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Vector calculus formulas, Study notes of Accelerator Physics

University of Mindanao (UM)Accelerator Physics

Vector Formulas used for electromagnetics and other Vector related fields

Typology: Study notes

2015/2016

Uploaded on 07/19/2016

james_nunieza 🇵🇭

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1 document

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COORDINATE CONVERSION FORMULAS

Rectangular to Cylindrical, P(x, y, z)→P(ρ,φ, z )

ρ=!x2+y2,

φ="tan−1#y

x$,if x>0

tan−1#y

x$+ 180o,if x<0

Cylindrical to Rectangular, P(ρ,φ, z)→P(x, y, z )

x=ρcos φ,

y=ρsin φ

Rectangular to Spherical, P(x, y, z)→P(r, θ,φ)

r=!x2+y2+z2

θ= cos−1z

!x2+y2+z2

φ="tan−1#y

x$,if x>0

tan−1#y

x$+ 180o,if x<0

Spherical to Rectangular, P(r, θ,φ)→P(x, y, z)

x=rsin θcos φ

y=rsin θsin φ

z=rcos θ

Cylindrical to Spherical, P(ρ,φ, z)→P(r, θ,φ)

r=!ρ2+z2

θ= cos−1z

!ρ2+z2

Spherical to Cylindrical, P(r, θ,φ)→P(ρ,φ, z)

ρ=rsin θ

z=rcos θ

VECTOR CONVERSION FORMULAS

Rectangular to Cylindrical, $

Arec(x, y , z)→$

Acyl(ρ,φ, z )

$

Acyl =$

Trec→cyl $

Arec

%

&

Aρ

Aφ

Az'

(=%

&

cos φsin φ0

−sin φcos φ0

0 0 1 '

(%

&

Ax

Ay

Az'

(

Cylindrical to Rectangular, $

Acyl(ρ,φ, z )→$

Arec(x, y , z)

$

Arec =$

Tcyl→rec $

Acyl

%

&

Ax

Ay

Az'

(=%

&

cos φ−sin φ0

sin φcos φ0

0 0 1 '

(%

&

Aρ

Aφ

Az'

(

Rectangular to Spherical, $

Arec(x, y , z)→$

Asph(r, θ,φ)

$

Asph =$

Trec→sph $

Arec

%

&

Ar

Aθ

Aφ'

(=%

&

sin θcos φsin θsin φcos θ

cos θcos φcos θsin φ−sin θ

−sin φcos φ0'

(%

&

Ax

Ay

Az'

(

Spherical to Rectangular, $

Asph(r, θ,φ)→$

Arec(x, y , z)

$

Arec =$

Tsph→rec $

Asph

%

&

Ax

Ay

Az'

(=%

&

sin θcos φcos θcos φ−sin φ

sin θsin φcos θsin φcos φ

cos θ−sin θ0'

(%

&

Ar

Aθ

Aφ'

(

Cylindrical to Spherical, $

Acyl(ρ,φ, z )→$

Asph(r, θ,φ)

$

Asph =$

Tcyl→sph $

Acyl

%

&

Ar

Aθ

Aφ'

(=%

&

sin θ0 cos θ

cos θ0−sin θ

0 1 0 '

(%

&

Aρ

Aφ

Az'

(

Spherical to Cylindrical, $

Asph(r, θ,φ)→$

Acyl(ρ,φ, z )

$

Acyl =$

Tsph→cyl $

Asph

%

&

Aρ

Aφ

Az'

(=%

&

sin θcos θ0

0 0 1

cos θ−sin θ0'

(%

&

Ar

Aθ

Aφ'

(

1

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COORDINATE CONVERSION FORMULAS

Rectangular to Cylindrical, P (x, y, z) → P (ρ, φ, z)

ρ =

x

2

y

2 ,

φ =

tan

− 1

y

x

, if x > 0

tan

− 1

y

x

o , if x < 0

Cylindrical to Rectangular, P (ρ, φ, z) → P (x, y, z)

x = ρ cos φ,

y = ρ sin φ

Rectangular to Spherical, P (x, y, z) → P (r, θ, φ)

r =

x

2

y

2

z

2

θ = cos

− 1

z

!

x

2

y

2

z

2

φ =

tan

− 1

y

x

, if x > 0

tan

− 1

y

x

o , if x < 0

Spherical to Rectangular, P (r, θ, φ) → P (x, y, z)

x = r sin θ cos φ

y = r sin θ sin φ

z = r cos θ

Cylindrical to Spherical, P (ρ, φ, z) → P (r, θ, φ)

r =

ρ

2

z

2

θ = cos

− 1

z

!

ρ

2

z

2

Spherical to Cylindrical, P (r, θ, φ) → P (ρ, φ, z)

ρ = r sin θ

z = r cos θ

VECTOR CONVERSION FORMULAS

Rectangular to Cylindrical,

Arec (x, y, z) →

Acyl (ρ, φ, z)

A

cyl

T

rec→cyl

A

rec

%

A

ρ

A

φ

A

z

cos φ sin φ 0

− sin φ cos φ 0

A

x

A

y

A

z

Cylindrical to Rectangular,

A

cyl

(ρ, φ, z) →

A

rec

(x, y, z)

A (^) rec =

Tcyl→rec

Acyl

Ax

Ay

Az

cos φ − sin φ 0

sin φ cos φ 0

Aρ

Aφ

Az

Rectangular to Spherical,

A

rec

(x, y, z) →

A

sph

(r, θ, φ)

Asph =

Trec→sph

Arec

Ar

Aθ

Aφ

sin θ cos φ sin θ sin φ cos θ

cos θ cos φ cos θ sin φ − sin θ

− sin φ cos φ 0

Ax

Ay

Az

Spherical to Rectangular,

A

sph

(r, θ, φ) →

A

rec

(x, y, z)

Arec =

Tsph→rec

Asph

Ax

Ay

Az

sin θ cos φ cos θ cos φ − sin φ

sin θ sin φ cos θ sin φ cos φ

cos θ − sin θ 0

Ar

Aθ

Aφ

Cylindrical to Spherical,

A

cyl

(ρ, φ, z) →

A

sph

(r, θ, φ)

Asph =

Tcyl→sph

Acyl

Ar

Aθ

Aφ

sin θ 0 cos θ

cos θ 0 − sin θ

Aρ

Aφ

Az

Spherical to Cylindrical,

A

sph

(r, θ, φ) →

A

cyl

(ρ, φ, z)

Acyl =

Tsph→cyl

Asph

Aρ

Aφ

Az

sin θ cos θ 0

cos θ − sin θ 0

Ar

Aθ

Aφ

GRADIENT OF A SCALAR FIELD

Rectangular Coordinate System (x, y, z)

∇V =

∂V

∂x

ax +

∂V

∂y

ay +

∂V

∂z

az

Cylindrical Coordinate System (ρ, φ, z)

∇V =

∂V

∂ρ

a ρ

ρ

∂V

∂φ

a φ

∂V

∂z

a z

Spherical Coordinate System (r, θ, φ)

∇V =

∂V

∂r

a r

r

∂V

∂θ

a θ

r sin θ

∂V

∂φ

a φ

DIVERGENCE OF A VECTOR FIELD

Rectangular Coordinate System (x, y, z)

F =

∂Fx

∂x

∂Fy

∂y

∂Fz

∂z

Cylindrical Coordinate System (ρ, φ, z)

F =

ρ

∂(ρFρ )

∂ρ

ρ

∂Fφ

∂φ

∂Fz

∂z

Spherical Coordinate System (r, θ, φ)

F =

r

2

∂(r

2 F r

∂r

r sin θ

∂(sin θF θ

∂θ

r sin θ

∂F

φ

∂φ

CURL OF A VECTOR FIELD

Rectangular Coordinate System (x, y, z)

∇ ×

F =

ax ay az

∂

∂x

∂

∂y

∂

∂z

Fx Fy Fz

Cylindrical Coordinate System (ρ, φ, z)

∇ ×

F =

ρ

aρ ρaφ az

∂

∂ρ

∂

∂φ

∂

∂z

Fρ ρF (^) φ Fz

Spherical Coordinate System (r, θ, φ)

∇ ×

F =

r

2 sin θ

ar raθ (r sin θ)aφ

∂

∂r

∂

∂θ

∂

∂φ

F

r

rF θ

(r sin θ)F φ

LAPLACIAN OF A SCALAR FIELD

Rectangular Coordinate System (x, y, z)

2 V =

2 V

∂x

2

∂V

2

∂y

2

∂V

2

∂z

2

Cylindrical Coordinate System (ρ, φ, z)

2 V =

ρ

∂ρ

ρ

∂V

∂ρ

ρ

2

2 V

∂φ

2

2 V

∂z

2

Spherical Coordinate System (r, θ, φ)

2

V =

r

2

∂r

r

2

∂V

∂r

r

2 sin θ

∂θ

sin θ

∂V

∂θ

r

2 sin

2 θ

2 V

∂φ

2

DIVERGENCE THEOREM

S

F · d

S =

v

F

dv

STOKES’S THEOREM

C

F · d

L =

S

∇ ×

F

· d

S

MAXWELL’S EQUATIONS

Differential Form

E =

ρv

0

∇ ×

E = −

B

∂t

H = 0

∇ ×

H =

J +

D

∂t

Integral Form

S

E · d

S =

v

ρv

0

dv

C

E · d

L = −

∂t

S

B · d

S

H · d

S = 0

C

H · d

L =

S

J +

D

∂t

· d

S

TEX Typeset:jmmartinezjr

Vector calculus formulas, Study notes of Accelerator Physics

Related documents

Partial preview of the text

Download Vector calculus formulas and more Study notes Accelerator Physics in PDF only on Docsity!

COORDINATE CONVERSION FORMULAS

VECTOR CONVERSION FORMULAS

A

T

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

GRADIENT OF A SCALAR FIELD

∇V =

∂V

∂V

∂V

∇V =

∂V

∂V

∂V

∇V =

∂V

∂V

∂V

DIVERGENCE OF A VECTOR FIELD

F =

F =

F =

∂F

CURL OF A VECTOR FIELD

∇ ×

F =

∇ ×

F =

∇ ×

F =

F

LAPLACIAN OF A SCALAR FIELD

∂V

∂V

∂V

∂V

∂V

DIVERGENCE THEOREM

S =

F

STOKES’S THEOREM

L =

∇ ×

F

S

MAXWELL’S EQUATIONS

E =

∇ ×

E = −

B

H = 0

∇ ×

H =

J +

D

S =

L = −

S

S = 0

L =