Երկու
∇
{\displaystyle \nabla }
համարժեքություն (երկրորդ կարգի օպերատոր)
խմբագրել
∇
×
(
∇
ψ
)
=
0
{\displaystyle \ \nabla \times (\nabla \psi )=0}
r
o
t
(
g
r
a
d
ψ
)
=
0
{\displaystyle \ \mathbf {rot} (\mathbf {grad} \ \psi )=0}
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \ \nabla \cdot (\nabla \times \mathbf {A} )=0}
d
i
v
(
r
o
t
A
)
=
0
{\displaystyle \ \mathbf {div} \ (\mathbf {rot} \ \mathbf {A} )=0}
Δ
ψ
=
∇
⋅
(
∇
ψ
)
=
∇
2
ψ
{\displaystyle \ \Delta \ \psi =\nabla \cdot (\nabla \psi )=\nabla ^{2}\psi }
Δ
ψ
=
d
i
v
(
g
r
a
d
ψ
)
{\displaystyle \ \Delta \ \psi =\mathbf {div} \ (\mathbf {grad} \ \psi )}
∇
×
∇
×
A
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \ \nabla \times \nabla \times \mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
r
o
t
(
r
o
t
A
)
=
g
r
a
d
(
d
i
v
A
)
−
Δ
A
{\displaystyle \ \mathbf {rot} \ (\mathbf {rot} \ \mathbf {A} )=\mathbf {grad} \ (\mathbf {div} \ \mathbf {A} )-\Delta \mathbf {A} }
Դաշտերի դիֆերենցում
խմբագրել
∇
⋅
(
ψ
A
)
=
A
⋅
∇
ψ
+
ψ
∇
⋅
A
{\displaystyle \nabla \cdot (\psi \mathbf {A} )=\mathbf {A} \cdot \nabla \psi +\psi \nabla \cdot \mathbf {A} }
d
i
v
(
ψ
A
)
=
A
⋅
g
r
a
d
ψ
+
ψ
d
i
v
A
{\displaystyle \mathbf {div} (\psi \mathbf {A} )=\mathbf {A} \cdot \mathbf {grad} \psi +\psi \ \mathbf {div} \mathbf {A} }
∇
×
(
ψ
A
)
=
∇
ψ
×
A
+
ψ
∇
×
A
{\displaystyle \nabla \times (\psi \mathbf {A} )=\nabla \psi \times \mathbf {A} +\psi \nabla \times \mathbf {A} }
r
o
t
(
ψ
A
)
=
g
r
a
d
ψ
×
A
+
ψ
r
o
t
A
{\displaystyle \mathbf {rot} (\psi \mathbf {A} )=\mathbf {grad} \psi \times \mathbf {A} +\psi \ \mathbf {rot} \mathbf {A} }
∇
(
A
⋅
B
)
=
(
A
⋅
∇
)
B
+
(
B
⋅
∇
)
A
+
{\displaystyle \nabla (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +}
+
A
×
(
∇
×
B
)
+
B
×
(
∇
×
A
)
{\displaystyle +\mathbf {A} \times (\nabla \times \mathbf {B} )+\mathbf {B} \times (\nabla \times \mathbf {A} )}
g
r
a
d
(
A
⋅
B
)
=
(
A
⋅
∇
)
B
+
(
B
⋅
∇
)
A
+
{\displaystyle \ \mathbf {grad} (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +}
+
A
×
r
o
t
B
+
B
×
r
o
t
A
{\displaystyle +\mathbf {A} \times \mathbf {rot} \mathbf {B} +\mathbf {B} \times \mathbf {rot} \mathbf {A} }
1
2
∇
A
2
=
A
×
(
∇
×
A
)
+
(
A
⋅
∇
)
A
{\displaystyle {\frac {1}{2}}\nabla A^{2}=\mathbf {A} \times (\nabla \times \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {A} }
1
2
g
r
a
d
A
2
=
A
×
(
r
o
t
A
)
+
(
A
⋅
∇
)
A
{\displaystyle {\frac {1}{2}}\ \mathbf {grad} A^{2}=\mathbf {A} \times (\mathbf {rot} \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {A} }
∇
⋅
(
A
×
B
)
=
B
⋅
∇
×
A
−
A
⋅
∇
×
B
{\displaystyle \nabla \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot \nabla \times \mathbf {A} -\mathbf {A} \cdot \nabla \times \mathbf {B} }
d
i
v
(
A
×
B
)
=
B
⋅
r
o
t
A
−
A
⋅
r
o
t
B
{\displaystyle \mathbf {div} \ (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot \mathbf {rot} \ \mathbf {A} -\mathbf {A} \cdot \mathbf {rot} \ \mathbf {B} }
∇
×
(
A
×
B
)
=
A
(
∇
⋅
B
)
−
B
(
∇
⋅
A
)
+
{\displaystyle \ \nabla \times (\mathbf {A} \times \mathbf {B} )=\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )+}
+
(
B
⋅
∇
)
A
−
(
A
⋅
∇
)
B
{\displaystyle \;+(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B} }
r
o
t
(
A
×
B
)
=
A
(
d
i
v
B
)
−
B
(
d
i
v
A
)
+
{\displaystyle \ \mathbf {rot} (\mathbf {A} \times \mathbf {B} )=\mathbf {A} \ (\mathbf {div} \ \mathbf {B} )-\mathbf {B} \ (\mathbf {div} \ \mathbf {A} )+}
+
(
B
⋅
∇
)
A
−
(
A
⋅
∇
)
B
{\displaystyle \;+(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B} }
[1] [2]
Ծանոթագրություններ
խմբագրել
↑ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lecture on Physics . Addison-Wesley. Vol II, p. 27–4. ISBN 0-8053-9049-9 .
↑ Kholmetskii, A. L.; Missevitch, O. V. (2005). «The Faraday induction law in relativity theory». arXiv :physics/0504223 . ;