Vector operators

10.6 SCALAR AND VECTOR FIELDS
A normal n to the surface at this point is then given by
i
j
∂r
∂r
n=
×
= a cos θ cos φ a cos θ sin φ
∂θ
∂φ −a sin θ sin φ a sin θ cos φ
k
−a sin θ
0
= a2 sin θ(sin θ cos φ i + sin θ sin φ j + cos θ k),
which has a magnitude of a2 sin θ. Therefore, the element of area at P is, from (10.23),
dS = a2 sin θ dθ dφ,
and the total surface area of the sphere is given by
π 2π
A=
dθ
dφ a2 sin θ = 4πa2 .
0
0
This familiar result can, of course, be proved by much simpler methods! 10.6 Scalar and vector fields
We now turn to the case where a particular scalar or vector quantity is defined
not just at a point in space but continuously as a field throughout some region
of space R (which is often the whole space). Although the concept of a field is
valid for spaces with an arbitrary number of dimensions, in the remainder of this
chapter we will restrict our attention to the familiar three-dimensional case. A
scalar field φ(x, y, z) associates a scalar with each point in R, while a vector field
a(x, y, z) associates a vector with each point. In what follows, we will assume that
the variation in the scalar or vector field from point to point is both continuous
and differentiable in R.
Simple examples of scalar fields include the pressure at each point in a fluid
and the electrostatic potential at each point in space in the presence of an electric
charge. Vector fields relating to the same physical systems are the velocity vector
in a fluid (giving the local speed and direction of the flow) and the electric field.
With the study of continuously varying scalar and vector fields there arises the
need to consider their derivatives and also the integration of field quantities along
lines, over surfaces and throughout volumes in the field. We defer the discussion
of line, surface and volume integrals until the next chapter, and in the remainder
of this chapter we concentrate on the definition of vector differential operators
and their properties.
10.7 Vector operators
Certain differential operations may be performed on scalar and vector fields
and have wide-ranging applications in the physical sciences. The most important
operations are those of finding the gradient of a scalar field and the divergence
and curl of a vector field. It is usual to define these operators from a strictly
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VECTOR CALCULUS
mathematical point of view, as we do below. In the following chapter, however, we
will discuss their geometrical definitions, which rely on the concept of integrating
vector quantities along lines and over surfaces.
Central to all these differential operations is the vector operator ∇, which is
called del (or sometimes nabla) and in Cartesian coordinates is defined by
∇≡i
∂
∂
∂
+j
+k .
∂x
∂y
∂z
(10.25)
The form of this operator in non-Cartesian coordinate systems is discussed in
sections 10.9 and 10.10.
10.7.1 Gradient of a scalar field
The gradient of a scalar field φ(x, y, z) is defined by
grad φ = ∇φ = i
∂φ
∂φ
∂φ
+j
+k .
∂x
∂y
∂z
(10.26)
Clearly, ∇φ is a vector field whose x-, y- and z- components are the first partial
derivatives of φ(x, y, z) with respect to x, y and z respectively. Also note that the
vector field ∇φ should not be confused with the vector operator φ∇, which has
components (φ ∂/∂x, φ ∂/∂y, φ ∂/∂z).
Find the gradient of the scalar field φ = xy 2 z 3 .
From (10.26) the gradient of φ is given by
∇φ = y 2 z 3 i + 2xyz 3 j + 3xy 2 z 2 k. The gradient of a scalar field φ has some interesting geometrical properties.
Let us first consider the problem of calculating the rate of change of φ in some
particular direction. For an infinitesimal vector displacement dr, forming its scalar
product with ∇φ we obtain
∂φ
∂φ
∂φ
+j
+k
· (i dx + j dy + k dx) ,
∇φ · dr = i
∂x
∂y
∂z
∂φ
∂φ
∂φ
dx +
dy +
dz,
=
∂x
∂y
∂z
= dφ,
(10.27)
which is the infinitesimal change in φ in going from position r to r + dr. In
particular, if r depends on some parameter u such that r(u) defines a space curve
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10.7 VECTOR OPERATORS
∇φ
a
Q
θ
P
dφ
in the direction a
ds
φ = constant
Figure 10.5 Geometrical properties of ∇φ. P Q gives the value of dφ/ds in
the direction a.
then the total derivative of φ with respect to u along the curve is simply
dφ
dr
= ∇φ · .
du
du
(10.28)
In the particular case where the parameter u is the arc length s along the curve,
the total derivative of φ with respect to s along the curve is given by
dφ
= ∇φ · t̂,
ds
(10.29)
where t̂ is the unit tangent to the curve at the given point, as discussed in
section 10.3.
In general, the rate of change of φ with respect to the distance s in a particular
direction a is given by
dφ
= ∇φ · â
ds
(10.30)
and is called the directional derivative. Since â is a unit vector we have
dφ
= |∇φ| cos θ
ds
where θ is the angle between â and ∇φ as shown in figure 10.5. Clearly ∇φ lies
in the direction of the fastest increase in φ, and |∇φ| is the largest possible value
of dφ/ds. Similarly, the largest rate of decrease of φ is dφ/ds = −|∇φ| in the
direction of −∇φ.
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VECTOR CALCULUS
For the function φ = x2 y + yz at the point (1, 2, −1), find its rate of change with distance
in the direction a = i + 2j + 3k. At this same point, what is the greatest possible rate of
change with distance and in which direction does it occur?
The gradient of φ is given by (10.26):
∇φ = 2xyi + (x2 + z)j + yk,
= 4i + 2k
at the point (1, 2, −1).
The unit vector in the direction of a is â = √114 (i + 2j + 3k), so the rate of change of φ
with distance s in this direction is, using (10.30),
10
dφ
1
= ∇φ · â = √ (4 + 6) = √ .
ds
14
14
From the above discussion, at the point √
(1, 2, −1) dφ/ds will be greatest in the direction
of ∇φ = 4i + 2k and has the value |∇φ| = 20 in this direction. We can extend the above analysis to find the rate of change of a vector
field (rather than a scalar field as above) in a particular direction. The scalar
differential operator â · ∇ can be shown to give the rate of change with distance
in the direction â of the quantity (vector or scalar) on which it acts. In Cartesian
coordinates it may be written as
â · ∇ = ax
∂
∂
∂
+ ay
+ az .
∂x
∂y
∂z
(10.31)
Thus we can write the infinitesimal change in an electric field in moving from r
to r + dr given in (10.20) as dE = (dr · ∇)E.
A second interesting geometrical property of ∇φ may be found by considering
the surface defined by φ(x, y, z) = c, where c is some constant. If t̂ is a unit
tangent to this surface at some point then clearly dφ/ds = 0 in this direction
and from (10.29) we have ∇φ · t̂ = 0. In other words, ∇φ is a vector normal to
the surface φ(x, y, z) = c at every point, as shown in figure 10.5. If n̂ is a unit
normal to the surface in the direction of increasing φ(x, y, z), then the gradient is
sometimes written
∇φ ≡
∂φ
n̂,
∂n
(10.32)
where ∂φ/∂n ≡ |∇φ| is the rate of change of φ in the direction n̂ and is called
the normal derivative.
Find expressions for the equations of the tangent plane and the line normal to the surface
φ(x, y, z) = c at the point P with coordinates x0 , y0 , z0 . Use the results to find the equations
of the tangent plane and the line normal to the surface of the sphere φ = x2 + y 2 + z 2 = a2
at the point (0, 0, a).
A vector normal to the surface φ(x, y, z) = c at the point P is simply ∇φ evaluated at that
point; we denote it by n0 . If r0 is the position vector of the point P relative to the origin,
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10.7 VECTOR OPERATORS
z
n̂0
(0, 0, a)
z=a
O
a
y
φ = x 2 + y 2 + z 2 = a2
x
Figure 10.6 The tangent plane and the normal to the surface of the sphere
φ = x2 + y 2 + z 2 = a2 at the point r0 with coordinates (0, 0, a).
and r is the position vector of any point on the tangent plane, then the vector equation of
the tangent plane is, from (7.41),
(r − r0 ) · n0 = 0.
Similarly, if r is the position vector of any point on the straight line passing through P
(with position vector r0 ) in the direction of the normal n0 then the vector equation of this
line is, from subsection 7.7.1,
(r − r0 ) × n0 = 0.
For the surface of the sphere φ = x2 + y 2 + z 2 = a2 ,
∇φ = 2xi + 2yj + 2zk
= 2ak
at the point (0, 0, a).
Therefore the equation of the tangent plane to the sphere at this point is
(r − r0 ) · 2ak = 0.
This gives 2a(z − a) = 0 or z = a, as expected. The equation of the line normal to the
sphere at the point (0, 0, a) is
(r − r0 ) × 2ak = 0,
which gives 2ayi − 2axj = 0 or x = y = 0, i.e. the z-axis, as expected. The tangent plane
and normal to the surface of the sphere at this point are shown in figure 10.6. Further properties of the gradient operation, which are analogous to those of
the ordinary derivative, are listed in subsection 10.8.1 and may be easily proved.
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VECTOR CALCULUS
In addition to these, we note that the gradient operation also obeys the chain
rule as in ordinary differential calculus, i.e. if φ and ψ are scalar fields in some
region R then
∇ [φ(ψ)] =
∂φ
∇ψ.
∂ψ
10.7.2 Divergence of a vector field
The divergence of a vector field a(x, y, z) is defined by
div a = ∇ · a =
∂ax
∂ay
∂az
+
+
,
∂x
∂y
∂z
(10.33)
where ax , ay and az are the x-, y- and z- components of a. Clearly, ∇ · a is a scalar
field. Any vector field a for which ∇ · a = 0 is said to be solenoidal.
Find the divergence of the vector field a = x2 y 2 i + y 2 z 2 j + x2 z 2 k.
From (10.33) the divergence of a is given by
∇ · a = 2xy 2 + 2yz 2 + 2x2 z = 2(xy 2 + yz 2 + x2 z). We will discuss fully the geometric definition of divergence and its physical
meaning in the next chapter. For the moment, we merely note that the divergence
can be considered as a quantitative measure of how much a vector field diverges
(spreads out) or converges at any given point. For example, if we consider the
vector field v(x, y, z) describing the local velocity at any point in a fluid then ∇ · v
is equal to the net rate of outflow of fluid per unit volume, evaluated at a point
(by letting a small volume at that point tend to zero).
Now if some vector field a is itself derived from a scalar field via a = ∇φ then
∇ · a has the form ∇ · ∇φ or, as it is usually written, ∇2 φ, where ∇2 (del squared)
is the scalar differential operator
∇2 ≡
∂2
∂2
∂2
+ 2 + 2.
2
∂x
∂y
∂z
(10.34)
∇2 φ is called the Laplacian of φ and appears in several important partial differential equations of mathematical physics, discussed in chapters 20 and 21.
Find the Laplacian of the scalar field φ = xy 2 z 3 .
From (10.34) the Laplacian of φ is given by
∇2 φ =
∂2 φ ∂2 φ ∂2 φ
+ 2 + 2 = 2xz 3 + 6xy 2 z. ∂x2
∂y
∂z
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10.7 VECTOR OPERATORS
10.7.3 Curl of a vector field
The curl of a vector field a(x, y, z) is defined by
curl a = ∇ × a =
∂az
∂ay
−
∂y
∂z
i+
∂ax
∂az
−
∂z
∂x
j+
∂ay
∂ax
−
∂x
∂y
k,
where ax , ay and az are the x-, y- and z- components of a. The RHS can be
written in a more memorable form as a determinant:
i
j
k ∂
∂
∂ (10.35)
∇ × a = ,
∂x ∂y ∂z ax ay az where it is understood that, on expanding the determinant, the partial derivatives
in the second row act on the components of a in the third row. Clearly, ∇ × a is
itself a vector field. Any vector field a for which ∇×a = 0 is said to be irrotational.
Find the curl of the vector field a = x2 y 2 z 2 i + y 2 z 2 j + x2 z 2 k.
The curl of a is given by
i
j
∂
∂
∇φ = ∂y
∂x
x2 y 2 z 2 y 2 z 2
k
∂
∂z
x2 z 2
= −2 y 2 zi + (xz 2 − x2 y 2 z)j + x2 yz 2 k . For a vector field v(x, y, z) describing the local velocity at any point in a fluid,
∇ × v is a measure of the angular velocity of the fluid in the neighbourhood of
that point. If a small paddle wheel were placed at various points in the fluid then
it would tend to rotate in regions where ∇ × v = 0, while it would not rotate in
regions where ∇ × v = 0.
Another insight into the physical interpretation of the curl operator is gained
by considering the vector field v describing the velocity at any point in a rigid
body rotating about some axis with angular velocity ω. If r is the position vector
of the point with respect to some origin on the axis of rotation then the velocity
of the point is given by v = ω × r. Without any loss of generality, we may take
ω to lie along the z-axis of our coordinate system, so that ω = ω k. The velocity
field is then v = −ωy i + ωx j. The curl of this vector field is easily found to be
i
∂
∇ × v = ∂x
−ωy
j
∂
∂y
ωx
353
k
∂
∂z
0
= 2ωk = 2ω.
(10.36)