Migrate more math content and the thesis

3 files changed, 55 insertions, 7 deletions

diff --git a/site/content/math/angle_between_two_vectors.typ b/site/content/math/angle_between_two_vectors.typ
index b779a5f..043d65a 100644
--- a/site/content/math/angle_between_two_vectors.typ
+++ b/site/content/math/angle_between_two_vectors.typ
@@ -3,6 +3,7 @@
 #show: post.with(
     title: "Angle between two vectors",
     date: datetime(year: 2020, month: 4, day: 17),
+    updated: datetime(year: 2026, month: 6, day: 14),
 )
 
 #figure(
@@ -67,16 +68,19 @@ which is why the dot product doesn't carry enough information to compute $psi$.
 
 == In two dimensions
 
-In any coordinate system:
+$
+  psi(bold(u), bold(v)) &= op("atan2")(bold(u) times_2 bold(v), bold(u) dot bold(v))
+$
+
+where the "two-dimensional cross product" is defined as:
 
 $
-  psi(bold(u), bold(v)) &= op("atan2")(bold(u) times_2 bold(v), bold(u) dot bold(v)) \
-  &= op("atan2")(u_1 v^2 - u_2 v^1, u_1 v^1 + u_2 v^2)
+  bold(u) times_2 bold(v) = u_1 v_2 - u_2 v_1 = u v sin theta
 $
 
 == Derivation: in two dimensions
 
-Using the geometric definitions of the #link-self("math/2d_cross_product")[two-dimensional cross product] and the dot product:
+Using the geometric definitions of the two-dimensional cross product (above) and the dot product:
 
 $
   cases(
@@ -94,7 +98,7 @@ $
   )
 $
 
-Solving for $theta$ (see #link-self("math/atan2")[Systems of equations involving the cosine and the sine of an unknown]):
+Solving for $theta$ (see #link-self("math/atan2#systems")[Systems of equations involving the cosine and the sine of an unknown]):
 
 $
   theta &= op("atan2")(1 / (u v) bold(u) times_2 bold(v), 1 / (u v) bold(u) dot bold(v)) \
diff --git a/site/content/math/atan2.typ b/site/content/math/atan2.typ
index 4442889..7fa77c8 100644
--- a/site/content/math/atan2.typ
+++ b/site/content/math/atan2.typ
@@ -1,4 +1,4 @@
-#import "/common.typ": post, static, img
+#import "/common.typ": post, static, img, anchor
 
 #show: post.with(
     title: "atan2",
@@ -45,7 +45,8 @@ $
 
 Interestingly, the function $tan^(-1)(y / x)$ has the same partial derivatives.
 
-= Systems of equations involving the cosine and the sine of an unknown
+#anchor("systems")
+= Systems of equations involving the cosine and the sine of an unknown <systems>
 
 #figure(
   img(static("math/polar-coords.png")),
diff --git a/site/content/math/index.typ b/site/content/math/index.typ
new file mode 100644
index 0000000..a1a6c4a
--- /dev/null
+++ b/site/content/math/index.typ
@@ -0,0 +1,43 @@
+#import "/common.typ": post, link-self
+
+#show: post.with(
+    title: "Math notes",
+)
+
+This part of the site is a loose collection of my math notes.
+
+- #link-self("math/angle_between_two_vectors")[Angle between two vectors]
+- #link-self("math/atan2")[atan2]
+- #link-self("math/integrating_sin2x")[Integrating sin² x]
+- #link-self("math/integrating_cos2x")[Integrating cos² x]
+- #link-self("math/ray_vs_sphere")[Ray vs. sphere intersection]
+- #link-self("math/rotational_symmetries_bravais_lattice")[Possible rotational symmetries in a Bravais lattice]
+
+= Notation
+
+Since there are so many conventions in linear algebra, I've decided to pick one and be consistent. Below is what I use throughout the site.
+
+== Vectors
+
+#table(
+    columns: 2,
+    [$bold(x)$], [A vector (bold font).],
+    [$x$], [The norm of the vector $bold(x)$ (a scalar), if it appears in the text. Otherwise, it's some scalar.],
+    [$x^2$], [The inner product of a vector with itself (a scalar). Equivalently, the norm of the vector squared.],
+    [$|bold(a) + bold(b) times bold(c)|$], [The norm of a vector expression (a scalar).],
+    [$bold(a) dot bold(b)$], [The inner product of two vectors (a scalar).],
+    [$bold(a) times bold(b)$], [The cross product of two vectors (a vector).],
+)
+
+== Matrices
+
+#table(
+    columns: 2,
+    [$hat(M)$], [A matrix.],
+    [$hat(A) hat(B)$], [A matrix product (a matrix). The former is *premultiplying* the latter or, equivalently, the latter is *postmultiplying* the former.],
+    [$hat(M) bold(x)$], [A matrix premultiplying a vector or, equivalently, operating on it (a vector).],
+    [$hat(A) times.circle hat(B)$], [The Kronecker product of two matrices (a matrix).],
+    [$hat(M)^(-1)$], [The inverse of an invertible square matrix (a matrix).],
+    [$op("det") hat(M)$], [The determinant of a square matrix (a scalar).],
+    [$op("Tr") hat(M)$], [The trace of a square matrix (a scalar).],
+)