electricidad

Needs "VectorAnalysis`"
SetCoordinates Spherical r, Θ, Φ
Spherical r, Θ, Φ
V r_, Θ_, Φ_
2

r Cos Θ

,

r Cos Θ

r Cos Θ Sin Θ , 3 r

^ 2,

r Cos Θ Sin Θ , 3 r

Vr :

Sin Θ Cos Φ , Sin Θ Sin Φ , Cos Θ

Vx :

1, 0, 0

VX r, Θ, Φ
2

r Cos Θ

V r, Θ, Φ .Vx Vr
Cos Φ Sin Θ , r Cos Θ

2

Sin Θ Sin Φ , r Cos Θ

Cos Θ Cos Φ , Cos Θ Sin Φ ,

VΘ :
Vy :

3

Sin Φ0, 1, 0

VY r, Θ, Φ
2

r Cos Θ

V r, Θ, Φ .Vy VΘ
Cos Φ Sin Θ ,

r Cos Θ

2

Sin Θ Sin Φ , r Cos Θ Sin Θ Sin Φ

Sin Φ , Cos Φ , 0

VΦ :
Vz :

0, 0, 1

VZ r, Θ, Φ

V r, Θ, Φ .Vz VΦ

3 r Sin Φ , 3 r Cos Φ , 0
W r_, Θ_, Φ_
2 r Cos Θ
Vr . Θ

2

Π

VX r, Θ, Φ
Cos Φ Sin Θ

2 .Φ

VY r, Θ, Φ

VZ r, Θ, Φ

3 r Sin Φ , 3 r Cos Φ , r Cos Θ

3

r Cos Θ Sin ΘSin Φ

0

1, 0, 0
Dot W r, Π

2, 0 , Vx

0
Integrate W r, Π

P0

2, 0 .Vx, r, 0, 1

0
Sin Φ , Cos Φ , 0

w:

Integrate W 1, Π

P1

2, Φ .w, Φ, 0, Π

2

3Π
2
Integrate W Tan Θ , Θ, Π

P2

ArcTan

2 . 0, 0,

Csc Θ

1
2

Csc Θ

2

Cos Θ

2

Sin Θ

Sin Θ

2

Θ

0

Simplify W z Tan Θ , Θ, Π
z 1

Cos Θ Cot Θ
2

z3

z

1

z2

B1z
0

2

2 . 0, 0,

Csc Θ

2

2

, Θ, 0, ArcTan 1

2

2

Julio Godoy[JRG]2.nb

3

2y

0

2 y2

B2

y
5 y2

1

6
5
Flujo

B2

4

3Π

5

2

B1

P1

P0

Needs "VectorAnalysis`"
SetCoordinates Spherical r, Θ, Φ
Spherical r, Θ, Φ
r2 Sin Θ , 4 r2 Cos Θ , r2 Tan Θ

V r_, Θ_, Φ_ :
VΘ :

0, 1, 0
V r, Π

P0

6, Φ .VΘ

3 r2

2

RP0 r Sin Π

P1

6

r

0

3 R4
4
2Π

P1 Φ

P5
0

1

3 Π R4

2
VR :

1, 0, 0
V R, Θ, Φ .VR

P2

R2 Sin Θ
Π 6

P2 R2 Sin Θ

P3

Θ

0

1
3

2 Π R4

3

24
2Π

P3 Φ

P4
0

1
3

Π

3

2 Π R4

12
Simplify Resp
1
Π 3

3

P4

2 Π R4

12
P7

Div V r, Θ, Φ

4 r Cos Θ Cot Θ

P5

Julio Godoy[JRG]2.nb

R

P7 r2 Sin ΘP8

r

0

R4 Cos Θ

2

Π 6

P8 Θ

P9
0

1
3

2 Π R4

3

24
2Π

P9 Φ

Resp1
0

1
Π 3

3

2 Π R4

12
FullSimplify Resp

Resp1

0

SetCoordinates Cartesian x, y, z
Cartesian x, y, z

a.
Gh
v

Grad v x, y, z
1,0,0

0,1,0

x, y, z , v

0,0,1

x, y, z , v

x, y, z

Curl Gh
0, 0, 0
SetCoordinates Cylindrical Ρ, Φ, z
Cylindrical Ρ, Φ, zGh1
U

Grad U Ρ, Φ, z

1,0,0

U

0,1,0

Ρ, Φ, z

Ρ, Φ, z ,

,U

0,0,1

Ρ, Φ, z

Ρ
Curl Gh1
0, 0, 0
SetCoordinates Spherical r, Θ, Φ
Spherical r, Θ, Φ
Gh2
W1

Grad W1 r, Θ, Φ
1,0,0

W1

0,1,0

r, Θ, Φ

r, Θ, Φ ,
r

Curl Gh2
0, 0, 0

b.
SetCoordinates Cartesian x, y, z
Cartesian x, y, z

A

x, y, z

x, y, z

Csc Θ W1

0,0,1

,
r

r, Θ, Φ3

4

Julio Godoy[JRG]2.nb

RA x_, y_, z_

Curl A

0, 0, 0
DR x_, y_, z_

Div RA x, y, z

0
SetCoordinates Cylindrical Ρ, Φ, z
Cylindrical Ρ, Φ, z
V1

Ρ, Φ, z

Ρ, Φ, z
RV Ρ_, Φ_, z_

Curl V1

Φ
0, 0,
Ρ
DRV Ρ_, Φ_, z_

Div RV Ρ, Φ, z

0
SetCoordinates Spherical r, Θ, Φ
Spherical r, Θ, Φ
R2

r, Θ, Φ

r, Θ, Φ
RR2 r_, Θ_, Φ_
Φ Cot Θ

Curl R2

Φ
,

rΘ
,

r

r

DR2 r_, Θ_, Φ_

Div RR2 r, Θ, Φ

0

SetCoordinates Cartesian x, y, z

Cartesian x, y, z
A1 x, y, z :
B12 x, y, z :
h11

A1x x, y, z , A1y x, y, z , A1z x, y, z
B12x x, y, z , B12y x, y, z , B12z x, y, z

Grad A1 x, y, z .B12 x, y, z

B12x x, y, z A1x

1,0,0
1,0,0

x, y, z

B12z x, y, z A1z
x, y, z
A1y x, y, z B12y 1,0,0 x, y, z
B12x x, y, z A1x0,1,0 x, y, z
B12z x, y, z A1z 0,1,0 x, y, z
A1y x, y, z B12y 0,1,0 x, y, z
B12x x, y, z A1x 0,0,1 x, y, z
B12z x, y, z A1z 0,0,1 x, y, z
A1y x, y, z B12y 0,0,1 x, y, z

B12y x, y, z A1y
A1x
A1z
B12y
A1x
A1z
B12y
A1x
A1z

x, y, z
x, y, z
x, y, z
x, y, z
x, y, z
x, y, z
x, y, z
x, y, z

1,0,0
1,0,0

x, y, z

B12x
x, y, z
B12z 1,0,0 x, y, z ,
A1y 0,1,0 x, y, z
B12x...