矢量与张量初步
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外积
A × a = Aijeie j × ak ek = ε jkl Aij ak eiel = −ε klj Ail ak eie j
a × A = aiei × Ajk e jek = ε ijl Ajk aiel ek = ε kli Alj ak eie j
a × A = −(AT × a)T
AT = A a × A = −(A × a)T
AT = −A a × A = (A × a)T
A × a = −ε ijkωk eie j × al el = aω − (a ⋅ ω)I
向量分析
Hamilton 算子
∇
=
ei
∂ ∂xi
r = xiei = xi′e′i x j = Cij xi′
∂ ∂xi′
∂q3
∇×a =
1 H1H 2 H 3
H1e1 ∂ / ∂q1 H1a1
H 2e2 ∂ / ∂q2 H 2a2
H 3e3 ∂ / ∂q3 H 3a3
∇ 2ϕ = 1 [ ∂ ( H 2 H 3 ∂ϕ ) H1H 2 H 3 ∂q1 H1 ∂q1
+
∂
( H1H 3 ∂ϕ ) +
∂
( H1H 2
∂ϕ )]
Gauss 公式
∫∫∫∇ ⋅ A dV = ∫∫ A ⋅ dS
Ω
∂Ω
Stokes 公式
∫∫ (∇ × A) ⋅ dS = ∫ A ⋅ dr
S
∂S
各向同性张量
H ′ = H i1i2Lin
i1i2Lin
零阶张量:标量 (a)
一阶张量:向量(a = 0)
二阶张量: H ij = λδ ij
三阶张量: H ijk = λε ijk
= ∂2 ∂xi ∂xi
Gauss 公式 ∫∫∫∇ ⋅ a dV = ∫∫a ⋅ dS
Ω
∂Ω
Stokes 公式 ∫∫ (∇ × a) ⋅ dS = ∫ a ⋅ dr
S
∂S
无旋场与保守场
张量与张量之单重积
A ⋅ B = Aijeie j ⋅ Bkl ek el = Aij B jl eiel = Aik Bkjeie j
面元: dAi = H j H k dq j dqk
体元: dV = H1H 2 H 3dq1dq2dq3
∇ϕ
=
1 H1
∂ϕ ∂q1
e1
+
1 H2
∂ϕ ∂q2
e2
+
1 H3
∂ϕ ∂q3
e3
∇⋅a =
1
[ ∂(a1H
2H
3
)
+
∂(a2
H1H3
)
+
∂(a3 H1 H
2
) ]
H1H 2 H 3
∂q1
∂q2
坐标变换(正交坐标变换)
e′i = Cij e j ei = C jie′j
A = Aij eie j = Aij CkiClj e′k e′l
Ai′j = Cik C jl Akl
Aij = CkiClj Ak′l
n = 0 p′ = p
n =1
α p′ = p i1
i1 ji j1
α α n = 2 p′ = p i1i2
⎜⎝ − ω2 ω1
0 ⎟⎠
⎪⎧ Aij = −ε ω ijk k
⎪⎩⎨ωi
=
1 2
ε
ijk
A jk
轴向量 ω = ωiei
A = 1 (A + AT ) + 1 (A − AT )
2
2
单位张量 I = δ ijeie j 迹 tr(A) = Aii
内积
向量与张量之积
A ⋅ a = Aij eie j ⋅ ak e k = Aij a jei
=
∂ ∂x j
∂x j ∂xi′
= Cij
∂ ∂x j
函数梯度
∇ϕ
=
ei
∂ϕ ∂xi
向量散度
∇⋅a
=
ei
∂ ∂xi
⋅aje j
=
∂ai ∂xi
向量旋度
∇×a
=
ei
∂ ∂xi
× aje j
= ε ijk
∂a j ∂xi
ek
Laplace 算子
Δ = ∇2
= ∇ ⋅∇ = ei
∂ ∂xi
⋅e j
∂ ∂x j
向量梯度
张量分析
∇a = ei
∂ ∂xi
aje j
= a j,ieie j
张量散度
∇ ⋅ A = ei
∂ ∂xi
⋅ Ajk e j e k
=
Aik ,ie k
∇ ⋅ (ϕ
I)
= ei
∂ ∂xi
⋅ϕ
e je j
=
∂ϕ ∂xi
ei
= ∇ϕ
张量旋度
∇× A = ei
∂ ∂xi
× Ajk e jek
=
A jk ,iε ijl el e k
e′i = Cij e j Cij Cik = δ jk
a = ai′e′i = ai′Cij e j = aiei
a j = Cij ai′ ai = C ji a′j
ei = C jie′j ai′ = Cij a j
Kronecker 记号
δ ij
= ei
⋅ej
=
⎧1 ⎩⎨0
i= j i≠ j
四阶张量:
H ijkl = μ (δikδ ij + δ ilδ jk ) + γδ ijδ kl
正交曲线坐标系
r = r(q1, q2 , q3 )
dr = H1dq1e1 + H 2dq2e2 + H 3dq3e3
弧元: dsi = H i dqi (i = 1, 2, 3)
式中 H1, H 2 , H 3 称为 Lame 系数。在正
a ⋅ A = aiei ⋅ A jk e je k = Aik aie k = A ji a jei
a⋅ A = AT ⋅a
AT = A a⋅ A = A⋅a
AT = −A a ⋅ A = −A ⋅a
A ⋅ a = −ε ijkωk eie j ⋅ al el = ε ω kji k a jei = ω × a
置换符号(Levi-Civita)
⎧
ε ijk
=
⎪ ⎨
⎪⎩ 0
1 Even −1 Odd (i − j)( j − k)(k − i) = 0
ei × e j = ε ijk e k
ε − δ 恒等式
ε ε ijk ist = δ jsδ kt − δ jtδ ks
张量代数 定义(并矢基)
a = aiei ⇒ A = Aij eie j
定义
向量代数
r = x i+ y j+ z k
r = x1e1 + x2e2 + x3e3
Einstein 约定求和法则
a = a1e1 + a2e2 + a3e3 = aiei
坐标变换(正交坐标变换)
e1 e2 e3 e1′ C11 C12 C13 e′2 C21 C22 C23 e′3 C31 C32 C33
∂q2 H 2 ∂q2 ∂q3 H 3 ∂q3
交曲线坐标系中,沿坐标轴的微元弧长等于 该坐标轴的微元增量乘以相应的拉梅系数。 其几何意义为坐标轴的单位变化引起的微 元弧长增量。 在直角坐标系中
H1 = 1 H2 = 1 H3 = 1
在柱坐标来自百度文库中
H1 = 1 H2 = r H3 = 1
在球坐标系中
H1 = 1 H 2 = R H 3 = R sinθ
i1 j1 i2 j2 j1 j2
二阶张量的矩阵表示
A ⇔ A = ( Aij ) A = CT A′C
加法: A + B = ( Aij + Bij ) eie j
转置: AT = Ajieie j
对称张量
AT = A
反对称张量 AT = −A
反对称张量的轴向量
⎜⎛ 0 − ω3 ω2 ⎟⎞
A = ⎜ ω3 0 − ω1 ⎟
A × B = Aijeie j × Bkl ek el = Aij Bklε jkpeie pel = Aip Blk ε plj eie j e k
∇ × a = 0 ⇔ ∃ ϕ, a = ∇ϕ
无源场与向量场
∇ ⋅ a = 0 ⇔ ∃ b, a = ∇ × b
Helmholtz 分解
∀a, ∃ϕ,b,∇ ⋅ b = 0 ⇒ a = ∇ϕ + ∇ × b
A × a = Aijeie j × ak ek = ε jkl Aij ak eiel = −ε klj Ail ak eie j
a × A = aiei × Ajk e jek = ε ijl Ajk aiel ek = ε kli Alj ak eie j
a × A = −(AT × a)T
AT = A a × A = −(A × a)T
AT = −A a × A = (A × a)T
A × a = −ε ijkωk eie j × al el = aω − (a ⋅ ω)I
向量分析
Hamilton 算子
∇
=
ei
∂ ∂xi
r = xiei = xi′e′i x j = Cij xi′
∂ ∂xi′
∂q3
∇×a =
1 H1H 2 H 3
H1e1 ∂ / ∂q1 H1a1
H 2e2 ∂ / ∂q2 H 2a2
H 3e3 ∂ / ∂q3 H 3a3
∇ 2ϕ = 1 [ ∂ ( H 2 H 3 ∂ϕ ) H1H 2 H 3 ∂q1 H1 ∂q1
+
∂
( H1H 3 ∂ϕ ) +
∂
( H1H 2
∂ϕ )]
Gauss 公式
∫∫∫∇ ⋅ A dV = ∫∫ A ⋅ dS
Ω
∂Ω
Stokes 公式
∫∫ (∇ × A) ⋅ dS = ∫ A ⋅ dr
S
∂S
各向同性张量
H ′ = H i1i2Lin
i1i2Lin
零阶张量:标量 (a)
一阶张量:向量(a = 0)
二阶张量: H ij = λδ ij
三阶张量: H ijk = λε ijk
= ∂2 ∂xi ∂xi
Gauss 公式 ∫∫∫∇ ⋅ a dV = ∫∫a ⋅ dS
Ω
∂Ω
Stokes 公式 ∫∫ (∇ × a) ⋅ dS = ∫ a ⋅ dr
S
∂S
无旋场与保守场
张量与张量之单重积
A ⋅ B = Aijeie j ⋅ Bkl ek el = Aij B jl eiel = Aik Bkjeie j
面元: dAi = H j H k dq j dqk
体元: dV = H1H 2 H 3dq1dq2dq3
∇ϕ
=
1 H1
∂ϕ ∂q1
e1
+
1 H2
∂ϕ ∂q2
e2
+
1 H3
∂ϕ ∂q3
e3
∇⋅a =
1
[ ∂(a1H
2H
3
)
+
∂(a2
H1H3
)
+
∂(a3 H1 H
2
) ]
H1H 2 H 3
∂q1
∂q2
坐标变换(正交坐标变换)
e′i = Cij e j ei = C jie′j
A = Aij eie j = Aij CkiClj e′k e′l
Ai′j = Cik C jl Akl
Aij = CkiClj Ak′l
n = 0 p′ = p
n =1
α p′ = p i1
i1 ji j1
α α n = 2 p′ = p i1i2
⎜⎝ − ω2 ω1
0 ⎟⎠
⎪⎧ Aij = −ε ω ijk k
⎪⎩⎨ωi
=
1 2
ε
ijk
A jk
轴向量 ω = ωiei
A = 1 (A + AT ) + 1 (A − AT )
2
2
单位张量 I = δ ijeie j 迹 tr(A) = Aii
内积
向量与张量之积
A ⋅ a = Aij eie j ⋅ ak e k = Aij a jei
=
∂ ∂x j
∂x j ∂xi′
= Cij
∂ ∂x j
函数梯度
∇ϕ
=
ei
∂ϕ ∂xi
向量散度
∇⋅a
=
ei
∂ ∂xi
⋅aje j
=
∂ai ∂xi
向量旋度
∇×a
=
ei
∂ ∂xi
× aje j
= ε ijk
∂a j ∂xi
ek
Laplace 算子
Δ = ∇2
= ∇ ⋅∇ = ei
∂ ∂xi
⋅e j
∂ ∂x j
向量梯度
张量分析
∇a = ei
∂ ∂xi
aje j
= a j,ieie j
张量散度
∇ ⋅ A = ei
∂ ∂xi
⋅ Ajk e j e k
=
Aik ,ie k
∇ ⋅ (ϕ
I)
= ei
∂ ∂xi
⋅ϕ
e je j
=
∂ϕ ∂xi
ei
= ∇ϕ
张量旋度
∇× A = ei
∂ ∂xi
× Ajk e jek
=
A jk ,iε ijl el e k
e′i = Cij e j Cij Cik = δ jk
a = ai′e′i = ai′Cij e j = aiei
a j = Cij ai′ ai = C ji a′j
ei = C jie′j ai′ = Cij a j
Kronecker 记号
δ ij
= ei
⋅ej
=
⎧1 ⎩⎨0
i= j i≠ j
四阶张量:
H ijkl = μ (δikδ ij + δ ilδ jk ) + γδ ijδ kl
正交曲线坐标系
r = r(q1, q2 , q3 )
dr = H1dq1e1 + H 2dq2e2 + H 3dq3e3
弧元: dsi = H i dqi (i = 1, 2, 3)
式中 H1, H 2 , H 3 称为 Lame 系数。在正
a ⋅ A = aiei ⋅ A jk e je k = Aik aie k = A ji a jei
a⋅ A = AT ⋅a
AT = A a⋅ A = A⋅a
AT = −A a ⋅ A = −A ⋅a
A ⋅ a = −ε ijkωk eie j ⋅ al el = ε ω kji k a jei = ω × a
置换符号(Levi-Civita)
⎧
ε ijk
=
⎪ ⎨
⎪⎩ 0
1 Even −1 Odd (i − j)( j − k)(k − i) = 0
ei × e j = ε ijk e k
ε − δ 恒等式
ε ε ijk ist = δ jsδ kt − δ jtδ ks
张量代数 定义(并矢基)
a = aiei ⇒ A = Aij eie j
定义
向量代数
r = x i+ y j+ z k
r = x1e1 + x2e2 + x3e3
Einstein 约定求和法则
a = a1e1 + a2e2 + a3e3 = aiei
坐标变换(正交坐标变换)
e1 e2 e3 e1′ C11 C12 C13 e′2 C21 C22 C23 e′3 C31 C32 C33
∂q2 H 2 ∂q2 ∂q3 H 3 ∂q3
交曲线坐标系中,沿坐标轴的微元弧长等于 该坐标轴的微元增量乘以相应的拉梅系数。 其几何意义为坐标轴的单位变化引起的微 元弧长增量。 在直角坐标系中
H1 = 1 H2 = 1 H3 = 1
在柱坐标来自百度文库中
H1 = 1 H2 = r H3 = 1
在球坐标系中
H1 = 1 H 2 = R H 3 = R sinθ
i1 j1 i2 j2 j1 j2
二阶张量的矩阵表示
A ⇔ A = ( Aij ) A = CT A′C
加法: A + B = ( Aij + Bij ) eie j
转置: AT = Ajieie j
对称张量
AT = A
反对称张量 AT = −A
反对称张量的轴向量
⎜⎛ 0 − ω3 ω2 ⎟⎞
A = ⎜ ω3 0 − ω1 ⎟
A × B = Aijeie j × Bkl ek el = Aij Bklε jkpeie pel = Aip Blk ε plj eie j e k
∇ × a = 0 ⇔ ∃ ϕ, a = ∇ϕ
无源场与向量场
∇ ⋅ a = 0 ⇔ ∃ b, a = ∇ × b
Helmholtz 分解
∀a, ∃ϕ,b,∇ ⋅ b = 0 ⇒ a = ∇ϕ + ∇ × b