B.1 Derivatives
605
with V = L 3 for a cubic region. Next, the operations of forming the
differential and carrying out the triple integration over the region may be
interchanged, to give dg as
dg =
+
1
2
−
1
2
+
1
2
−
1
2
+
1
2
−
1
2
d(Vf ) dξ dηdζ ,
or equivalently, as
dg =
+
L
2
−
L
2
+
L
2
−
L
2
+
L
2
−
L
2
1
L 3 d(Vf ) dxdydz .
Finally, this expression for dg may be rewritten in the form
dg =
V
1
V
d(Vf ) d
3 r .
B.1.3 Inverse Differential Operator: Antiderivative
Amongst the operators defined in finite-difference calculus are: 2 the difference
operator, , the shifting operator, E, the differential operator, D, and the integral
operator, J . These four operators are defined in terms of their effects on a function
f (x), specifically, via
Ef (x) = f (x + h) ,
(B.1.16a)
(x) = f (x + h) − f (x) ,
(B.1.16b)
Df (x) = f
(x) ,
(B.1.16c)
J f (x) =
x+h
x
f (x) dx .
(B.1.16d)
Positive integer powers of these operators are defined by iteration, and the zeroth
power of any operator is the identity operator, 1 , that leaves any function unchanged.
All four of these operators are linear operators, which means that if L is a linear
operator, then L[af (x) + bg(x)] = aLf (x) + bLg(x).
Two operators O 1 and O 2 are said to be equal if O 1 f (x) = O 2 f (x) for
all functions f (x) for which the operations are defined. Each of the four linear
operators defined above possesses the commutative, distributive, and associative
2 See, for example, F. Hildebrand, Introduction to Numerical Analysis, 2nd Ed. (McGraw–Hill,
New York, 1974).