Vector Analysis day 1

Electromagnetics is the study of electric and magnetic fields and their interaction with the environment.
There are three types of coordinate system

1) Rectangular coordinate system:

It is represented as P(x,y,z). its unit vector is ax, ay, az. the vector is represented as A=Axax+Ayay+Azaz.

For differential volume dv=dxdydz.

2) Cylindrical Coordinate System:

The three coordinates (ρ, φ, z) of a point P are defined as:

*The radial distance ρ is the Euclidean distance from the z axis to the point P.

*The azimuth φ is the angle between the reference direction on the chosen plane and the line from         the origin to the projection of P on the plane.

*The height z is the signed distance from the chosen plane to the point P.

The volume element is

The surface element in a surface 

             





3) spherical coordinate system:
Spherical coordinates is represented as P(r, θ, φ) as often used in mathematics: radial distance r, azimuthal angle θ, and polar angle φ. The meanings of θ and φ have been swapped compared to the physics conventi

Any spherical coordinate triplet (r, θ, φ) specifies a single point of three-dimensional space. On the other hand, every point has infinitely many equivalent spherical coordinates. One can add or subtract any number of full turns to either angular measure without changing the angles themselves, and therefore without changing the point. It is also convenient, in many contexts, to allow negative radial distances, with the convention that (−r, θ, φ) is equivalent to (r, θ + 180°, φ) for any r, θ, and φ. Moreover, (r, −θ, φ) is equivalent to (r, θ, φ + 180°)on.
Its volume element is: 

dot products of unit vector in spherical and rectangular coordinate system.

                   ar                   aθ                aφ
ax          sinθcosφ        cosθcosφ        -sinφ
ay          sinθsinφ         cosθsinφ          cosφ
az           cosθ                 -sinθ                 0