1. Ω !
" Ω Ω !
#
$%$%$%%$%
→Ω & %
$ Ω !
' ( ( ( !
% ≥ $
∂
∂
∂
∂
∂
∂
=
∂
∂
$%$%$%$% )
* # !
' !
→Ω Ω∈= $%
$%
$%$%$%$%
∂
∂
+
∂
∂
+
∂
∂
=
+ $% # Ω ( * $% !
→Ω Ω∈= $% !
, $% # ( (
⋅=∧∈∀ $%$%
!
- ( # ( #
$% ∧ !
+.
$%= !
$%$% −−−=∧ !
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
=∧
$%$%$%$%$%$%
$%$%$%$%$%$%$%
touscours.net

2. /
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
= $%$%$%$%$%$%$%
' $% # Ω ( * $% !
!
∇ %0 - $ 1 2
∂
∂
∂
∂
∂
∂
- (
⋅∇=$% ∧∇=$%
+ →Ω 3 - ( ∇= !
" #
Ω →Ω !
' ( (
( = !
4 5*5
∂
∂
∂
∂
∂
∂
=Ω∈∀ $% !
6 7
→Ω #
! →Ω 8$% =Ω∈∀
∂
∂
∂
∂
∂
∂
= !
8$%
//////
=
∂∂
∂
−
∂∂
∂
∂∂
∂
−
∂∂
∂
∂∂
∂
−
∂∂
∂
= # 7
7 9 :!
( % $
Ω 4 5*5 ( # . Ω∈8 (
Ω⊂Ω∈∀ ;< 8
→Ω !
# !
touscours.net

3. " $
% $ , !
%/$
/
8$% = ∆=
∂
∂
+
∂
∂
+
∂
∂
= /
/
/
/
/
/
$% %, $
/
8$% = % 9 :$
% $ =
!!$% +=
⋅+= $%$%$!%
∧+= $%$%$!%
- (
∧∇+∧∇=∧∇
⋅∇+⋅∇=∇
∇+∇=∇
=∧∇⋅∇
∆=∇=∇⋅∇
=∇∧∇
$%$%$!%
$%$%$!%
$%$!%$%
8$%
$%
8$%
/
touscours.net