The document defines and explains the operator del (nabla), and how it can act in three ways: on a scalar field to give the gradient, on a vector field to give the divergence, and on a vector field to give the curl. It also summarizes Maxwell's equations, Gauss's law, Ampere's law, and other concepts in electromagnetism such as divergence theorem, Stokes' theorem, and the continuity equation.

1. Electromagnetism

2. The Operator Del 
  x
x
 y
y
 z
z

     
It gives the space rate of variation of a scalar field or vector field

3. The Operator Del 
Three ways Operator Del can act
1. On a scalar function T : (Gradient)
(Divergence)
2. On a vector function v:
3.
T
.V
On a vector function v :  x V (Curl)

4. Gradient
If T(r) = T(x, y, z) is a scalar field, ie a scalar function of position r = [x, y,
z] in 3 dimensions, then its gradient at any point is defined in Cartesian co-
ordinates by
It is a vector quantity
It gives the maximum rate of change of function at a point and its direction
is that which the rate of change is maximum.

6. Divergence

 
x y z
 y  z . v x  v y  v z
 
x y z
 
 
.v  x
         
Divergence is flux-density, or flux per unit volume. It tells you
how much the vector (function) spreads out from the point in
question.
vy
vx vz
.v 
x

y

z


7. Divergence
Positive divergence Negative divergence
Zero Divergence

8. The
Curl

 
x y z
 y  z  v x  v y  v z
 
x y z
 
 
  v  x
         
Curl is circulation-density, or circulation per unit area. It tells
you how much the vector (function) curls about the point in
question.
vy    vy
  v   y

z  x   z

x  y   x

y  z
 
   
   vz  vx vz   vx  

9. The Curl
Curl pointing in the Z-direction Zero Curl

10. Maxwell’s Equations
1
ρ
ε0
.B  0
t

.E 



B
t
 
  E  


E
 
  B  μ0J+μ0ε0

11. Gauss Law for electricity
• The electric flux out of any closed surface is proportional to
the total charge enclosed within the surface.
The divergence of the electric field at a point in space is equal to the
charge density divided by the permittivity of space.

12. Gauss Law for magnetism
• The net magnetic flux out of any closed surface is zero.
Magnetic monopoles do not exist

13. Faradays Law

14. Amperes law

15. Displacement current

17. Maxwells equation in vacuum

18. Wave equation from Maxwells equation

20. Line Integeral
• A line integral is an expression of the form
• Where v is the vector function and dl is the
infinitesimal displacement vector.
• The integeral is to be carried out in a prescribed path
(i.e) from a to b.
• If it’s a closed path a=b then the equation
• Work done by a force (F): W= 𝐹 . 𝑑𝑙

21. Surface Integerals
• A surface integral is an expression of form
• where v is the vector function and its over a surface
S, da- is the infinitesimal patch of area, with
directions perpendicular to any surface .
• For a closed surface
• If V describes the flow of liquid then 𝑣. 𝑑𝑎 - total mass
per unit time passing through the surface (flux)

22. Volume Integral
• A volume integral is an expression of form
• Where T is a scalar function and dτ is volume
element. In Cartesian co-ordinates dτ=dx dy dz.
• If T is the density, volume integeral will give the total
mass.
• For a vector function,

23. Divergence Theorem/Gauss’s
Theorem/Green’s theorem
• The volume integral of the divergence of any vector
field over any volume is equal to the net flow across
the volume’s boundary or it is equal to the surface
integral of a vector field taken over the closed
surface enclosing the volume.

24. Stokes theorem
• The integeral of a derivative (curl) over a region
(surface S ) is equal to the value of the function at
the boundary (perimeter of patch, P)

25. Continuity Equation