
10-6 Calculus – Differentiation and Integration
Proof:
∇ = ∇ + ∇ = ∇ + = ∇
∴ ∇ = ∇
( )
( ) .
kf k f f k k f f k f
kf k f
0
THEOREM 10.4 GRADIENT OF QUOTIENT OF TWO SCALAR POINT FUNCTIONS If f and g are two
scalar point functions, then
grad
i.e.,
f
g
g f f g
g
f
g
g f
=
−
∇
=
∇
grad grad
2
−− ∇
≠
f g
g
g
2
0where .
Proof: By definition,
grad
f
g
i
x
f
g
i
g
f
x
f
g
x
g
= ∑
∂
∂
= ∑
∂
∂
−
∂
∂
2
=
∑
∂
∂
− ∑
∂
∂
=
−
g i
f
x
f i
g
x
g
f f g
g
2
2
g grad grad
.
Corollary
∇
=
−
∇
1 1
2
g
g
g.
Proof:
∇
=
∇ − ∇
=
−∇
=
−
∇
1 1 1
0 1
2
2 2
g
g g
g
g g
g g
g.
10.2.1 Geometrical Interpretation of Grad f
THEOREM 10.5 Grad f is a vector normal to the surface ...