Bài giảng Vật lý điện tử __vector-analysis-formulas-(electromagnetic-fields) | Môn Vật lý điện tử | Trường Đại học Bách Khoa Hà Nội

Bài giảng Vật lý điện tử __vector-analysis-formulas-(electromagnetic-fields) | Môn Vật lý điện tử | Trường Đại học Bách Khoa Hà Nội. Tài liệu gồm 2 trang giúp bạn tham khảo ôn tập đạt kết quả cao trong kỳ thi sắp tới. Mời bạn đọc đón xem.

39 20 lượt tải Tải xuống
Chia sẻ Báo cáo tài liệu

Môn: Vật lý điện tử

Trường: Đại học Bách Khoa Hà Nội

Thông tin:

2 trang 3 tháng trước

Tác giả:

Profile Trịnh Thảo Anh
PHY2206 (Electromagnetic Fields) Vector Analysis Formulae
1
Vector Analysis Formulae
Identities
1
• A × B
()
⋅C=A⋅B×C
()
2•A×B×C
()
=BA⋅C
()
−CA⋅B
()
3 A×B
()
⋅C×D
()
=A⋅C
()
B⋅D
()
−A⋅D
()
B⋅C
()
4 A×B
()
×C×D
()
=CA⋅B×D
()
{}
−DA⋅B×C
()
{}
5A×B
()
×C×D
()
=BA⋅C×D
()
{}
−AB⋅C×D
()
{}
6•∇fg
()
=f∇g+g∇f
7 ∇fg
()
=1g
()
∇f−fg
2
()
∇g
8 ∇A⋅B
()
=B⋅∇
()
A+A⋅∇
()
B+B×∇×A
()
+A×∇×B
()
9 ∇⋅ fA
()
=∇f
()
⋅A+f∇⋅A
()
10 • ∇⋅ A×B
()
=B⋅∇×A
()
−A⋅∇×B
()
11 • ∇⋅∇
()
f=∇
2
f
12 • ∇× ∇f
()
=0
13 • ∇⋅ ∇×A
()
=0
14 ∇× fA
()
=∇f
()
×A+f∇×A
()
15 ∇× A×B
()
=B⋅∇
()
A−A⋅∇
()
B+∇⋅B
()
A−∇⋅A
()
B
16a • ∇× ∇×A
()
=∇∇⋅A
()
−∇
2
A
16b ∇
2
A =∇⋅ ∇⋅A
()
−∇× ∇×A
()
17 • ∇ 1/r
()
=−
ˆ
r/r
2
If S is the closed surface that encloses the volume V and C is the closed curve that bounds an open
surface A then:
18
• ∇f
()
⋅dl
a
b
∫
=fb
()
−fa
()
19 ∇f
()
dV
V
∫
=fdS
S
∫
20 ∇×B
()
dV
V
∫
=− B×dS
S
∫
21 • B⋅dS
S
∫
=∇⋅B
()
dV
V
∫
(The Divergence Theorem)
22 • B⋅dl
C
∫
=∇×B
()
⋅dA
A
∫
(Stokes’s Theorem)
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu duong than cong . com
PHY2206 (Electromagnetic Fields) Vector Analysis Formulae
2
© Copyright CDH Williams
Exeter 1996, CW000127/1
Special Coordinate Systems
Cartesian Coordinates x,y,z
()
•∇f=
∂
f
∂
x
ˆ
x+
∂
f
∂
y
ˆ
y+
∂
f
∂
z
ˆ
z
•∇⋅A=
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
• ∇×A=
∂
A
z
∂
y
−
∂
A
y
∂
z
ˆ
x+
∂
A
x
∂
z
−
∂
A
z
∂
x
ˆ
y+
∂
A
y
∂
x
−
∂
A
x
∂
y
ˆ
z
• ∇
2
f =
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
• ∇
2
A=∇
2
A
x
ˆ
x+∇
2
A
y
ˆ
y+∇
2
A
z
ˆ
z=∇∇⋅A
()
−∇× ∇×A
()
Cylindrical Polar Coordinates r,
θ
,z
()
∇f=
∂
f
∂
r
ˆ
r+
1
r
∂
f
∂θ
ˆ
θθ
+
∂
f
∂
z
ˆ
z
∇⋅A=
1
r
∂
∂
r
rA
r
()
+
1
r
∂
A
θ
∂θ
+
∂
A
z
∂
z
∇×A=
1
r
∂
A
z
∂θ
−
∂
A
θ
∂
z
ˆ
r+
∂
A
r
∂
z
−
∂
A
z
∂
r
ˆ
θθ
+
1
r
∂
∂
r
rA
θ
()
−
∂
A
r
∂θ
ˆ
z
∇
2
f=
1
r
∂
∂
r
r
∂
f
∂
r
+
1
r
2
∂
2
f
∂θ
2
+
∂
2
f
∂
z
2
Spherical Polar Coordinates r,
θ
,
ϕ
()
∇f=
∂
f
∂
r
ˆ
r+
1
r
∂
f
∂θ
ˆ
θθ
+
1
rsin
θ
∂
f
∂ϕ
ˆ
ϕϕ
∇⋅A=
1
r
2
∂
∂
r
r
2
A
r
()
+
1
rsin
θ
∂
∂
θ
A
θ
sin
θ
()
+
1
rsin
θ
∂
A
ϕ
∂ϕ
∇×A=
1
rsin
θ
∂
∂
θ
A
ϕ
sin
θ
()
−
∂
A
θ
∂ϕ
ˆ
r+
1
r
1
sin
θ
∂
A
r
∂ϕ
−
∂
∂
r
rA
ϕ
()
ˆ
θ
+
1
r
∂
∂
r
rA
θ
()
−
∂
A
r
∂θ
ˆ
ϕ
∇
2
f=
1
r
2
∂
∂
r
r
2
∂
f
∂
r
+
1
r
2
sin
θ
∂
∂
θ
sin
θ
∂
f
∂θ
+
1
r
2
sin
2
θ
∂
2
f
∂ϕ
2
CuuDuongThanCong.com https://fb.com/tailieudientucntt
cuu duong than cong . com
| 1/2
| | Tải xuống

Có thể bạn quan tâm:

Preview text:

PHY2206 (Electromagnetic Fields) Vector Analysis Formulae Vector Analysis Formulae Identities 1
• (A × B)⋅C = A⋅(B × C) 2
• A × (B × C) = B(A⋅C) − C(A⋅B) 3
(A × B)⋅(C × D) = (A⋅C)(B⋅D) − (A⋅D)(B⋅C) 4
(A × B) × (C × D) = C A
{ ⋅(B× D)}− D A { ⋅(B× C)} 5
(A × B) × (C × D) = B A
{ ⋅(C× D)}− A B { ⋅(C× D)} 6
• ∇( fg) = f ∇g + g∇f 7 ∇( f g) = 1
( g)∇f − f g2 ( )∇g 8
∇(A⋅B) = (B⋅∇)A + (A⋅∇)B + B × (∇ × A) + A × (∇ × B) 9
∇⋅( fA) = (∇f )⋅A + f (∇⋅A)
10 • ∇⋅(A × B) = B⋅(∇ × A) − A ⋅(∇ × B)
11 • (∇⋅ ∇) f = ∇2 f
12 • ∇ × (∇f ) = 0
13 • ∇⋅(∇ × A) = 0 14
∇ × ( fA) = (∇f ) × A + f (∇ × A) 15
∇ × (A × B) = (B⋅∇)A − (A ⋅∇)B + (∇ ⋅ B)A − (∇ ⋅ A)B
16a • ∇ × (∇ × A) = ∇(∇ ⋅ A) − ∇2A 16b
∇2A = ∇ ⋅(∇ ⋅ A) − ∇ × (∇ × A) 17 • ∇ 1
( / r) = −ˆr / r2 cuu duong than cong . com
If S is the closed surface that encloses the volume V and C is the closed curve that bounds an open surface A then: b 18
• ∫ (∇f )⋅ dl = f(b) − f(a) a 19
∫ (∇f) dV = f dS ∫ V S 20
∫ (∇ × B)dV = − B× dS ∫ V S 21 • B⋅ dS ∫
= ∫ (∇⋅B) dV (The Divergence Theorem) S V 22 • B⋅ dl ∫
= ∫ (∇ × B)⋅dA (Stokes’s Theorem) C A 1 CuuDuongThanCong.com
https://fb.com/tailieudientucntt
PHY2206 (Electromagnetic Fields) Vector Analysis Formulae Special Coordinate Systems
Cartesian Coordinates (x, y, z)
• ∇f = ∂f ˆ ˆ ˆ
∂ x + ∂f y + ∂f z x ∂y ∂z ∂ • ∇ ⋅ A A = ∂Ax + y + ∂Az ∂x ∂y ∂z  ∂  ∂ •   ∇ × A   A A = ∂Az − y − ∂Az y − ∂Ax  ∂y
∂z  ˆx + ∂Ax  ∂z
∂x  ˆy +  ∂x ∂y  ˆz • ∂2 ∂2 ∂2 ∇2 f f f f = + + ∂x2 ∂y2 ∂z2
• ∇2A = ∇2A ˆx + ∇2A ˆy + ∇2A ˆz = ∇(∇ ⋅ A) − ∇ × (∇ × A) x y z
Cylindrical Polar Coordinates (r,θ, z) ∇ ∂f f = ∂f ˆ ˆθ + ∂f ˆ ∂ r + 1 z r r ∂θ ∂z ∇⋅ ∂ ∂A A = 1
(rA )+ 1 θ + ∂Az r ∂r r r ∂θ ∂z ∇ ×  ∂A    ∂  A = 1 z − ∂Aθ ˆ ( )   − ∂Az θ + 1 rAθ − ∂Ar r ∂θ
∂z  ˆr + ∂Ar  ∂z ∂r r ∂r ∂θ ˆz ∇ ∂  ∂ ∂2 2 f f f = 1 r + ∂2 f
r ∂r  ∂r  + 1 r2 ∂θ 2 ∂z2
Spherical Polar Coordinates (r,θ,ϕ) ∇ ∂f ∂f f = ∂f ˆ ˆθ + 1 ˆ ϕ ∂ r + 1 r r ∂θ r sin θ ∂ϕ cuu duong than cong . com ∂ ∇ ⋅ ∂ ∂ Aϕ A = 1 (r2A )+ 1 (Aθsinθ) + 1 r2 ∂r r r sin θ ∂θ r sin θ ∂ϕ ∇ ×  ∂   1 ∂A   ∂  A = 1 A ( r ϕ sin θ ) − ∂Aθ − ∂ rA ( ϕ) (rAθ)− ∂Ar r sin θ ∂θ
∂ϕ ˆr + 1r sin  θ ∂ϕ ∂r
ˆθ + 1r ∂r ∂θ  ˆϕ ∇ ∂  ∂ ∂  ∂ ∂2 2 f f f f = 1 r2 sin θ r2 ∂r  ∂r + 1 r2 sin θ ∂θ  ∂θ  + 1 r2 sin2 θ ∂ϕ 2 © Copyright CDH Williams Exeter 1996, CW000127/1 2 CuuDuongThanCong.com
https://fb.com/tailieudientucntt