VECTOR CALCULUS
ais a vector field, these four combinations are grad(gradφ), div(diva), curl(diva)
and grad(curla). 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(gradφ), for
example, grad acts on gradφ, 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
theouter productof 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 gradφ=∇×∇φ= 0 , (10.37)div curla=∇·(∇×a)=0. (10.38)From (10.37), we see that ifais derived from the gradient of some scalar function
such thata=∇φthen it is necessarily irrotational (∇×a= 0). We also note
that ifais an irrotational vector field then another irrotational vector field is
a+∇φ+c,whereφis any scalar field andcis a constant vector. This follows
since
∇×(a+∇φ+c)=∇×a+∇×∇φ= 0.Similarly, from (10.38) we may infer that ifbis the curl of some vector fielda
such thatb=∇×athenbis solenoidal (∇·b= 0). Obviously, ifbis solenoidal
andcis any constant vector thenb+cis also solenoidal.
The three remaining combinations of grad, div and curl arediv gradφ=∇·∇φ=∇^2 φ=∂^2 φ
∂x^2+∂^2 φ
∂y^2+∂^2 φ
∂z^2, (10.39)grad diva=∇(∇·a),=(
∂^2 ax
∂x^2+∂^2 ay
∂x∂y+∂^2 az
∂x∂z)
i+(
∂^2 ax
∂y∂x+∂^2 ay
∂y^2+∂^2 az
∂y∂z)
j+(
∂^2 ax
∂z∂x+∂^2 ay
∂z∂y+∂^2 az
∂z^2)
k, (10.40)curl curla=∇×(∇×a)=∇(∇·a)−∇^2 a, (10.41)where (10.39) and (10.40) are expressed in Cartesian coordinates. In (10.41), the
term∇^2 ahas 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.