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Folkert Kevelam 2026-02-11 13:41:39 +01:00
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Courses/EE1M2_Calculus_and_Linear_Algebra.typ
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@ -90,7 +90,14 @@ are referred to as *grid curves*.
== Tangent vectors and normal vectors == Tangent vectors and normal vectors
#proposition[ #proposition[
Consider the vector function $r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r$ 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[ #remark[
@ -100,12 +107,17 @@ are referred to as *grid curves*.
] ]
#corollary[ #corollary[
The vectors $r_u (u_0, v_0)$ and $r_v (u_0, v_0)$ ar 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[ #definition[
Let $P$ be a point in a surface $S$ that is not on the boundary of $S$. Then 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 $r_u "cross" r_v eq.not 0$ $S$ 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[ #definition[
@ -116,7 +128,7 @@ are referred to as *grid curves*.
#theorem[ #theorem[
Suppose a parametric surface $S$, given by $r(u,v)$, is smooth ath the point 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 $r(u_0, v_0)$. Then the vector $n=r_u (u_0, v_0) times r_v (u_0, v_0)$ is a normal
vector of $S$ at $r(u_0, v_0)$ vector of $S$ at $r(u_0, v_0)$, provided it is nonzero.
] ]
== Areas of parametric surfaces == Areas of parametric surfaces
@ -135,3 +147,17 @@ are referred to as *grid curves*.
"area"(S) = integral.double_S d S = attach(limits(integral.double),b:D) |r_u times r_v| d u d v "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 $integral.surf_cal(S) f d cal(S)$.
]