
Differential Operators 10-11
⇒
∂
∂
= + + ⋅
= ⋅ = ⋅
−
−
−
f
x
x y z x
nx r nx r
n
n
n
2
2
2 2 2
1
2
1
2
2
2
( )
( ) .
Similarly,
∂
∂
= ⋅
∂
∂
= ⋅
=
∂
∂
+
∂
∂
+
∂
∂
=
− −
f f
f
f f f
y
ny r
z
nz r
i
x
j
y
k
z
n
n n2 2
and
grad
(
xxi y j zk r nr r
n n
+ + = ⋅
− −
) .
2 2
Example 10.5 Show that
grad ( )a r a⋅ =
Solution:
a r a x a y a z
a r i
x
a r
i
x
a x a y a z
a
⋅ = + +
⋅ = ∑
∂
∂
⋅
= ∑
∂
∂
+ +
( )
=
1 2 3
1 2 3
grad ( ) ( )
11 2 3
i a j a k
a
+ +
= .
10.3 DIVERGENCE AND CURL OF A VECTOR POINT FUNCTION
If f be a differentiable vector point function, then the divergence of f written as div f f= ∇⋅( ) is
defined as:
div f i
f
x
j
f
y
k
f
z
i
x
j
y
k
z
= ⋅
∂
∂
+ ⋅
∂
∂
+ ⋅
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂
⋅⋅
= ∇⋅
= ∇⋅ = ∑ ⋅
∂
f
f
f f i
f