diff --git a/Courses/EE1M2_Calculus_and_Linear_Algebra.typ b/Courses/EE1M2_Calculus_and_Linear_Algebra.typ
index 63562b0..fb8fa73 100644
--- a/Courses/EE1M2_Calculus_and_Linear_Algebra.typ
+++ b/Courses/EE1M2_Calculus_and_Linear_Algebra.typ
@@ -90,7 +90,14 @@ are referred to as *grid curves*.
 == Tangent vectors and normal vectors
 
 #proposition[
-  Consider the vector function $r(u,v) = chevron.l x(u,v), y(u,v), z(u,v) chevron.r$
+  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[
@@ -100,12 +107,17 @@ are referred to as *grid curves*.
 ]
 
 #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[
   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[
@@ -116,7 +128,7 @@ are referred to as *grid curves*.
 #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 $r(u_0, v_0)$
+  vector of $S$ at $r(u_0, v_0)$, provided it is nonzero.
 ]
 
 == 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
   $
 ]
+
+== 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 $integral.surf_cal(S) f d cal(S)$.
+]