0 −Az Ay
Az 0 −Ax
−Ay Ax 0
As a direct application of this result, we consider another useful operator to be added to the previous list. It is ˆr × I = ˆr × Itr:
ˆr × I ↔
0 0 0
0 0 −1
0 1 0
↔ ˆϕ ˆϑ − ˆϑ ˆϕ
where we have used the spherical basis and clearly ˆr ↔ (1 0 0)T in this basis.
A.2 Calculus of vector fields
In this section we review the basic concepts of calculus, i.e. derivatives and integrals, applied to vector fields. Let us start with a scalar field h(x,y), where h is a smooth function of only two variables for simplicity. The partial derivatives
∂h
∂x
∂h
∂y
evaluated in the point P ↔ (x0,y0) have the meaning of local rate of change of h for small increments of the coordinates dx, dy in the x, y directions around point P , respectively. A quantity of interest is the directional derivative of h in the direction ˆs, which has the meaning of local rate of change of h for small displacements in the ˆs direction. According to the chain rule,
∂h
∂s = ∂h
∂x dx ds+∂h
∂y dy ds
x y
sˆ
sx
sy
dy α
dx ds
Figure A.4. Direction along which the directional derivative is computed
With reference to Fig. A.4, we find dx
ds = cos α = sx dy
ds = sin α = sy
so that
∂h
∂s =∂h
∂xsx+∂h
∂ysy
This formula can be considered as a scalar product of the vector gradh = ∇h = ∂h
∂xx +ˆ ∂h
∂yˆy
called gradient of h, times the unit vector ˆs. The gradient of a scalar function is a vector field.
The magnitude of the gradient is the maximum rate of change of h when the direction ˆs is allowed to vary. The direction in which his happens is the direction of the gradient. If h is the height of the ground above the sea level, so that h(x,y) can be interpreted as the local height of a hill, the direction of the gradient is that of the steepest slope. Note, however, that the gradient is a vector belonging to the x, y plane and it is not tangent to the surface. When ˆs is orthogonal to the gradient, the directional derivative is zero: in that direction h is not changing, hence the gradient is always orthogonal to contour lines (lines on which h= const). Fig. A.5 shows an example of a scalar field h(x,y). Fig. A.6 shows a contour plot of the same function with the vector field gradh, computed numerically by finite differences. Note that the gradient is zero in the extrema of the graph of h, i.e. maxima, minima and saddle points. Also, the arrows are orthogonal to the contours.
If the scalar field h(x,y,z) depends on the three space coordinates, the gradient gradh is defined as
gradh = ∇h = ∂h
∂xˆx +∂h
∂yy +ˆ ∂h
∂zˆz (A.28)
If the scalar field is given in a non cartesian system of coordinates the expression of the gradient are more complicated, essentially because the unit vectors are not constant. It can be proved that
• in cylindrical coordinates
gradh(ρ,ϕ,z) = ∇h = ∂h
∂ρρ +ˆ 1 ρ
∂h
∂ϕϕ +ˆ ∂h
∂zˆz (A.29)
−3
−2
−1 0
1 2
3
−3
−2
−1 0 1 2 3
−8
−6
−4
−2 0 2 4 6 8 10
z=h(x,y)
y x z
Figure A.5. Scalar field z = h(x,y)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
x
y
grad h(x,y) and contours
Figure A.6. Contours of z = h(x,y) and vector field gradh(x,y)
Figure A.7. Closed surface Σ for the definition of the flux of a vector field.
• in spherical coordinates
gradh(r,ϑ,ϕ) = ∇h = ∂h
∂rˆr +1 r
∂h
∂ϑϑ +ˆ 1 r sin ϑ
∂h
∂ϕϕˆ (A.30)
We turn now to a vector field A(r) and define its flux across a surface, which may be either open or closed. Fig. A.7 shows a closed surface Σ enclosing the volume V , with the outward normal unit vector ˆν. The outward flux of A through Σ is defined by the scalar
ΦΣ(A) = I
Σ
A · ˆνdΣ Recalling Gauss Theorem Z
V
∇ · AdV = I
Σ
A · ˆνdΣ (A.31)
we can interpret this equation as defining an “average divergence”
average{∇ · A} = 1 V
Z
V
∇ · AdV = 1 V
I
Σ
A · ˆνdΣ
In the limit of the volume reducing to a point we obtain the divergence as flux per unit volume across a small closed surface surrounding the point. It can be shown that the explicit expressions for the divergence of a vector field are
• in cartesian coordinates
div A = ∇ · A =∂Ax
∂x +∂Ay
∂y +∂Az
∂z (A.32)
• in cylindrical coordinates
div A = ∇ · A =1 ρ
∂
∂ρ(ρAρ) +1 ρ
∂Aϕ
∂ϕ +∂Az
∂z (A.33)
• in spherical coordinates
div A = ∇ · A = 1 r2
∂
∂r(r2Ar) + 1 r sin ϑ
∂
∂ϑ(sin ϑAϑ) + 1 r sin ϑ
∂Aϕ
∂ϕ (A.34)
In conclusion, the divergence of a vector field is a particular combination of derivatives of the field components, whose meaning is clarified by Gauss theorem. Consider a flow of charges across a closed surface so that A = J = ρv where v is the velocity field and ρ the volume charge density.
The units of J are A/m2, hence it is a current density. The flux ΦΣ(J) has the units of A and it yields the amount of charge coming out every second from the total volume V , i.e. the total current. The source of this outflow is the divergence of J, which tells us how much charge per unit volume is coming out every second from each point of V . The flux of J provides a global information, the divergence of J a local one.
Finally it is useful to write Gauss theorem (A.31) for the one dimensional domain [a,b], with boundary consisting of the points x = a (with outward normal −ˆx) and x = b (with outward
normal ˆx) Z b
a
∂Ax
∂x dx = Ax(b) − Ax(a) Clearly, this is just the fundamental formula of integral calculus.
Another type of integral that we can form with a vector field is a line integral Z
Γ
A · ˆτ ds
where the curve Γ may be either open or closed. Assume it is closed, so that the integral is called the circulation of A. Stokes theorem states that
Z
Σo
∇ × A · ˆνdΣo= I
Γ
A · ˆτ ds (A.35)
where Σois an open surface with Γ as boundary; the orientation of the tangent vector ˆτ is related
Figure A.8. Geometry for the application of Stokes theorem
to that of ˆν by the right-hand-rule, as shown in Fig. A.8. For the purpose of the definition, it is convenient to limit the generality of the theorem and consider a planar curve Γ so that Σo can
be taken as the part of plane inside Γ, with constant normal vector ˆν. This equation allows us to define the ˆν component of an “average curl”
average{∇ × A} · ˆν = 1
Obviously, by choosing three loops with linearly independent normals we can define completely the “total amount of rotation” or average curl. By letting the size of the loops go to zero, we can define the curl in a point as circulation per unit area in the neighborhood of a single point. By carrying out this prescription in different systems of coordinates, it can be proved that the explicit expressions of the curl of a vector field are
• in cartesian coordinates
curl A = ∇ × A =
• in cylindrical coordinates curl A = ∇ × A =
• in spherical coordinates curl A = ∇ × A =
If we examine the expressions of gradient, divergence and curl in cartesian coordinates we can identify a kind of formal vector ∇ defined by
∇ = ˆx ∂
∂x+ ˆy ∂
∂y + ˆz∂
∂z (A.39)
such that the various differential operators can be imagined to be formed by means of standard product, scalar product and vector product of ∇ times a scalar field or a vector field. It is to be remarked that such an interpretation (due to the American physicist W. Gibbs 1839-1903) is possible only in cartesian coordinates. Hence, for instance in spherical coordinates, even if the curl of A is written usually as ∇ × A, this expression has to be taken as a single symbol (whose meaning is given by (A.2)) and not interpreted as the product of two factors.
First order differential operators can combined to form second order operators.
• The gradient of a scalar field is a vector, so that we can compute the curl and divergence of it. However
∇ × ∇h = 0
identically for any smooth h(r). To remember the property, we can note that the vector product of two equal vectors is zero.
• The divergence of a gradient produces the Laplace operator or laplacian
∇ · ∇h = ∇2h = 4h
We recall only the expression of the laplacian in cartesian coordinates
∇2h = ∂2h
∂x2 +∂2h
∂y2 +∂2h
∂z2
• The divergence of a vector field A(r) is a scalar, hence its gradient can be computed
∇(∇ · A)
• The curl of a vector field A(r) is a vector, hence its divergence and curl can be computed.
However
∇ · (∇ × A) = 0
identically for any smooth A(r). To remember the property, we can note that the triple mixed product with two equal factors is zero.
• As for the curl of the curl, the following identity is to be noted
∇ × (∇ × A(r)) = ∇(∇ · A) − ∇2A
Often multidimensional integrals have to be computed. It is useful to remember from Fig. A.2 that the elementary volume in spherical coordinates is a cube with edges of size dr (along ˆr), rdϑ (along ˆϑ) and r sin ϑdϕ (along ˆϕ), thus
dV = r2sin ϑdrdϑdϕ
Concerning surface integrals in spherical coordinates, the elementary patch is a square with sides rdϑ (along ˆϑ) and r sin ϑdϕ (along ˆϕ), thus
dΣ = r2sin ϑdϑdϕ
Connected with this is the concept of solid angle, displayed in Fig. A.9
The natural measurement unit of a plane angle is radian (rad). A one radian angle is the one with the vertex at the center of a circle of radius r that subtends an arc whose length is equal to the radius r. Thus the radian measure of an angle is equal to the ratio between the length of the subtended arc and the radius: for instance, the measure of a full round angle is 2π radians.
Similarly, the solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point; it is measured in steradians (sr). A one steradian solid angle is the one with the vertex at the center of a sphere of radius r that subtends a patch whose area is equal to r2. Since the surface of a sphere is 4πr2, the measure of the total solid angle around a point is 4π steradians. As another example, the solid angle defined by x ≥ 0, y ≥ 0, z ≥ 0 has a measure of 4π/8 = π/2 steradians. The elementary solid angle subtended by a patch of area dΣ is, in spherical coordinates
dΩ = dΣ
r2 = sin ϑdϑdϕ
Figure A.9. Definition of radian for a plane angle and of steradian for a solid angle