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+#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 $r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r$
+]
+
+#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 $r_v (u_0, v_0)$ ar
+]
+
+#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 $r_u "cross" r_v eq.not 0$
+]
+
+#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 $r(u_0, v_0)$
+]
+
+== 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
+  $
+]