Add Lecture 1

Courses/EE1P1_Electricity_and_Magnetism.typ

@@ -1,3 +1,6 @@
+#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(
@@ -6,4 +9,56 @@
   author: "Folkert Kevelam"
 )
 
-= Lecture 1:
+= 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: $rho, phi, theta$