ˆ + ez
∂vy ∂vx − ∂x ∂y
x2 + y 2 + z 2 ∂ x2 + y 2 + z 2 ∂ ˆ ˆ ex + ey + ∇r = ∂x ∂y x y z r ˆ ˆ ˆ ˆ ∇r = ex + ey + ez = = = er r r r r ∂
x2 + y 2 + z 2 ˆ ez ∂z
4
Let there be light 4. (cross product, vector product)
ˆ A × B = |A||B| sin θ n A × B = −B × A A × (B + C) = A × B + A × C A×A=0
ˆ ˆ ˆ A = Axex + Ay ey + Az ez e e e A + B = (Ax + Bx)ˆx + (Ay + By )ˆy + (Az + Bz )ˆz A · B = AxBx + Ay By + Az Bz A= A·A= A2 + A2 + A2 x y z
13
Let there be light r ˆ ∇r = = er , r df ∇f (u) = ∇u du d A(u) ∇ × A(u) = (∇u) × du
d A(u) , ∇ · A(u) = (∇u) · du ∇r2 = 2r∇r = 2r 1 1 1 r ˆ ∇ = − 2 ∇r = − 2 er = − 3 r r r r ∂ ∂ ∂ ˆ e e ∇ · r = ex +ˆy +ˆz ∂x ∂y ∂z ∇×r = ˆ ˆ ˆ ex ey ez
C(f g) = g Cf = f Cg ∇ ∇
A f
∇f ∇ALeabharlann C · (f A) = A · (Cf ) = f (C · A)
∇ ∇ ∇ ∇
16
Let there be light
∇ × (f A) = ∇f × (f A) + ∇A × (f A)
∇ 1. ∇ f A 2. ∇ ∇
A f
∇f ∇A
9
Let there be light
§ 1.2
∂T ∂T ∂T ˆ ˆ ˆ ∇T = ex + ey + ez ∂x ∂y ∂z ∂T ∂T ∂T dx + dy + dz dT = ∂x ∂y ∂z = (∇ T ) · (d l) = |∇T | |d l| cos θ ∇T T
T
10
Let there be light
Let there be light
• § 1.1 • § 1.2 • § 1.3 • § 1.4 • § 1.5 • § 1.6 Dirac delta • § 1.7
2
Let there be light
§ 1.1
v m 1. commutative A+B =B+A associative (A + B) + C = A + (B + C) r q
2
∇2 f
∂ 2f ∂ 2f ∂ 2f = 2+ 2+ 2 ∂x ∂y ∂z
Laplacian
∂2 ∂2 ∂2 ˆ ˆ ˆ ∇2 A = + 2 + 2 (Axex + Ay ey + Az ez ) 2 ∂x ∂y ∂z ∂ 2Ax ∂ 2Ax ∂ 2Ax ∂ 2Ay ∂ 2Ay ∂ 2Ay ˆ ˆ = ex + ey + + + + 2 2 2 2 2 2 ∂x ∂y ∂z ∂x ∂y ∂z ∂ 2Az ∂ 2Az ∂ 2Az ˆ ez + + + Laplacian 2 2 2 ∂x ∂y ∂z ∇2A = (∇2Ax)ˆx + (∇2Ay )ˆy + (∇2Az )ˆz e e e
20
Let there be light
∇(A · B) = ∇A(A · B) + ∇B (A · B)
∇ A 1. ∇ B 2. ∇ B A ∇A ∇B
= B × (∇A × A) + (B · ∇A)A + A × (∇B × B) + (A · ∇B )B
∇
B × (C × A) = C(A · B) − (B · C)A ⇓ C(A · B) = B × (C × A) + (B · C)A C(A · B) = A × (C × B) + (A · C)B
8
A × (B × C) + B × (C × A) + C × (A × B) = 0
Let there be light
ˆ ˆ r = xˆx + y ey + z ez e √ r = |r| = r · r = x2 + y 2 + z 2 ˆ er = r/r
ˆ ˆ ˆ dl ≡ dr = dx ex + dy ey + dz ez R ≡ r − r = (x − x )ˆx + (y − y )ˆy + (z − z )ˆz e e e r field point source point r R = |R|= (x − x )2 + (y − y )2 + (z − z )2
=⇒ =⇒ =⇒
∇T ∇·v ∇×v
(gradient) (divergence) (curl)
∂ ∂ ∂ ˆ e e ∇ · v = ex +ˆy +ˆz ∂x ∂y ∂z ∂vx ∂vy ∂vz + + = ∂x ∂y ∂z
ˆ ˆ ˆ · (vxex + vy ey + vz ez )
12
Let there be light
∂ ∂x ∂ ∂y ∂ ∂z
ˆ ˆ · (xˆx + y ey + z ez ) = 3 e
=0
x
y
z
14
Let there be light
∇
∇ (f + g) = ∇ f + ∇ g ∇ · (A + B) = ∇ · A + ∇ · B ∇ × (A + B) = ∇ × A + ∇ × B k ∇ (kf ) = k ∇ f ∇ · (k A) = k ∇ · A ∇ × (k A) = k ∇ × A
∇ A 1. ∇ B 2. ∇ B A ∇A ∇B
= B · (∇A × A) − A · (∇B × B)
∇
C · (A × B) = B · (C × A) = −A · (C × B)
∇ ∇ ∇ ∇
∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
18
Let there be light
∂ ∂ ∂ ˆ e e ∇ × v = ex +ˆy +ˆz ∂x ∂y ∂z = ˆ ˆ ˆ ex ey ez vx vy vz ˆ = ex
∂ ∂x ∂ ∂y ∂ ∂z
ˆ ˆ ˆ × (vxex + vy ey + vz ez )
∂vz ∂vy − ∂y ∂z
ˆ + ey
∂vx ∂vz − ∂z ∂x
∇ × (A × B) = ∇A × (A × B) + ∇B × (A × B)
∇ A 1. ∇ B 2. ∇ B A ∇A ∇B
= (B · ∇A)A − B(∇A · A) + A(∇B · B) − (A · ∇B )B
∇
C × (A × B) = (B · C)A − B(C · A) = A(C · B) − (A · C)B
6
Let there be light Levi-Civita
1. 2. 3. εijk εmnk = δimδjn − δinδjm =
k
ˆ ˆ ei × ej =
k
ˆ εijk ek
εijk = −εikj δim δin δjm δjn
4. εijk εmjk
j, k
n=j = δimδjj − δij δmj = 3δim − δim = 2δim m=i εijk εijk
∇
∂T ∂T ∂ ∂ ∂ ∂T ˆ ˆ ˆ ˆ ex + ey + ez = ex +ˆy +ˆz e e T ∇T = ∂x ∂y ∂z ∂x ∂y ∂z del ∇ ∂ ∂ ∂ ˆ ∇ = ex +ˆy +ˆz e e ∂x ∂y ∂z
11
Let there be light
Aa A·B A×B
i, j , k
7
5.
= 2δii = 6
Let there be light
1.
(scalar triple product
)
A·(B×C) = B·(C×A) = C·(A×B) A x Ay Az Bx By Bz Cx Cy Cz
A · (B × C) =
2.
(vector triple product) A × (B × C) = (A · C)B − (A · B)C A × (B × C) = (A × B) × C not associative
15
Let there be light f g, A · B, f A, A × B
∇(f g) = ∇f (f g) + ∇g (f g)