diff options
Diffstat (limited to 'site/content')
| -rw-r--r-- | site/content/math/atan2.typ | 6 | ||||
| -rw-r--r-- | site/content/math/rotational_symmetries_bravais_lattice.typ | 2 |
2 files changed, 4 insertions, 4 deletions
diff --git a/site/content/math/atan2.typ b/site/content/math/atan2.typ index 4e78e56..4442889 100644 --- a/site/content/math/atan2.typ +++ b/site/content/math/atan2.typ @@ -43,13 +43,13 @@ $ & (partial)/(partial x) = -y / (x^2 + y^2) $ -Interesingly, the function $tan^(-1)(y / x)$ has the same partial derivatives. +Interestingly, the function $tan^(-1)(y / x)$ has the same partial derivatives. = Systems of equations involving the cosine and the sine of an unknown #figure( img(static("math/polar-coords.png")), - caption: [A geometric intepretation: θ = atan2(y,x).], + caption: [A geometric interpretation: θ = atan2(y,x).], ) Let $theta, x, y in RR$. Consider the following system of equations: @@ -73,7 +73,7 @@ Because $op("atan2")(sin theta, cos theta) = theta$ (up to a multiple of $2 pi$) = C's atan2 -The definition at the beginning of this article is suitable for mathematical analysis. However, a more intricate definition is needed for floating-point computation, one that takes signed zeros and infinites into account. The following appears in the C standard: +The definition at the beginning of this article is suitable for mathematical analysis. However, a more intricate definition is needed for floating-point computation, one that takes signed zeros and infinities into account. The following appears in the C standard: $ op("atan2")(y, x) = cases( diff --git a/site/content/math/rotational_symmetries_bravais_lattice.typ b/site/content/math/rotational_symmetries_bravais_lattice.typ index 6ad701d..13eb75b 100644 --- a/site/content/math/rotational_symmetries_bravais_lattice.typ +++ b/site/content/math/rotational_symmetries_bravais_lattice.typ @@ -71,7 +71,7 @@ $ (cos theta = 1) &=> (theta = 0) $ -These are the angles from the table in the beginning of the article and their multiplies. For any $theta$ not in this set of solutions, the trace of $hat(B)^(-1) hat(R)_theta hat(B)$ is a non-integer, the matrix itself necessarily has non-integer entries and the $bold(n)'$ vector ends up with non-integer entries. This contradicts the premise, hence the restriction on possible $theta$'s. +These are the angles from the table in the beginning of the article and their multiples. For any $theta$ not in this set of solutions, the trace of $hat(B)^(-1) hat(R)_theta hat(B)$ is a non-integer, the matrix itself necessarily has non-integer entries and the $bold(n)'$ vector ends up with non-integer entries. This contradicts the premise, hence the restriction on possible $theta$'s. = Proof in two dimensions |