电磁场与电磁波第5讲旋度和旋度定理零恒等式亥姆霍兹定理
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Circulation of A around contour C c A dl
Since circulation as defined in Eq. is a line integral of a dot product, its value obviously depends on the orientation of the contour C relative to the vector A. In order to define a point function, which is a measure of the strength of a vortex source, we must make C very small and orient it in such a way that the circulation is maximum.
7
In words, Eq. states that the curl of a vector field A, denoted by curl A or A, is a vector whose magnitude is the maximum net circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the net circulation maximum.
P1P2
(
3 2
0)aˆx
(1.5 2)aˆy
(1 3)aˆz ;
A P1P2 1.5 4 2.5
answer A P1P2 5 1.12
P1P2
2
2
Review
1. Gradient of a Scalar Field
GradV V
dV dn
an
V
V x
aˆx
V y
aˆ y
Az
aˆR
Raˆ
R sin aˆ
球坐标系
1
A R2 sin R
AR
RA
R sin A
9
旋度运算规则
(A B) A B (C A) C A ( A) A A
10
Example 2-21(P57-58)
11
A curl-free vector field is called an irrotational or a conservative field. A divergenceless field is called a solenoidal field.
M
z 正最大
z
x z
零
x
y x
z
y
x
次之
y x
z
零
y
z 负最大
y
y
x
注意：在每一点P处都有无穷多的方向环量，且大小可能不等。
6
1. Curl of a Vector Field
Flow source; vortex source; vortex sink
the (net ) circulation
Field and Wave Electromagnetic 电磁场与电磁波
2011. 3. 2
P.2-15
x r cos y r sin zz
x 3 cos(600 ) 3 2
y 3 sin(600 ) 3 2
z 1
P1(0, 2,3); P2 (
3 , 1.5,1); 2
A 3aˆy 2aˆz ;
S (Curl A) dS S ( A) dS C A dl
Stokes’s theorem converts a surface integral of the curl of a vector to a line integral of the vector, and vice versa. It always implies an open surface with a rim. We remind ourselves here that the directions of dl and ds(an) follow the right-hand rule.
12
2. Stokes’s Theorem
The surface integral of the curl of a vector field over an open surface is equal to the closed line integral of the vector along the contour bounding the surface.
4
矢量场的旋度
1).矢量场的环量
矢量场A沿有向闭合曲线 l 的线积分称为矢量场A沿该曲 线的环量，以表示为
l A dl
>0, <0,=0
如
B c
dl
0
c内
I
0J ds
2A
2A
10A
3A
5A
3A
如何显示源的
分布特性？？
5
lim l A dl
S0 S
en
l
S
en1 en2
称为矢量A对于方向en的环量强度
CurlA A
lim 1 s0 S
aˆn
C
A
dl
max
The component of A in any other direction au is au ·(A),
which can be determined from the circulation per unit area
normal to au as the area approaches zero.
V z
aˆz
2. Divergence of a Vector Field
div A lim S A dS V 0 v
divA A= Ax Ay Az x y z
3. Divergence Theorem
V divAdV S A d S
3
Main topic
1. Curl of a Vector Field 2. Stokes’s Theorem 3. Two Null Identities 4. Helmholtz’s Theorem
(
A)u
aˆu
(
A)
lim
s0
1 Su
Cu
A
dl
8
直角坐标系
aˆx
aˆ y
aˆ z
CurlA A x
y
z
Ax
Ay
Az
＝
aˆx
Az y
－
Ay z
－aˆ
y
Az x
－
Ax z
Fra Baidu bibliotek
aˆ z
Ay x
Ax y
柱坐标系
aˆr
raˆ
aˆ z
A 1
r r z
Ar
rA
Since circulation as defined in Eq. is a line integral of a dot product, its value obviously depends on the orientation of the contour C relative to the vector A. In order to define a point function, which is a measure of the strength of a vortex source, we must make C very small and orient it in such a way that the circulation is maximum.
7
In words, Eq. states that the curl of a vector field A, denoted by curl A or A, is a vector whose magnitude is the maximum net circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the net circulation maximum.
P1P2
(
3 2
0)aˆx
(1.5 2)aˆy
(1 3)aˆz ;
A P1P2 1.5 4 2.5
answer A P1P2 5 1.12
P1P2
2
2
Review
1. Gradient of a Scalar Field
GradV V
dV dn
an
V
V x
aˆx
V y
aˆ y
Az
aˆR
Raˆ
R sin aˆ
球坐标系
1
A R2 sin R
AR
RA
R sin A
9
旋度运算规则
(A B) A B (C A) C A ( A) A A
10
Example 2-21(P57-58)
11
A curl-free vector field is called an irrotational or a conservative field. A divergenceless field is called a solenoidal field.
M
z 正最大
z
x z
零
x
y x
z
y
x
次之
y x
z
零
y
z 负最大
y
y
x
注意：在每一点P处都有无穷多的方向环量，且大小可能不等。
6
1. Curl of a Vector Field
Flow source; vortex source; vortex sink
the (net ) circulation
Field and Wave Electromagnetic 电磁场与电磁波
2011. 3. 2
P.2-15
x r cos y r sin zz
x 3 cos(600 ) 3 2
y 3 sin(600 ) 3 2
z 1
P1(0, 2,3); P2 (
3 , 1.5,1); 2
A 3aˆy 2aˆz ;
S (Curl A) dS S ( A) dS C A dl
Stokes’s theorem converts a surface integral of the curl of a vector to a line integral of the vector, and vice versa. It always implies an open surface with a rim. We remind ourselves here that the directions of dl and ds(an) follow the right-hand rule.
12
2. Stokes’s Theorem
The surface integral of the curl of a vector field over an open surface is equal to the closed line integral of the vector along the contour bounding the surface.
4
矢量场的旋度
1).矢量场的环量
矢量场A沿有向闭合曲线 l 的线积分称为矢量场A沿该曲 线的环量，以表示为
l A dl
>0, <0,=0
如
B c
dl
0
c内
I
0J ds
2A
2A
10A
3A
5A
3A
如何显示源的
分布特性？？
5
lim l A dl
S0 S
en
l
S
en1 en2
称为矢量A对于方向en的环量强度
CurlA A
lim 1 s0 S
aˆn
C
A
dl
max
The component of A in any other direction au is au ·(A),
which can be determined from the circulation per unit area
normal to au as the area approaches zero.
V z
aˆz
2. Divergence of a Vector Field
div A lim S A dS V 0 v
divA A= Ax Ay Az x y z
3. Divergence Theorem
V divAdV S A d S
3
Main topic
1. Curl of a Vector Field 2. Stokes’s Theorem 3. Two Null Identities 4. Helmholtz’s Theorem
(
A)u
aˆu
(
A)
lim
s0
1 Su
Cu
A
dl
8
直角坐标系
aˆx
aˆ y
aˆ z
CurlA A x
y
z
Ax
Ay
Az
＝
aˆx
Az y
－
Ay z
－aˆ
y
Az x
－
Ax z
Fra Baidu bibliotek
aˆ z
Ay x
Ax y
柱坐标系
aˆr
raˆ
aˆ z
A 1
r r z
Ar
rA