
10-16 Calculus – Differentiation and Integration
=
∑
⋅
∂
∂
+
∂
∂
=
∑
⋅
∂
∂
+
∑
⋅
∂
∂
= +
i
f
x
g
x
i
f
x
i
g
x
f g
div div .
In a similar way, by replacing ‘×’ (dot) by ´ (cross), we get
curl ( )f g f g+ = +curl curl .
Identity 2
If f is a differential vector function and f is a differentiable scalar function, then
div ( ) grad div
i.e.,
f f f
f f f
f f f
f f f
= ⋅ +
∇⋅
( )
= ∇ ⋅ + ∇⋅
( )
( )
( ) .
Proof:
div f f
f
f
f
f i
x
f
i
x
f
f
x
i
x
( )
=
∑
⋅
∂
∂
( )
=
∑
⋅
∂
∂
+
∂
∂
=
∂
∂
∑
⋅ + ⋅
∂
∂
∑
= ∇
( )
⋅ + ∇⋅
( )
= ⋅ +
f i
f
x
f f
f f
f
f f
f f
( ) ( ).grad div
Identity 3
If
f is a differentiable vector function and f is a differentiable scalar function, then
curl( ) grad curlf f ff f f=