#import "../template/lib.typ" : *
#set page(paper: "a4")

#show: notes.with(
  title: [EE1M2],
  subtitle: [Calculus and Linear Algebra],
  author: "Folkert Kevelam"
)

= Lecture 1 - Parametric surfaces

== Parametric surfaces

There are multiple ways to parametrize a surface. Given the paraboloid $z = x^2 + y^2$,
it can be paramaterized by the functions: $r(x,y,z) = chevron.l x,y,x^2 + y^2 chevron.r$ or
$r(theta, z) = chevron.l sqrt(z)cos(theta),sqrt(z)sin(theta),z chevron.r$.

#definition[
  A *parametric surface* $S$ in $RR^3$ is a 2-dimensional set of points whose
  positions are given by the continuous function:

  $
    r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r "with" (u,v) in D
  $

  where $D$ is a region in the uv-plane. Here $u$ and $v$ are called *parameters*
  and $r(u,v)$ is called the parameterization of $S$.
]

There are special curves that correspond to constant values of $u$ and $v$, and
are referred to as *grid curves*.

#theorem[
  Let $S$ be the graph of a function $f$ of two variables, $x$ and $y$, with
  domain $D$. Then, $S$ can be parameterized by

  $
    r(x,y) = chevron.l x, y, f(x,y) chevron.r
  $

  with $(x,y)$ in $D$.
]

#remark[
  Not each surface in $RR^3$ is the graph of a function. Example: a sphere
]

#example[
  Parameterization of a half-sphere:

  $
    r(x,y) &= chevron.l x, y, sqrt(1-x^2-y^2) chevron.r "with" x^2 + y^2 + z^2 <= 1 \
    r(theta,r) &= chevron.l r cos(theta), r sin(theta), sqrt(1-r^2) chevron.r "with" 0 <= r <= 1, 0<=theta<= 2 pi\
    r(theta,phi,rho) &= chevron.l rho cos(theta) sin(phi), rho sin(theta) sin(phi), rho cos(phi) chevron.r "with" 0 <= rho <= 1, 0 <= phi <= pi/2, 0 <= theta <= 2 pi
  $
]

#definition[
  A surface in $RR^3$ is called a *closed surface*, if it is the boundary
  of a solid region in $RR^3$.
]

#example[
  Parameterization of a shifted half-sphere:

  $
    x^2 + y^2 + (z-a)^2 <= rho^2 \
    x^2 + y^2 + z^2 - 2 a z + a^2 <= rho^2 \
    (x^2 + y^2 + z^2) - 2 a z + a^2 <= rho^2
  $
]

#definition[
  A surface $S$ in $RR^3$ that is generated by rotating a curve about an
  axis, is called a *surface of revolution*
]

#theorem[
  Let $S$ be the surface obtained by rotating a parametric curve $C$, with
  $r_c(t) = chevron.l x(t), y(t), o chevron.r$ and $a <= t <= b$, about the
  x-axis. Then, $S$ can be parameterized by

  $
    r_S(t,theta) = chevron.l x(t), y(t) cos(theta), y(t) sin(theta) chevron.r
  $

  with $a<= t <= b$ and $0 <= theta <= 2 pi$
]

== Tangent vectors and normal vectors

#proposition[
  Consider the vector function $r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r$,
  then

  $
    r_u(u,v) = chevron.l x_u (u,v),y_u (u,v),z_u (u,v) chevron.r.
  $

  Similarly for $r_v (u,v)$
]

#remark[
  Consider the surface $S$ parameterized by a vector function $r(u,v)$. The
  vectors $r_u (u_0, v_0)$ and $r_v (u_0, v_0)$ are tangent to the grid curves
  $r(u, v_0)$ and $r(u_0, v)$ respectively.
]

#corollary[
  The vectors $r_u (u_0, v_0)$ and $r_v (u_0, v_0)$ are tangent to $S$ at the
  position $r(u_0, v_0)$.
]

#definition[
  Let $P$ be a point in a surface $S$ that is not on the boundary of $S$. Then
  $S$ is called *smooth* at $P$ if $r_u "cross" r_v eq.not 0$

  if $S$ is smooth at the point $P = r(u_0, v_0)$, then the *tangent plane*
  to $S$ at the point $P$ is the plane spanned by the tangent vectors $r_u (u_0, v_0)$
  and $r_v (u_0, v_0)$ through $P$.
]

#definition[
  A nonzero vector $n$ that is perpendicular to the tangent plane of $S$ at $P$
  is called a *normal vector* of $S$ at $P$.
]

#theorem[
  Suppose a parametric surface $S$, given by $r(u,v)$, is smooth ath the point
  $r(u_0, v_0)$. Then the vector $n=r_u (u_0, v_0) times r_v (u_0, v_0)$ is a normal
  vector of $S$ at $r(u_0, v_0)$, provided it is nonzero.
]

== Areas of parametric surfaces

#theorem[
  Suppose $S$ is a smooth surface given by

  $
    r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r,  (u,v) in D
  $

  where $S$ is covered once as $(u,v)$ ranges through parameter domain $D$.
  The *surface area* of $S$ is

  $
    "area"(S) = integral.double_S d S = attach(limits(integral.double),b:D) |r_u times r_v| d u d v
  $
]

== Lecture 2 - Surface Integrals

*Orientable surfaces* are surfaces which have a "top" and a "bottom".

#definition[
  The *surface integral* of $f$ over a surface $cal(S)$ in $RR^3$ is

  $
    attach(limits(integral.double), b: cal(S)) f d cal(S) = lim_(m arrow infinity) lim_(n arrow infinity) sum_(i=1)^(m) sum_(j=1)^n f(P_(i j)) Delta cal(S)_(i j)
  $

  A surface integral over a *closed surface* $cal(S)$ is denoted as $integral.surf_cal(S) f d cal(S)$.
]