Some comments on Grad, Div, Curl in Chap. 8
- Steepest Descent Method (Gradient Method) for optimization problems see next slides
- Some basic formulas for grad, div, curl
0 ) v (
0 ) f (
) v (
u ) u (
v ) v u (
v f
v f )
v f (
g f g f
2 f g ) fg (
f f
f g g
f f g g
f ) g f (
w v
) v ( w ) w v ( f
v v f
) v f (
) g f f g )(
g / 1 ( ) g / f (
f g g f ) fg (
2 2
2 2
2 2
=
×
∇
⋅
∇
=
∇
×
∇
×
∇
⋅
−
×
∇
⋅
=
×
⋅
∇
×
∇ +
×
∇
=
×
∇
∇ +
∇
⋅
∇ +
∇
=
∇
∇
⋅
∇
=
∇
∇
⋅
∇ +
∇
=
∇
⋅
∇ +
∇
⋅
∇
=
∇
⋅
∇
∇
⋅
=
⋅
∇
=
⋅
∇
∇
⋅
=
⋅
∇
=
⋅
∇
∇
−
∇
=
∇
∇ +
∇
=
∇
- The form of grad, div, curl in curvilinear coordinates (see Appendix A3.4) Special cases: cylindrical, spherical coordinates
(a) Cylindrical coordinates:
z z ), x / y ( tan ,
y x
r
z z , sin r y , cos r x
1 2
2 + = =
=
=
=
=
θ
−θ θ
z y
x z
z y
x z
y x
z y
x z
y x
r
) 1 ( )
0 ( )
0 (
) 0 ( )
(cos )
sin z (
s y
s x
s
) 0 ( )
(sin )
z (cos r y
r x
r
δ δ
δ δ
δ δ
θ δ
θ δ
δ δ
δ
δ δ
θ δ
θ δ
δ δ
δ
θ θ
θ θ
+ +
=
+ +
−
∂ = + ∂
∂ + ∂
∂
= ∂
+ +
∂ = + ∂
∂ + ∂
∂
= ∂
) z 1 ( )
0 r ( ) 0 z (
) z 0 ( r )
(cos ) r
y (sin
) z 0 ( r )
( sin ) r
x (cos
∂ + ∂
∂ + ∂
∂
= ∂
∂
∂
∂ + ∂
∂ + ∂
∂
= ∂
∂
∂ ∂
+ ∂
∂
− ∂
∂ +
= ∂
∂
∂
θ
θ θ θ
θ θ θ
z r
z
z r
y
z r
x
) 1 ( )
0 ( )
0 (
) 0 ( )
(cos )
(sin
) 0 ( )
sin ( )
(cos
δ δ
δ δ
δ δ
θ δ
θ δ
δ δ
θ δ
θ δ
θ
θ θ
+ +
=
+ +
=
+
− +
=
θ rd
θ
ds =z 0 , z 0
, z 0
0 ,
,
r 0 , r 0
, r 0
z r
z r
r
z r
∂ =
= ∂
∂
= ∂
∂
∂
∂ =
− ∂
∂ =
= ∂
∂
∂
∂ =
= ∂
∂
= ∂
∂
∂
δ δ
δ
θ δ δ
θ δ δ
θ δ
δ δ
δ
θ θ θ
θ
z r
1 r
z y
x
z r
z y
x
∂ + ∂
∂ + ∂
∂
= ∂
∂ + ∂
∂ + ∂
∂
= ∂
∇
θ δ δ
δ
δ δ
δ
θ
( )
z v r
v v
r 1 r v
v v
z v r
1 v r
z r
r
z z r r
z r
∂ + ∂
∂ + + ∂
∂
= ∂
+ +
⋅
∂ + ∂
∂ + ∂
∂
= ∂
⋅
∇
θ
δ δ
δ θ δ
δ δ
θ
θ θ θ
(b) Spherical coordinates:
) x / y ( tan ),
z / y x
( tan ,
z y
x r
cos r z , sin sin
r y , cos sin
r x
1 2
2 1
2 2
2 + + = − + = −
=
=
=
=
φ θ
θ φ
θ φ
θ
φ θ
θ θ
φ θ φ θ
φ φ θ
θ
φ θ φ θ
φ φ θ
θ
∂ + ∂
∂
− ∂
∂ +
= ∂
∂
∂
∂ + ∂
∂ + ∂
∂
= ∂
∂
∂
∂
− ∂
∂ + + ∂
∂
= ∂
∂
∂
) 0 ( r )
( sin ) r
z (cos
sin ) r ( cos r )
sin (cos
) r sin y (sin
sin ) r ( sin r )
cos (cos
) r cos x (sin
y θ P
φ r
x z
θ δ
θ δ
φ δ θ
δ φ δ
θ δ
φ δ
θ δ δ
θ δ δ
θ δ
δ δ
δ
θ φ
φ θ
φ
φ θ
θ
φ θ
cos sin
, cos ,
sin
0 ,
,
r 0 , r 0
, r 0
r r
r r
r
−
−
∂ =
= ∂
∂
= ∂
∂
∂
∂ =
− ∂
∂ =
= ∂
∂
∂
∂ =
= ∂
∂
= ∂
∂
∂
= ???
∇
???
v =
⋅
∇
z y
x
z y
x
z y
x r
) 0 ( )
(cos )
sin (
) sin ( )
sin (cos
) cos (cos
) (cos )
sin (sin
) cos (sin
δ δ
φ δ
φ δ
δ θ δ
φ θ
δ φ θ
δ
δ θ δ
φ θ
δ φ θ
δ
φ θ
+ +
−
=
− + +
=
+ +
=
φ θ
φ θ
φ θ
δ δ
θ δ
θ δ
δ φ δ
φ θ
δ φ θ
δ
δ φ δ
φ θ
δ φ θ
δ
) 0 ( )
sin ( )
(cos
) (cos )
sin (cos )
sin (sin
) sin ( )
cos (cos
) cos (sin
r z
r y
r x
+
− +
=
+ +
=
− + +
=
φ θ
θ
φθ rd , ds rsin d
ds = =
