ABSTRACT
Let u(x) be a vector-valued function of position vector x, and consider a vector-valued function F(u(x), u′(x), x), in which u′(x)=∂u/∂x. Furthermore, let v(x) be a function such that v(x)=0 when u(x)=0 and v′(x)=0 when u′(x)=0, but which is otherwise arbitrary. The differential dF measures how much F changes if x changes. The variation δF measures how much F changes if u and u′ change at fixed x. Following Ewing, we introduce the vector-valued function as follows (Ewing, 1985):
Φ(e:F)=F(u(x)+ev(x), u′(x)+ev′(x), x)−F(u(x), u′(x), x) (3.1)
The variation δF is defined by
(3.2)
with x fixed. Elementary manipulation demonstrates that
(3.3)
in which If F=u, then δF=δu=ev. If F=u′, then δF=δu′= ev′. This suggests the form
(3.4)
The variational operator exhibits five important properties:
1. δ(.) commutes with linear differential operators and integrals. For example, if S denotes a prescribed contour of integration:
2. δ(f) vanishes when its argument f is prescribed.
