164 lines
4.5 KiB
Plaintext
164 lines
4.5 KiB
Plaintext
#import "../template/lib.typ" : *
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#set page(paper: "a4")
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#show: notes.with(
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title: [EE1M2],
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subtitle: [Calculus and Linear Algebra],
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author: "Folkert Kevelam"
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)
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= Lecture 1 - Parametric surfaces
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== Parametric surfaces
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There are multiple ways to parametrize a surface. Given the paraboloid $z = x^2 + y^2$,
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it can be paramaterized by the functions: $r(x,y,z) = chevron.l x,y,x^2 + y^2 chevron.r$ or
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$r(theta, z) = chevron.l sqrt(z)cos(theta),sqrt(z)sin(theta),z chevron.r$.
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#definition[
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A *parametric surface* $S$ in $RR^3$ is a 2-dimensional set of points whose
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positions are given by the continuous function:
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$
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r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r "with" (u,v) in D
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$
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where $D$ is a region in the uv-plane. Here $u$ and $v$ are called *parameters*
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and $r(u,v)$ is called the parameterization of $S$.
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]
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There are special curves that correspond to constant values of $u$ and $v$, and
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are referred to as *grid curves*.
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#theorem[
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Let $S$ be the graph of a function $f$ of two variables, $x$ and $y$, with
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domain $D$. Then, $S$ can be parameterized by
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$
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r(x,y) = chevron.l x, y, f(x,y) chevron.r
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$
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with $(x,y)$ in $D$.
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]
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#remark[
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Not each surface in $RR^3$ is the graph of a function. Example: a sphere
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]
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#example[
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Parameterization of a half-sphere:
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$
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r(x,y) &= chevron.l x, y, sqrt(1-x^2-y^2) chevron.r "with" x^2 + y^2 + z^2 <= 1 \
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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\
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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
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$
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]
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#definition[
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A surface in $RR^3$ is called a *closed surface*, if it is the boundary
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of a solid region in $RR^3$.
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]
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#example[
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Parameterization of a shifted half-sphere:
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$
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x^2 + y^2 + (z-a)^2 <= rho^2 \
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x^2 + y^2 + z^2 - 2 a z + a^2 <= rho^2 \
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(x^2 + y^2 + z^2) - 2 a z + a^2 <= rho^2
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$
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]
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#definition[
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A surface $S$ in $RR^3$ that is generated by rotating a curve about an
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axis, is called a *surface of revolution*
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]
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#theorem[
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Let $S$ be the surface obtained by rotating a parametric curve $C$, with
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$r_c(t) = chevron.l x(t), y(t), o chevron.r$ and $a <= t <= b$, about the
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x-axis. Then, $S$ can be parameterized by
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$
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r_S(t,theta) = chevron.l x(t), y(t) cos(theta), y(t) sin(theta) chevron.r
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$
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with $a<= t <= b$ and $0 <= theta <= 2 pi$
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]
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== Tangent vectors and normal vectors
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#proposition[
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Consider the vector function $r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r$,
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then
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$
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r_u(u,v) = chevron.l x_u (u,v),y_u (u,v),z_u (u,v) chevron.r.
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$
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Similarly for $r_v (u,v)$
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]
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#remark[
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Consider the surface $S$ parameterized by a vector function $r(u,v)$. The
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vectors $r_u (u_0, v_0)$ and $r_v (u_0, v_0)$ are tangent to the grid curves
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$r(u, v_0)$ and $r(u_0, v)$ respectively.
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]
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#corollary[
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The vectors $r_u (u_0, v_0)$ and $r_v (u_0, v_0)$ are tangent to $S$ at the
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position $r(u_0, v_0)$.
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]
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#definition[
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Let $P$ be a point in a surface $S$ that is not on the boundary of $S$. Then
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$S$ is called *smooth* at $P$ if $r_u "cross" r_v eq.not 0$
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if $S$ is smooth at the point $P = r(u_0, v_0)$, then the *tangent plane*
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to $S$ at the point $P$ is the plane spanned by the tangent vectors $r_u (u_0, v_0)$
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and $r_v (u_0, v_0)$ through $P$.
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]
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#definition[
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A nonzero vector $n$ that is perpendicular to the tangent plane of $S$ at $P$
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is called a *normal vector* of $S$ at $P$.
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]
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#theorem[
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Suppose a parametric surface $S$, given by $r(u,v)$, is smooth ath the point
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$r(u_0, v_0)$. Then the vector $n=r_u (u_0, v_0) times r_v (u_0, v_0)$ is a normal
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vector of $S$ at $r(u_0, v_0)$, provided it is nonzero.
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]
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== Areas of parametric surfaces
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#theorem[
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Suppose $S$ is a smooth surface given by
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$
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r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r, (u,v) in D
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$
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where $S$ is covered once as $(u,v)$ ranges through parameter domain $D$.
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The *surface area* of $S$ is
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$
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"area"(S) = integral.double_S d S = attach(limits(integral.double),b:D) |r_u times r_v| d u d v
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$
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]
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== Lecture 2 - Surface Integrals
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*Orientable surfaces* are surfaces which have a "top" and a "bottom".
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#definition[
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The *surface integral* of $f$ over a surface $cal(S)$ in $RR^3$ is
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$
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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)
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$
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A surface integral over a *closed surface* $cal(S)$ is denoted as $integral.surf_cal(S) f d cal(S)$.
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]
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