand side of (12.68) becomes 
, implying
In other words, 
at a point r is the circulation per unit area around the boundary of an infinitesimal surface ds containing the point r. For example, let us again consider a vector field V = ρv, where ρ is the density and v is the velocity field of a fluid. Then for a uniform flow pattern, such as that shown in Figure 12.12a, 
and V is said to be irrotational. On the other hand, at the centre of a vortex, like that shown in Figure 12.12b, clearly 
. It is also non-zero in a non-uniform parallel motion, as shown in Figure 12.12c, since the velocities on either side of a point are different. Essentially, 
when there is rotational motion in addition to, or opposed to, translational motion. A practical viewpoint is to consider what would happen if one inserted a small ‘paddle wheel’, which is free to rotate about its axis. In the flow pattern of Figure 12.12a, where 
, it would not rotate: the motion is irrotational. In Figures 12.12b and 12.12c, where 
, it would rotate.
Figure 12.12 Flow of a fluid. The coloured lines are the flow lines; the arrows show the direction of the vector field V. Their lengths show the relative magnitudes of V.
In the rest of this section we will first derive Stokes' theorem, and then consider some applications.
12.4.1 Proof of Stokes' theorem
We start by considering a closed curve C surrounding a plane surface S parallel to the x–y plane, so that z is constant. Then
and
But by Green's theorem in the plane (11.20), we have
so that
where 
, a unit vector in the z-direction. Furthermore, in the limit ds → 0, where the variation of 
over the surface can be neglected, we obtain
As there is nothing special about the z-direction – we may choose it in any direction we like – it follows that (12.70) and (12.71) hold for any finite or infinitesimal planar surface, respectively, where 
is the normal defined in the usual sense. We will now use this result to derive Stokes' theorem.
Consider an open surface, which must be two-sided, divided into small regions ds, as shown in Figure 12.13a. As ds → 0, each element, irrespective of its shape, approaches ever more closely to an element of the plane tangential to the surface at the centre of the surface ds. Therefore, (12.71) implies (12.69), and (12.70) becomes
as ds → 0, where the circulation is around the boundary of ds. If we sum over all ds,
and from the enlarged section shown in Figure 12.13b it is clear that all interior contributions to the circulation will vanish, resulting in
Figure 12.13 Construction to derive Stokes' theorem.
This is Stokes' theorem as required. It is worth emphasising that the right-hand side is an integral over any surface that is bounded by the curve C. Note also the direction of the circulation, which is ‘right-handed’ relative to the directions ds, as discussed in Section 12.2.3 [cf. Figure 12.6]. In the following subsections we will consider some applications of this theorem.
*12.4.2 Curl in curvilinear co-ordinates
Having derived Stokes' theorem (12.68) and its corollary (12.69), the latter can be regarded as an alternative definition of curl, independent of the co-ordinate system. Here we shall use it to obtain the expression for curl in an arbitrary system of orthogonal linear co-ordinates.7
To do this, we consider the infinitesimal surface element ds = ds e3 swept out when u1 → u1 + du1 and u2 → u2 + du2 at constant u3, as shown in Figure 12.14. Then from (12.69), we have
where C is the contour ABCD shown in Figure 12.4. We now write
where e1 and e2 are unit vectors along the directions AB and AD, respectively, and use the fact that in the limit that du1, du2 tend to zero the corresponding lines may be approximated by straight lines, and ABCD may be approximated by a rectangle, since e1 and e2 are orthogonal. Hence the contribution of V1 to the line integral arises solely from the arcs AB and DC and is
Figure 12.14 Construction to derive Stokes' theorem in curvilinear co-ordinates.
Similarly, the contribution from V2 is
and ds = h1h2u1u2, so that on substituting into (12.72) we obtain
This identical to the e3 component given in (12.20) and analogous results follows for the other components.
*12.4.3 Applications to electromagnetic fields
Finally we illustrate the use of Stokes' theorem by applying it to the behaviour of electric and magnetic fields, starting with the electric field E. In free space, this is determined by the fundamental equations
where B is the magnetic field intensity, ρ is the charge density, and ϵ0 is the electric permittivity. Of these, (12.73a) was discussed in Section 12.3.3, where we saw that it expressed the fact that charge is the source of electric flux. However, in contrast to electrostatics, if there are time-dependent magnetic fields present, 
no longer vanishes. Hence E is not in general a conservative field and loop integrals of the form
no longer vanish. Rather, by Stokes' theorem and (12.73), we have
where S is any open surface spanning the loop C. This is Faraday's law of induction that states that the ‘emf’ ϵC induced around a loop C is equal to minus the rate of change of the magnetic flux through the loop. We also note that the argument can be reversed: if (12.74) holds, then Stokes' theorem gives
which can only hold for an arbitrary open surface S if (12.74) is satisfied. Equation (12.73b) and (12.74) are the differential and integral forms of Faraday's law.
Equations (12.73a) and (12.73b) are the first two Maxwell's equations in free space. The remaining two are
(12.75) 
where j is the electric current density, μ0 is the magnetic permeability of free space, and the speed of light c = (μ0ϵ0)− 1/2. On comparing with (12.73a), we see that (12.75a) reflects the experimental observation that there are no free magnetic charges. The second equation (12.75b) indicates that non-zero magnetic fields can be generated by currents or time-dependent electric fields. In the absence of the latter, it becomes
By Stokes' theorem
giving
where S is any surface spanning the loop C. This is called Ampère's law and it states that the line integral of B around a closed loop is equal to μ0 times the total current Iencl flowing through the loop. It enables the magnetic field to be calculated quickly in symmetrical situations, as we shall illustrate.
Figure 12.15 The circuit C used to derive (12.79), where the current I at the centre is directed out of the page.
Problems 12
12.1 A scalar field electrostatic potential is given by φ = x2 − y2 and the associated electric field E is given by
. What is the magnitude and direction of E at (2, 1)? In what direction does φ increase most rapidly at the point (−3, 2) and what is the rate of change of φ at the point (1, 2) in the direction 3i − j?12.2 Given the scalar function ψ = x2 − y2z, find (a)
at (1, 1, 1); (b) the derivative of ψ at (1, 1, 1) in the direction i − 2j + k; (c) the equation of the normal to the surface ψ = x2 − y2z = 0 at (1, 1, 1).12.3 If A = 2xz2i − yz j + 3xz3k and S = x2yz, find in Cartesian co-ordinates (a) curl A, (b)
, (c) 
, (d) 
and (e) 
.12.4 Given a scalar field ψ and a vector field V, show (a) that
Hence show (b) that if
, where α is a scalar field, then
12.5 Show, without explicitly writing out the components, that
12.6 If ψ = 2yz and
, express
in spherical polar co-ordinates.
12.7 Directly evaluate the line integral
where
around the circle (x2 + y2) = a2 in the x–y plane. Verify your result using Green's theorem in the plane.
12.8 A force field is
Find the work done in moving a particle round a closed curve from the origin to the point (x, y, z) = (0, 0, 2π) along the path
and then back to the origin along the z-axis.
-
12.9 Find the work done by a force F given by
when moving a particle clockwise along a semicircle of unit radius in the x–y plane from x = −1 to x = 1 with y ≥ 0.
12.10 Find the work done by a force F = (x2 + y2)j when moving between the points A(x = a, y = 0, z = 0) and B(x = 0, y = a, z = 0) along a path C, where C is (a) along the x-axis to the origin, then along the y-axis to B, and (b) along the arc of the circle x2 + y2 = a2, z = 0, in the positive quadrant.
12.11 A force
moves around a closed loop starting at the origin along the curve 
to (2, 1), then parallel to the x axis to (0, 1) and finally returning to the origin along the y axis. Use Green's theorem in the plane to calculate the work done by the force.12.12 Show that
is a conservative field and find a scalar potential φ, such that
.12.13 A force field
where a, b, c, are constants. For what values of a, b, c, is F a conservative field? Find the scalar potential in this case.
12.14 Let
be that part of the surface of the cylinder
for which x > 0, y > 0. What is the value of the surface integral
if A = 6yi + (2x + z)j − xk and S is that part of
that lies on the curved surface of the cylinder?12.15 Evaluate the integral
where
and S is the surface of a unit cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 , without using the divergence theorem. The vector s is defined in the outward direction from each face of the cube.12.16 Evaluate the integral
over the curved surface of a hemisphere of radius a with its centre at the origin and base in the x–y plane, where ds = |ds | and σ is a constant.
12.17 A sphere of uniform density ρ has mass M and radius a. Calculate its moment of inertia about (a) a tangent to the sphere and (b) an axis through the centre of the sphere.
12.18 A cylinder of uniform density ρ0 has mass M, radius a and length d. Calculate its moment of inertia about an axis that lies in the plane of the base of the cylinder and passes through the centre of the base.
12.19 Use the divergence theorem to evaluate
where F = 4xz i − y2 j + yx k and S is the surface of the cube bounded by
12.20 Scalar fields φi(i = 1, 2, …) are solutions of the equations
where the γi are constants, within a region Ω, subject to the boundary conditions φi = 0 on the closed surface S enclosing Ω. Show that
if i ≠ j γi ≠ γj.
12.21 Prove the identity
for any scalar field ψ. A scalar field ψ satisfies the conditions ψ = 0 on S and
in Ω, where S is the closed surface surrounding the region Ω. Show that ψ = 0 in Ω.*12.22 State Gauss' theorem for the gravitational field. A homogeneous spherical shell has mass M, inner radius a and outer radius b > a. Find an expression for the gravitational field due to the shell for (a) r > b, (b) r < a and (c) a < r < b, where r = |r |. Finally, calculate the potential at any point with r < a assuming the potential goes to zero as r → ∞.
- *12.23
- Prove the relation
where φ is a scalar field and E is a vector field.
Let ρ(r) be an electric charge density, which vanishes outside a finite region Ω1 enclosing the origin, with total charge
Write down an approximate value for the electrostatic field E and potential φ on a sphere centred at the origin with radius R, assuming that R is very large compared to the dimensions of Ω1.
Show that, in the same approximation,
where Ω2 is the interior of the sphere of radius R, and find the value of the constant c. [Hint: use Poisson's equation
.]
- Prove the relation
*12.24 In a homogeneous continuous medium, Maxwell's equations take the form
with
, and the constants ϵ and μ are the permittivity and permeability of the medium.- Show that Maxwell's equations imply that ρ is the density of a conserved charge.
- In a conductor, the current obeys Ohm's law j = σE, where σ is the conductivity. Find the charge density ρ as a function of time if ρ(r, t = 0) = ρ0(r).
12.25 Verify Stokes' theorem for the vector
where S is the surface of the hemisphere
and C is the boundary of S.
12.26 Use Stokes' theorem (12.68) to prove the relation
where φ is a scalar fields and S is an open surface bounded by a closed curve C. [Hint: apply Stokes' theorem to the vector field V = φ c, where c is a constant vector.]
12.27 A force field F = y2i + x2j acts on a particle. Write down a line integral corresponding to the work done by the field when the particle moves once round the circle x2 + y2 = a2, z = 0 in the anticlockwise direction. Evaluate this integral (a) directly and (b) by converting it to a surface integral. (c) Is the force field F conservative?