FLUXES OF ENERGY IN THE ELECTROMAGNETIC FIELD. 
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used when a moving particle is followed, q being its velocity. Thus, 
0 0B 0A 
f=r AB — A ^ + B ~?r- 
0/ dt dt 
= AB + BA + A.qV.B + B.qV.A. 
Here qV is a scalar differentiator given by 
cl d d 
qV = q 'dx + q *dy + q *~dz' ' 
• • (16) 
(17) 
so that A.qV.B is the scalar product of A and the vector qV.B ; the dots here again 
act essentially as separators. Otherwise, we may write it A (qV) B. 
The fictitious vector V given by 
V = iV 1 +jV 2 + kV 3 = i^ + j^ + k^.(18) 
is very important. Physical mathematics is very largely the mathematics of V. The 
name Nabla seems, therefore, ludicrously inefficient. In virtue of i, j, k, the operator 
V behaves as a vector. It also, of course, differentiates what follows it. 
Acting on a scalar P, the result is the vector 
VP = iVjP + jVoP + kV 3 P, 
(19) 
the vector rate of increase of P with length. 
If it act on a vector A, there is first the scalar product 
VA = ViAj + V 2 A 3 + V 3 A 3 = div A,.(20) 
or the divergence of A. Regarding a vector as a flux, the divergence of a vector is 
the amount leaving the unit volume. 
The vector product WA is 
YVA --= i (VoA s - V 3 A 2 ) + j (V 3 A! - VjAg) + k (y^ - V.A,), 
== curl A. (21) 
The line-integral of A round a unit area equals the component of the carl of A 
perpendicular to the area. 
We may also have the scalar and vector products NV and VHV, where the vector 
M is not differentiated. These operators, of course, require a function to follow them 
on which to operate; the previous qV.A of (16) illustrates. 
The Laplacean operator is the scalar product V 2 or VV : or 
V* = vy + Vo 3 + V 3 3 ; 
(22) 
