1. Vector Calculus

2. Vector Calculus
COORDINATE SYSTEMS
• RECTANGULAR or Cartesian
• CYLINDRICAL Choice is based on
• SPHERICAL symmetry of
problem
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL

3. Cylindrical Symmetry Spherical Symmetry

4. Cartesian Coordinates Or Rectangular Coordinates
P (x, y, z
z)
−∞ < x < ∞ P(x,y,z)
−∞ < y < ∞
y
−∞ < z < ∞
x
A vector A in Cartesian coordinates can be written as
( Ax , Ay , Az ) or Ax a x + Ay a y + Az a z
where ax,ay and az are unit vectors along x, y and z-directions.

5. Cylindrical Coordinates
z
P (ρ, Φ, z)
0≤ρ <∞
z
P(ρ, Φ, z)
0 ≤ φ < 2π
−∞ < z < ∞ x Φ
ρ y
A vector A in Cylindrical coordinates can be written as
( Aρ , Aφ , Az ) or Aρ aρ + Aφ aφ + Az a z
where aρ,aΦ and az are unit vectors along ρ, Φ and z-directions.
x= ρ cos Φ, y=ρ sin Φ, z=z
y
ρ = x + y , φ = tan
2 2 −1
,z = z
x

6. The relationships between (ax,ay, az) and (aρ,aΦ, az)are
a x = cos φaρ − sin φaφ
a y = sin φaρ − cos φaφ
az = az
or
aρ = cos φa x + sin φa y
aφ = − sin φa x + cos φa y
az = az
Then the relationships between (Ax,Ay, Az) and (Aρ, AΦ, Az)are
A = ( Ax cos φ + Ay sin φ )aρ + (− Ax sin φ + Ay cos φ )aφ + Az a z

7. Aρ = Ax cos φ + Ay sin φ
Aφ = − Ax sin φ + Ay cos φ
Az = Az
In matrix form we can write
 Aρ   cos φ sin φ 0  Ax 
 A  = − sin φ cos φ 0  Ay 
 φ   
 Az   0
   0 1  Az 
 

8. Spherical Coordinates
z
P (r, θ, Φ) 0≤r <∞ P(r, θ, Φ)
θ r
0 ≤θ ≤π
0 ≤ φ < 2π x Φ
y
A vector A in Spherical coordinates can be written as
( Ar , Aθ , Aφ ) or Ar ar + Aθ aθ + Aφ aφ
where ar, aθ, and aΦ are unit vectors along r, θ, and Φ-directions.
x=r sin θ cos Φ, y=r sin θ sin Φ, Z=r cos θ
x2 + y2 −1 y
r = x 2 + y 2 + z 2 , θ = tan −1 , φ = tan
z x

9. The relationships between (ax,ay, az) and (ar,aθ,aΦ)are
a x = sin θ cos φar + cos θ cos φaθ − sin φaφ
a y = sin θ sin φar + cos θ sin φaθ + cos φaφ
a z = cos θar − sin θaθ
or
ar = sin θ cos φa x + sin θ sin φa y + cos θa z
aθ = cos θ cos φa x + cos θ sin φa y − sin θa z
aφ = − sin φa x + cos φa y
Then the relationships between (Ax,Ay, Az) and (Ar, Aθ,and AΦ)are
A = ( Ax sin θ cos φ + Ay sin θ sin φ + Az cos θ )ar
+ ( Ax cos θ cos φ + Ay cos θ sin φ − Az sin θ )aθ
+ (− Ax sin φ + Ay cos φ )aφ

10. Ar = Ax sin θ cos φ + Ay sin θ sin φ + Az cos θ
Aθ = Ax cos θ cos φ + Ay cos θ sin φ − Az sin θ
Aφ = − Ax sin φ + Ay cos φ
In matrix form we can write
 Ar   sin θ cos φ sin θ sin φ cos θ   Ax 
  
 Aθ  = cos θ cos φ cos θ sin φ − sin θ   Ay 
 
 Aφ   − sin φ cos φ 0   Az 
    

11. z z
P(r, θ, Φ)
Cartesian Coordinates P(x,y,z)
θ r P(x, y, z) y
y x
x Φ
Spherical Coordinates Cylindrical Coordinates
P(r, θ, Φ) z P(ρ, Φ, z)
z
P(ρ, Φ, z)
r y
x Φ

12. Differential Length, Area and Volume
Cartesian Coordinates
Differential displacement
dl = dxa x + dya y + dza z
Differential area
dS = dydza x = dxdza y = dxdya z
Differential Volume
dV = dxdydz

13. Cylindrical Coordinates
ρ ρ
ρ ρ
ρ
ρ ρρ
ρ
ρ ρ

14. Differential Length, Area and Volume
Cylindrical Coordinates
Differential displacement
dl = dρa ρ + ρdφaφ + dza z
Differential area
dS = ρdφdza ρ = dρdzaφ = ρdρdφa z
Differential Volume
dV = ρdρdφdz

15. Spherical Coordinates

16. Differential Length, Area and Volume
Spherical Coordinates
Differential displacement
dl = drar + rdθaθ + r sin θdφaφ
Differential area
dS = r sin θdθdφar = r sin θdrdφaθ = rdrdθaφ
2
Differential Volume
dV = r sin θdrdθdφ
2

17. Line, Surface and Volume Integrals
Line Integral
∫ A.dl
L
Surface Integral
ψ = ∫ A.dS
S
Volume Integral
∫ p dv
V
v

18. Gradient, Divergence and Curl
The Del Operator
• Gradient of a scalar function is a
vector quantity.
∇f Vector
• Divergence of a vector is a scalar
∇. A
quantity.
• Curl of a vector is a vector
∇× A
quantity.
• The Laplacian of a scalar A ∇ A
2

19. Del Operator
Cartesian Coordinates
∂ ∂ ∂
∇ = ax + a y + az
∂x ∂y ∂z
Cylindrical Coordinates
∂ 1 ∂ ∂
∇= aρ + aφ + a z
∂ρ ρ ∂φ ∂z
Spherical Coordinates
∂ 1 ∂ 1 ∂
∇ = ar + aθ + aφ
∂r r ∂θ r sin θ ∂φ

20. Gradient of a Scalar
The gradient of a scalar field V is a vector that represents
both the magnitude and the direction of the maximum space
rate of increase of V.
∂V ∂V ∂V
∇V = ax + ay + az
∂x ∂y ∂z
∂V 1 ∂V ∂V
∇V = aρ + aφ + az
∂ρ ρ ∂φ ∂z
∂V 1 ∂V 1 ∂V
∇V = ar + aθ + aφ
∂r r ∂θ r sin θ ∂φ

21. Divergence of a Vector
The divergence of A at a given point P is the outward flux per
unit volume as the volume shrinks about P.
∫ A.dS
divA = ∇. A = lim S
∆v →0 ∆v
∂A ∂A ∂A
∇. A = + +
∂x ∂y ∂z
1 ∂ 1 ∂Aφ ∂Az
∇. A = ( ρAρ ) + +
ρ ∂ρ ρ ∂φ ∂z

22. Curl of a Vector
The curl of A is an axial vector whose magnitude is the
maximum circulation of A per unit area tends to zero and
whose direction is the normal direction of the area when the
area is oriented to make the circulation maximum.
 A.dl 
 ∫ 
curlA = ∇ × A =  lim L  an
 ∆s →0 ∆S 
 
  max
Where ΔS is the area bounded by the curve L and an is the unit
vector normal to the surface ΔS

23.  ax ay az   aρ ρaφ az 
∂ ∂ ∂ 1∂ ∂ ∂
∇× A =   ∇× A =  
 ∂x ∂y ∂z  ρ  ∂ρ ∂φ ∂z 
 Ax
 Ay Az 
  Aρ
 ρAφ Az 

Cartesian Coordinates Cylindrical Coordinates
 ar raθ r sin θaφ 
1 ∂ ∂ ∂ 
∇× A = 2  
r sin θ  ∂r ∂θ ∂φ 
 Ar
 rAθ r sin θAφ 

Spherical Coordinates

24. Divergence or Gauss’
Theorem
The divergence theorem states that the total outward flux of
a vector field A through the closed surface S is the same as the
volume integral of the divergence of A.
∫ A.dS = ∫ ∇. Adv
V

25. Stokes’ Theorem
Stokes’s theorem states that the circulation of a vector field A around a
closed path L is equal to the surface integral of the curl of A over the open
surface S bounded by L, provided A and × A
∇ are continuous on S
∫ A.dl = ∫ (∇ × A).dS
L S