
Chapter 1 2
Systems of Linear
Partial Differential Equations
12.1 Preliminary Remarks. Some Notation and Helpful
Relations
Let f and u = u e
1
+ v e
2
+ w e
3
be arbitrary sufficiently smooth scalar and vector func-
tions, and let e
1
, e
2
, and e
3
be the unit coordinate vectors corresponding to the Cartesian
coordinates x, y, and z. Then, by definition, w e have
∇f = f
x
e
1
+ f
y
e
2
+ f
z
e
3
,
div u = ∇ · u = u
x
+ v
y
+ w
z
,
curl u = ∇ ×u = (w
y
− v
z
)e
1
+ (u
z
− w
x
)e
2
+ (v
x
− u
y
)e
3
,
∆f = f
xx
+ f
y y
+ f
zz
,
where the subscripts x, y, and z stand for the derivatives. The following differential rela-
tions hold:
curl ∇f = 0, div curl u = 0, div ∇f = ∆ f,
curl curl u = ∇div u −∆u, ∆(xf ) = x