#import "@preview/cetz:0.4.2": canvas, draw
#import "@preview/cetz-plot:0.1.3": plot

#import "../template/lib.typ": *
#set page(paper: "a4")
#show: notes.with(
  title: [EE1P1],
  subtitle: [Electricity and Magnetism],
  author: "Folkert Kevelam"
)

= Lecture 1 - Mathematical Instruments

== Vectors

#definition[
  definition for a vector with a given source point by $arrow(r)_1$
  $
    arrow(r)_(1,2) &= (x_2-x_1)hat(x) + (y_2-y_1)hat(y) + (z_2-z_1)hat(z) \
    |arrow(r)_(1,2)| &= sqrt((x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2) \
    hat(r)_(1,2) &= arrow(r)_(1,2) / (|arrow(r)_(1,2)|)
  $

  Most of the time, vectors are defined from a source point to a observation
  point.

  $
    arrow(R) = arrow(r) - attach(limits(r), t: arrow prime)
  $
]

#definition[

  Scalar product: $RR^3 dot RR^3 arrow RR$, $arrow(a)dot arrow(b) = arrow(b) dot arrow(a)$

  Cross product: $RR^3 times RR^3 arrow RR^3$, $arrow(a) times arrow(b) = -arrow(b) times arrow(a)$

  $arrow(a) times arrow(b) = ||$

  $|arrow(a) times arrow(b)| = |arrow(a)||arrow(b)| sin(alpha)$

  $arrow(a) dot (arrow(b) times arrow(c)) = arrow(c) dot (arrow(a) times arrow(b))$

  $arrow(a) times (arrow(b) times arrow(c))$
]

== Integrals

=== Integrals over $cal(D) subset RR^3$

+ Compact form: $integral_cal(D) f(x,y,z) d V$
+ Extended form: $integral.triple_cal(D) f(x,y,z) d V$

=== Polar coordinates

+ EE1M1: $r, theta$, EE1P1: $rho, phi$

=== Cylindrical coordinates

+ EE1M1: $r, theta, z$, EE1P1: $rho, phi, z$

=== Spherical coordinates

+ EE1M1: $r, theta, phi$, EE1P1: $rho, phi, theta$