alexandria/documents/by-course/math-6a/course-notes/main.typ

#import "@youwen/zen:0.1.0": *
#import "@preview/cetz:0.3.1"

#show: zen.with(
  title: "Math 6A Course Notes",
  author: "Youwen Wu",
  date: "Winter 2025",
  subtitle: [Taught by Nathan Scheley],
)

#outline()

= Lecture #datetime(day: 7, month: 1, year: 2025).display()

== Review of fundamental concepts

You can parameterize curves.

#example[Unit circle][
  $
    x = cos(t) \
    y = sin(t)
  $
]

For an implicit equation
$ y = f(t) $
Parameterize it by setting
$ x = t \ y = f(t) $

Parameterize a line passing through two points $arrow(p)_1$ and $arrow(p)_2$ by
$ arrow(c)(t) = arrow(p)_1 + t (arrow(p)_2 - arrow(p)_1) $

Take the derivative of each component to find the velocity vector. The
magnitude of velocity is speed.

#example[
  $
    arrow(c)(t) = <5t, sin(t)> \
    arrow(v)(t) = <5, cos(t)>
  $
]

== Polar coordinates

Write a set of Cartesian coordinates in $RR^2$ as polar coordinates instead, by
a distance from origin $r$ and angle about the origin $theta$.

$ (x,y) -> (r, theta) $

= Lecture #datetime(day: 9, month: 1, year: 2025).display()

== Vectors

A dot product of two vectors is a generalization of the sense of size for a
point or vector.

#example[
  How far is the point $x_1, x_2, x_3$ from the origin? \
  Answer: $x_1^2 + x_2^2 + x_3^2$
]

#definition[
  For vectors $u$ and $v$, where
  $ v = vec(v_1, v_2, dots.v, n), u = vec(u_1, u_2, dots.v, n) $
  The dot product is defined as
  $ sum_(i=1)^n v_i dot u_i $
]

#proposition[
  The dot product of two vectors is the product of their magnitudes and the cosine of the angle between.

  $ arrow(v) dot arrow(w) = ||arrow(v)|| dot ||arrow(w)|| cos theta $
]

= Lecture #datetime(day: 23, month: 1, year: 2025).display()

Midterm is next Thursday in class!

== Arclength and curvature

Easy way of finding curvature: reparameterize curve with speed 1, then
curvature is acceleration. If we can't do that then we need some other
technique.

Given $arrow(c)(t) = <2t^(-1), 6, 2t>$, find the curvature $kappa(t)$.
$
  kappa (t) = (||arrow(c)'(t) times arrow(c)''(t)||) / (||arrow(c)'(t)||^3)
$

== Arclength parameterization

Find an arc-length parameterization of $arrow(c)(t) = <e^t sin(t), e^t cos(t), 5e^t>$.

Let $s = 0$ when $t = 0$ and let $s$ be the arc-length that has traveled along
the curve after $t$ seconds, then we can find $s$ by integrating the curve's
speed over $t$.

$
  s(t) = integral^t_0 ||arrow(c)'(u)|| dif u
$