This document discusses vector transformations and operations in three common coordinate systems: Cartesian, cylindrical, and spherical. It provides the formulas for differential length/volume elements, gradient, divergence, curl, and Laplacian in each system. Conversion formulas between the different coordinate representations of a vector are also outlined.

1. Formulas/transformations of vectors in three coordinates system
Cartesian Coordinates System(X,Y,Z):
∧ ∧ ∧
DIFFERENTIAL LENGTH VECTOR : dl = dx a x + dy a y + dz a z
DIFFERENTIAL VOLUME ELEMENT : dV = dx dy dz −∞ < X < ∞, − ∞ < Y < ∞, − ∞ < Z < ∞
∧
DIFFERENTIAL SURFACE ELEMENTS: dS x = dy dz a x − ∞ < Y < ∞, − ∞ < Z < ∞
∧
dS y = dx dz a y − ∞ < X < ∞, − ∞ < Z < ∞
∧
dS z = dy dz a x − ∞ < X < ∞, − ∞ < Y < ∞
DISTANCE BETWEEN TWO POINTS: [
d = ( x1 − x 2 ) 2 + ( y1 − y 2 ) 2 + ( z1 − z 2 ) 2 ]
1/ 2
∧ ∂V ∧ ∂V ∧ ∂V
GRADIENT OF SCALAR V : ∇ = ax
V +a y +az
∂x ∂y ∂z
∂Ax ∂A y ∂Az
DIVERGENCE OF VECTOR A : ∇• A = + +
∂x ∂y ∂Z
∧ ∧ ∧
ax ay az
∇ A
×
∂ ∂ ∂
CURL OF VECTOR A: =
∂x ∂y ∂z
Ax A y Az
∂2V ∂2V ∂2V
LAPLACIAN OF A SCALAR V: ∇2V = 2
+ 2
+
∂x ∂y ∂Z 2
A VECTOR A IS SAID TO BE SOLENOIDAL (OR DIVERGENCELESS ) if ∇ A =
• 0
A VECTOR A IS SAID TO BE IRROTATIONAL( OR POTENTIAL) IF ∇ A=
× 0
(BOTH STATEMENT ARE TRUE IN ALL THE COORDINATE SYSTEMS)
DIVERGENCE THEORM(GREEN'S THEORM) : ∫A •ds =∫∇•A
S V
dV
STOCK'S THEORM: ∫ A • dl
L
= ∫(∇×A ) •dS
S
COMPUTATION FORMULAS ON GRADIENT:
(a ) ∇ V +U ) =∇ +∇
( V U
(b) ∇ UV ) =V∇ +U∇
( U V
 V  U∇ −V∇
V U
(c ) ∇ =
U  U2
( d ) ∇ n = nV n −1∇
V V
where U and V are scalars and n is int eger

2. Cylindrical coordinates system ( ρ ,φ , z)
RELATIONSHIP BETWEEN (X,Y,Z) AND ( ( ρ , φ , z ) :
X = ρ cos φ y
φ = tan −1  
.Y = ρ sin φ x 0 ≤ ρ < ∞ , 0 ≤ φ < 2π , − ∞ < z < ∞
Z =Z ρ= 2
x +y 2
DIFFERENTIAL LENGTH VECTOR : dl = dρ aρ + ρ dφ aφ + dza z
ˆ ˆ ˆ
DIFFERENTIAL VOLUME ELEMENT : dv = ρ dρ dφ dz 0 ≤ ρ < ∞ , 0 ≤ φ < 2π , − ∞ < z < ∞
DIFFERENTIAL SURFACE ELEMENTS :
ds ρ = ρ dφ dz a ρ
ˆ 0 ≤ φ < 2π , − ∞ < z < ∞
dsφ = dρ dz aφ
ˆ 0 ≤ ρ < ∞, −∞ < z < ∞
ds z = ρ dφ dρ a z
ˆ 0 ≤ ρ < ∞ , 0 ≤ φ < 2π
DISTANCE BETWEEN TWO POINTS : d 2
= ρ1
2
+ ρ2
2
− 2 ρ1 ρ2 cos(φ1 −φ2 ) + ( z 2 − z1 ) 2
Transformation of A from cylindrical to cartesian coordinates system
 Aρ   cos φ − sin φ 0   Ax 
    
 Aφ  =  sin φ cos φ 0   A y 
A   0 0 1   Az 
 z   
Transformation of A from cartesian to cylinderical coordinates system
 Ax   cos φ sin φ 0   Aρ 
    
 A y  =  − sin φ cos φ 0   Aφ 
A   0 0 1   Az 
 z   
∂ ∧
V 1 ∂ ∧
V ∂V ∧
GRADIENT OF A SCALAR V: ∇ =
V aρ+ aφ + az
∂ρ ρ ∂φ ∂Z
1 ∂Aφ
DIVERGENCE OF A VECTOR A: ∇• A =
1 ∂
ρ ∂ρ
(
ρ Aρ +
ρ ∂φ
)+
∂Az
∂Z
aρ ρ Aφ Az
1∂ ∂ ∂
CURL OF A VECTOR A: ∇ × A=
ρ ∂ρ ∂φ ∂Z
Aρ ρ Aφ Az

3. 1 ∂  ∂V  1 ∂2V  ∂2V 
LAPLACIAN OF A SCALAR V: ∇2V = ρ + 2 + 
ρ ∂ρ  ∂ρ  ρ ∂φ 2  ∂z 2
  



4. Spherical Coordinate System (r ,θ, φ)
y
φ = tan −1  
X = r sin θ cos φ x
.Y = r sin θ sin φ r = x2 + y2 +z2 0 ≤ r < ∞ , - π ≤ θ < π , 0 < φ < 2π
Z =r cos θ
x2 +y 2
θ =tan −1
z
Differenial length vector : dl = dr a r + r dθ aθ + r sin θ dφ aφ
ˆ ˆ ˆ
DIFFERENTIAL VOLUME ELEMENT : dV = r 2 sin θ dr dθ dφ 0 ≤ r < ∞, - π ≤ θ < π , 0 < φ < 2
dIFFERENTIAL SURFACE ELEMENT :
ds r = r 2 sin θ dθ dφ a r
ˆ - π ≤θ < π, 0 < φ < 2π
dsθ = r sin θ dr dθ aφ
ˆ 0 ≤ r < ∞, 0 < φ < 2π
dsφ = r dr dθ aφ
ˆ 0 ≤ r < ∞, - π ≤ θ < π
DISTANCE BETWEEN TWO POINTS : d 2 = r12 + r2 + 2r1 r2 cos θ1 cos θ2 − 2 r1 r2 sin θ1 sin θ2 cos(θ1 −θ
2
Transformation of A from Cartesian to spherical coordinate system
 Ar   sin θ cos φ sin θ sin φ cos θ   Ax 
    
 Aθ  =  cos θ cos φ cos θ sin φ − sin θ   A y 

 Aφ   − sin φ


  cos φ 0   Az 
 
Transformation of A from spherical to cartesian coordinates system
 Ax   sin θ cos φ cos θ cos φ − sin φ   Ar 
    
 A y  =  sin θ sin φ cos θ sin φ cos φ   Aθ 
 A   sin θ  
 z  − sin θ 0   Aφ 
 
Transformation of A from spherical to cylindrical coordinates system
 Aρ   sin θ cos θ 0   Ar 
    
 Aφ  = 0 0 1   Aθ 
A   cos θ  
 z   − sin θ 0   Aφ 
 
Transformation of A from cylindrical to spherical coordinates system
 Ar   sin θ 0 cos θ   Aρ 
    
 Aθ  =  cos θ 0 − sin θ   Aφ 
 
 Aφ   0  A 
   1 0  z 
∂V ∧ 1 ∂ ∧
V 1 ∂ ∧
V
GRADIENT OF A SCALAR V: ∇ =
V ar + aθ + aφ
∂r r ∂θ r sin θ ∂φ
DIVERGENCE OF A VECTOR A: ∇• A =
r
1 ∂ 2
2 ∂r
(
r Ar +
1
) ∂
r sin ϑ ∂ϑ
( Aϑ Sin ϑ) + 1
∂Aφ
r Sin ϑ ∂φ
∧ ∧ ∧
a r r aϑ r Sinϑ aφ
1 ∂ ∂ ∂
CURL OF A VECTOR A: ∇ × A=
r 2 Sinϑ ∂ r ∂ ϑ ∂ φ
Ar rAφ r Sinϑ Aφ

5. LAPLACIAN OF A SCALER FIELD, V:
1 ∂  ∂V  1 ∂  ∂V  1  ∂2V 
∇2V = ρ + sin θ +  
ρ ∂ρ  ∂ρ  r 2 sin θ ∂θ 
  ∂θ  r 2 sin 2 θ  ∂φ 2




6. LAPLACIAN OF A SCALER FIELD, V:
1 ∂  ∂V  1 ∂  ∂V  1  ∂2V 
∇2V = ρ + sin θ +  
ρ ∂ρ  ∂ρ  r 2 sin θ ∂θ 
  ∂θ  r 2 sin 2 θ  ∂φ 2


