Vector operator formulae

VECTOR CALCULUS
∇(φ + ψ)
∇ · (a + b)
∇ × (a + b)
∇(φψ)
∇(a · b)
∇ · (φa)
∇ · (a × b)
∇ × (φa)
∇ × (a × b)
=
=
=
=
=
=
=
=
=
∇φ + ∇ψ
∇·a+∇·b
∇×a+∇×b
φ∇ψ + ψ∇φ
a × (∇ × b) + b × (∇ × a) + (a · ∇)b + (b · ∇)a
φ∇ · a + a · ∇φ
b · (∇ × a) − a · (∇ × b)
∇φ × a + φ∇ × a
a(∇ · b) − b(∇ · a) + (b · ∇)a − (a · ∇)b
Table 10.1 Vector operators acting on sums and products. The operator ∇ is
defined in (10.25); φ and ψ are scalar fields, a and b are vector fields.
Therefore the curl of the velocity field is a vector equal to twice the angular
velocity vector of the rigid body about its axis of rotation. We give a full
geometrical discussion of the curl of a vector in the next chapter.
10.8 Vector operator formulae
In the same way as for ordinary vectors (chapter 7), for vector operators certain
identities exist. In addition, we must consider various relations involving the
action of vector operators on sums and products of scalar and vector fields. Some
of these relations have been mentioned earlier, but we list all the most important
ones here for convenience. The validity of these relations may be easily verified
by direct calculation (a quick method of deriving them using tensor notation is
given in chapter 26).
Although some of the following vector relations are expressed in Cartesian
coordinates, it may be proved that they are all independent of the choice of
coordinate system. This is to be expected since grad, div and curl all have clear
geometrical definitions, which are discussed more fully in the next chapter and
which do not rely on any particular choice of coordinate system.
10.8.1 Vector operators acting on sums and products
Let φ and ψ be scalar fields and a and b be vector fields. Assuming these fields
are differentiable, the action of grad, div and curl on various sums and products
of them is presented in table 10.1.
These relations can be proved by direct calculation.
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10.8 VECTOR OPERATOR FORMULAE
Show that
∇ × (φa) = ∇φ × a + φ∇ × a.
The x-component of the LHS is
∂
∂az
∂ay
∂φ
∂φ
∂
(φaz ) − (φay ) = φ
+
az − φ
−
ay ,
∂y
∂z
∂y
∂y
∂z
∂z
∂az
∂φ
∂ay
∂φ
+
=φ
−
az −
ay ,
∂y
∂z
∂y
∂z
= φ(∇ × a)x + (∇φ × a)x ,
where, for example, (∇φ × a)x denotes the x-component of the vector ∇φ × a. Incorporating
the y- and z- components, which can be similarly found, we obtain the stated result. Some useful special cases of the relations in table 10.1 are worth noting. If r is
the position vector relative to some origin and r = |r|, then
∇φ(r) =
dφ
r̂,
dr
dφ(r)
,
dr
2
d φ(r) 2 dφ(r)
∇2 φ(r) =
,
+
dr 2
r dr
∇ × [φ(r)r] = 0.
∇ · [φ(r)r] = 3φ(r) + r
These results may be proved straightforwardly using Cartesian coordinates but
far more simply using spherical polar coordinates, which are discussed in subsection 10.9.2. Particular cases of these results are
∇r = r̂,
together with
∇ · r = 3,
∇ × r = 0,
1
r̂
= − 2,
r
r
r̂
1
∇ · 2 = −∇2
= 4πδ(r),
r
r
∇
where δ(r) is the Dirac delta function, discussed in chapter 13. The last equation is
important in the solution of certain partial differential equations and is discussed
further in chapter 20.
10.8.2 Combinations of grad, div and curl
We now consider the action of two vector operators in succession on a scalar or
vector field. We can immediately discard four of the nine obvious combinations of
grad, div and curl, since they clearly do not make sense. If φ is a scalar field and
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VECTOR CALCULUS
a is a vector field, these four combinations are grad(grad φ), div(div a), curl(div a)
and grad(curl a). In each case the second (outer) vector operator is acting on the
wrong type of field, i.e. scalar instead of vector or vice versa. In grad(grad φ), for
example, grad acts on grad φ, which is a vector field, but we know that grad only
acts on scalar fields (although in fact we will see in chapter 26 that we can form
the outer product of the del operator with a vector to give a tensor, but that need
not concern us here).
Of the five valid combinations of grad, div and curl, two are identically zero,
namely
curl grad φ = ∇ × ∇φ = 0,
div curl a = ∇ · (∇ × a) = 0.
(10.37)
(10.38)
From (10.37), we see that if a is derived from the gradient of some scalar function
such that a = ∇φ then it is necessarily irrotational (∇ × a = 0). We also note
that if a is an irrotational vector field then another irrotational vector field is
a + ∇φ + c, where φ is any scalar field and c is a constant vector. This follows
since
∇ × (a + ∇φ + c) = ∇ × a + ∇ × ∇φ = 0.
Similarly, from (10.38) we may infer that if b is the curl of some vector field a
such that b = ∇ × a then b is solenoidal (∇ · b = 0). Obviously, if b is solenoidal
and c is any constant vector then b + c is also solenoidal.
The three remaining combinations of grad, div and curl are
div grad φ = ∇ · ∇φ = ∇2 φ =
∂2 φ ∂2 φ ∂2 φ
+ 2 + 2,
∂x2
∂y
∂z
(10.39)
grad div a = ∇(∇ · a),
2
2
∂ ax
∂2 az
∂2 ay
∂ ax
∂2 ay
∂2 az
+
+
+
+
=
i
+
j
∂x2
∂x∂y ∂x∂z
∂y∂x
∂y 2
∂y∂z
2
∂ ax
∂2 ay
∂2 az
+
+
k,
(10.40)
+
∂z∂x ∂z∂y
∂z 2
curl curl a = ∇ × (∇ × a) = ∇(∇ · a) − ∇2 a,
(10.41)
where (10.39) and (10.40) are expressed in Cartesian coordinates. In (10.41), the
term ∇2 a has the linear differential operator ∇2 acting on a vector (as opposed to
a scalar as in (10.39)), which of course consists of a sum of unit vectors multiplied
by components. Two cases arise.
(i) If the unit vectors are constants (i.e. they are independent of the values of
the coordinates) then the differential operator gives a non-zero contribution
only when acting upon the components, the unit vectors being merely
multipliers.
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