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\section{Derivatives}
\scriptsize
$D_x e^x=e^x$\newline
$D_x \sin(x)=\cos(x)$\newline
$D_x \cos(x)=-\sin(x)$\newline
$D_x \tan(x)=\sec^2(x)$\newline
$D_x \cot(x)=-\csc^2(x)$\newline
$D_x \sec(x)=\sec(x)\tan(x)$\newline
$D_x \csc(x)=-\csc(x)\cot(x)$\newline
$D_x \sin^{-1}=\frac{1}{\sqrt{1-x^2}}, x \in [-1,1]$\newline
$D_x \cos^{-1}=\frac{-1}{\sqrt{1-x^2}}, x \in [-1,1]$\newline
$D_x \tan^{-1}=\frac{1}{1+x^2}, \frac{-\pi}{2}\le x \le \frac{\pi}{2}$\newline
$D_x \sec^{-1}=\frac{1}{\mid x \mid \sqrt{x^2-1}}, |x| > 1$\newline
$D_x \sinh(x)=\cosh(x)$\newline
$D_x \cosh(x)=\sinh(x)$\newline
$D_x \tanh(x)=sech^2(x)$\newline
$D_x \coth(x)=-csch^2(x)$\newline
$D_x sech(x)=-sech(x)\tanh(x)$\newline
$D_x csch(x)=-csch(x)\coth(x)$\newline
$D_x \sinh^{-1}=\frac{1}{\sqrt{x^2+1}}$\newline
$D_x \cosh^{-1}=\frac{-1}{\sqrt{x^2-1}}, x > 1$\newline
$D_x \tanh^{-1}=\frac{1}{1-x^2} -1 < x < 1$\newline
$D_x sech^{-1}=\frac{1}{x \sqrt{1-x^2}}, 0 < x < 1$
$ D_x \ln(x) = \frac{1}{x} $

\section{Integrals}
\scriptsize
$\int \frac{1}{x}dx = \ln|x|+c$\newline
$\int e^x dx = e^x+c $\newline
$\int a^x dx = \frac{1}{\ln a} a^x+c $\newline
$\int e^{ax} dx = \frac{1}{a} e^{ax}+c $\newline
$\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1}(x)+c $\newline
$\int \frac{1}{1+x^2} dx = \tan^{-1}(x)+c $\newline
$\int \frac{1}{x\sqrt{x^2-1}} dx = \sec^{-1}(x)+c $\newline
$\int \sinh(x) dx = \cosh(x)+c $\newline
$\int \cosh(x) dx = \sinh(x)+c $\newline
$\int \tanh(x) dx = \ln|\cosh(x)|+c $\newline
$\int \tanh(x)sech(x) dx = -sech(x)+c $\newline
$\int sech^2(x) dx = \tanh(x)+c $\newline
$\int csch(x)\coth(x) dx = -csch(x)+c $\newline
$\int \tan(x) dx = -\ln|\cos(x) |+c $\newline
$\int \cot(x) dx = \ln|\sin(x)|+c $\newline
$\int \cos(x) dx = \sin(x)+c $\newline
$\int \sin(x) dx = -\cos(x)+c $\newline
$\int \frac{1}{\sqrt{a^2-u^2}} dx = \sin^{-1}(\frac{u}{a})+c $\newline
$\int \frac{1}{a^2+u^2} dx = \frac{1}{a}\tan^{-1}\frac{u}{a}+c $\newline
$ \int ln(x) dx = (xln(x))-x+c $\newline

\textbf{U-Substitution}\newline
Let $ u=f(x) $ (can be more than one variable).\newline
Determine: $ du = \frac{f(x)}{dx}dx $ and solve for dx.\newline
Then, if a definite integral, substitute the bounds for $ u=f(x) $ at each bounds\newline
Solve the integral using u.\newline

\textbf{Integration by Parts}\newline
$ \int u dv = uv-\int v du $

\section{Fns and Identities}
$\sin(\cos^{-1}(x)) = \sqrt{1-x^2}$\newline
$\cos(\sin^{-1}(x)) = \sqrt{1-x^2} $\newline
$\sec(\tan^{-1}(x)) = \sqrt{1+x^2} $\newline
$\tan(\sec^{-1}(x)) \newline = (\sqrt{x^2-1} \: $if$ \: x \ge 1)\newline =(-\sqrt{x^2-1} \: if \: x \le -1)$
$\sinh^{-1}(x) = \ln{x+\sqrt{x^2+1}} $\newline
$\sinh^{-1}(x) = \ln{x+\sqrt{x^2-1}}, \: x \ge -1 $\newline
$\tanh^{-1}(x) = \frac{1}{2}\ln{x+\frac{1+x}{1-x}}, \: 1 < x < -1 $\newline
$sech^{-1}(x) = \ln[{\frac{1+\sqrt{1-x^2}}{x}}], \: 0 < x \le -1 $\newline
$\sinh(x) = \frac{e^{x}-e^{-x}}{2} $\newline
$\cosh(x) = \frac{e^{x}+e^{-x}}{2} $

\section{Trig Identities}
$ \sin^2(x)+\cos^2(x) = 1 $\newline
$ 1+\tan^2(x) = \sec^2(x) $\newline
$ 1+\cot^2(x) = \csc^2(x) $\newline
$ \sin(x\pm y) = \sin(x)\cos(y)\pm\cos(x)\sin(y) $\newline
$ \cos(x\pm y) = \cos(x)\cos(y)\pm\sin(x)\sin(y) $\newline
$ \tan(x\pm y) = \frac{\tan(x)\pm\tan(y)}{1 \mp \tan(x)\tan(y)} $\newline
$ \sin(2x) = 2\sin(x)\cos(x) $\newline
$ \cos(2x) = \cos^{2}(x) - \sin^{2}(x) $\newline
$ \cosh(n^{2}x)-\sinh^{2}x = 1 $\newline
$ 1+\tan^2(x) = \sec^2(x) $\newline
$ 1+\cot^2(x) = \csc^2(x) $\newline
$ \sin^2(x) = \frac{1-\cos(2x)}{2} $\newline
$ \cos^2(x) = \frac{1+\cos(2x)}{2} $\newline
$ \tan^2(x) = \frac{1-\cos(2x)}{1+\cos(2x)} $\newline
$ \sin(-x) = -\sin(x) $\newline
$ \cos(-x) = \cos(x) $\newline
$ \tan(-x) = -\tan(x) $

\section {Calculus 3 Concepts}

\subsection{Cartesian coords in 3D}
given two points:\newline
$ (x_1, y_1, z_1) $ and $ (x_2, y_2, z_2)$,\newline
Distance between them:\newline
$ \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2} $\newline
Midpoint:\newline
$ (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}) $\newline
Sphere with center (h,k,l) and radius r:\newline
$ (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 $

\subsection{Vectors}
Vector: $ \vec{u} $\newline
Unit Vector: $ \hat{u} $\newline
Magnitude: $ ||\vec{u}|| = \sqrt{u_1^2+u_2^2+u_3^2}$\newline
Unit Vector: $ \hat{u} = \frac{\vec{u}}{||\vec{u}||} $\newline

\textbf{Dot Product}\newline
$ \vec{u} \cdot \vec{v} $\newline
Produces a Scalar \newline
(Geometrically, the dot product is a vector projection)\newline
$\vec{u} = < u_1, u_2, u_3 >$\newline
$\vec{v} = < v_1, v_2, v_3 >$\newline
$ \vec{u} \cdot \vec{v} = \vec{0} $ means the two vectors are Perpendicular
$ \theta $ is the angle between them.\newline
$\vec{u} \cdot \vec{v} = ||\vec{u}||\:||\vec{v}||\cos(\theta) $\newline
$\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 $\newline
NOTE:\newline
$ \hat{u} \cdot \hat{v} = \cos(\theta) $\newline
$ ||\vec{u}||^2 = \vec{u} \cdot \vec{u} $\newline
$ \vec{u} \cdot \vec{v} = 0 $ when $ \bot $\newline
Angle Between $ \vec{u} $ and $ \vec{v} $:\newline
$ \theta = \cos^{-1}(\frac{\vec{u} \cdot \vec{v}}{||\vec{u}||\:||\vec{v}||}) $\newline
Projection of $ \vec{u} $ onto $ \vec{v} $:\newline
$ pr_{\vec{v}}\vec{u} = (\frac{\vec{u} \cdot \vec{v}}{||\vec{v}||^2})\vec{v} $\newline

\textbf{Cross Product}\newline
$\vec{u} \times \vec{v}$\newline
Produces a Vector\newline
(Geometrically, the cross product is the area of a paralellogram with sides $ ||\vec{u}|| $ and $ ||\vec{v}|| $)\newline
$\vec{u} = < u_1, u_2, u_3 >$\newline
$\vec{v} = < v_1, v_2, v_3 >$\newline
\[
\vec{u} \times \vec{v} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix}
\]\newline
$ \vec{u} \times \vec{v} = \vec{0} $ means the vectors are paralell

\subsection {Lines and Planes}
\textbf{Equation of a Plane}\newline
$ (x_0, y_0, z_0) $ is a point on the plane and $ <A,B,C> $ is a normal vector\newline

$A(x-x_0)+B(y-y_0)+C(z-z_0) = 0$\newline
$ <A,B,C> \cdot <x-x_0, y-y_0, z-z_0> = 0 $\newline
$ Ax+By+Cz = D $ where $ D=Ax_0+By_0+Cz_0 $\newline

\textbf{Equation of a line}\newline
A line requires a Direction Vector $ \vec{u}=<u_1,u_2,u_3> $ and a point $(x_1,y_1,z_1)$\newline
then,\newline a parameterization of a line could be:\newline
$ x = u_1t+x_1 $\newline
$ y = u_2t+y_1 $\newline
$ z = u_3t+z_1 $\newline

\textbf{Distance from a Point to a Plane}\newline
The distance from a point $(x_0,y_0,z_0)$ to a plane Ax+By+Cz=D can be expressed by the formula:\newline
$ d=\frac{|Ax_0+By_0+Cz_0-D|}{\sqrt{A^2+B^2+C^2}} $\newline

\subsection { Coord Sys Conv }
\textbf{Cylindrical to Rectangular}\newline
$ x=r\cos(\theta) $\newline
$ y=r\sin(\theta) $\newline
$ z=z $\newline
\textbf{Rectangular to Cylindrical}\newline
$ r=\sqrt{x^2+y^2} $\newline
$ \tan(\theta) = \frac{y}{x} $\newline
$ z=z $\newline
\textbf{Spherical to Rectangular}\newline
$ x=\rho\sin(\phi)\cos(\theta) $\newline
$ y=\rho\sin(\phi)\sin(\theta) $\newline
$ z=\rho\cos(\phi) $\newline
\textbf{Rectangular to Spherical}\newline
$ \rho=\sqrt{x^2+y^2+z^2} $\newline
$ \tan(\theta) = \frac{y}{x} $\newline
$ \cos(\phi) = \frac{z}{\sqrt{x^2+y^2+z^2}} $\newline
\textbf{Spherical to Cylindrical}\newline
$ r=\rho\sin(\phi) $\newline
$ \theta=\theta $\newline
$ z=\rho\cos(\phi) $\newline
\textbf{Cylindrical to Spherical}\newline
$ \rho=\sqrt{r^2+z^2} $\newline
$ \theta=\theta $\newline
$ \cos(\phi) = \frac{z}{\sqrt{r^2+z^2}} $

\subsection{Surfaces}
\textbf{Ellipsoid}\newline
$ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1 $\newline
\includegraphics[scale=0.15]{ellipsoid}\newline
\textbf{Hyperboloid of One Sheet}\newline
$ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2} = 1 $\newline
(Major Axis: z because it follows - )\newline
\includegraphics[scale=0.15]{hyperboloid1}\newline
\textbf{Hyperboloid of Two Sheets}\newline
$ \frac{z^2}{c^2}-\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $\newline
(Major Axis: Z because it is the one not subtracted)\newline
\includegraphics[scale=0.15]{hyperboloid2}\newline
\textbf{Elliptic Paraboloid}\newline
$ z=\frac{x^2}{a^2}+\frac{y^2}{b^2} $\newline
(Major Axis: z because it is the variable NOT squared)\newline
\includegraphics[scale=0.15]{elliptic_paraboloid}\newline
\textbf{Hyperbolic Paraboloid}\newline
(Major Axis: Z axis because it is not squared)\newline
$ z=\frac{y^2}{b^2}-\frac{x^2}{a^2} $\newline
\includegraphics[scale=0.15]{hyperbolic_paraboloid}\newline
\textbf{Elliptic Cone}\newline
(Major Axis: Z axis because it's the only one being subtracted)\newline
$ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2} = 0 $\newline
\includegraphics[scale=0.15]{elliptic_cone}\newline
\textbf{Cylinder}\newline
1 of the variables is missing\newline
OR\newline
$ (x-a)^2+(y-b^2) = c $\newline
(Major Axis is missing variable)

\subsection{Partial Derivatives}
Partial Derivatives are simply holding all other variables constant (and act like constants for the derivative) and only taking the derivative with respect to a given variable.\newline
Given z=f(x,y), the partial derivative of z with respect to x is:\newline
$f_x(x,y) = z_x = \frac{\partial z}{\partial x} = \frac{\partial f(x,y)}{\partial x}$\newline
likewise for partial with respect to y:\newline
$f_y(x,y) = z_y = \frac{\partial z}{\partial y} = \frac{\partial f(x,y)}{\partial y}$\newline
\textbf{Notation}\newline
For $ f_{xyy} $, work "inside to outside" $ f_{x} $ then $ f_{xy} $, then $ f_{xyy} $\newline
$f_{xyy} = \frac{\partial^3 f}{\partial^2 y \partial x}, $\newline
For $\frac{\partial^3 f}{\partial^2 y\partial x}$ , work right to left in the denominator

\subsection {Gradients}
The Gradient of a function in 2 variables is $ \nabla f = < f_x, f_y > $\newline
The Gradient of a function in 3 variables is $ \nabla f = < f_x, f_y, f_z > $

\subsection {Chain Rule(s)}

Take the Partial derivative with respect to the first-order variables of the function times the partial (or normal) derivative of the first-order variable to the ultimate variable you are looking for summed with the same process for other first-order variables this makes sense for.\newline
Example:\newline
let x = x(s,t), y = y(t) and z = z(x,y).\newline
z then has first partial derivative:\newline
$ \frac{\partial z}{\partial x} $ and $ \frac{\partial z}{\partial y} $ \newline
x has the partial derivatives:\newline
$ \frac{\partial x}{\partial s} $ and $ \frac{\partial x}{\partial t} $ \newline
and y has the derivative:\newline
$ \frac{dy}{dt} $ \newline
In this case (with z containing x and y as well as x and y both containing s and t), the chain rule for $ \frac{\partial z}{\partial s} $ is $ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} $\newline
The chain rule for $ \frac{\partial z}{\partial t} $ is $ \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{dy}{dt}$\newline
Note: the use of "d" instead of $ "\partial" $ with the function of only one independent variable

\subsection{Limits and Continuity}
\textbf{Limits in 2 or more variables}\newline
Limits taken over a vectorized limit just evaluate separately for each component of the limit.\newline
\textbf{Strategies to show limit exists}\newline
1. Plug in Numbers, Everything is Fine\newline
2. Algebraic Manipulation\newline
-factoring/dividing out\newline
-use trig identites\newline
3. Change to polar coords\newline
\:\:$if (x,y)\to(0,0)\Leftrightarrow r\to0$\newline
\textbf{Strategies to show limit DNE}\newline
1. Show limit is different if approached from different paths\newline
(x=y, $x=y^2,$ etc.)\newline
2. Switch to Polar coords and show the limit DNE.\newline
\textbf{Continunity}\newline
A fn, $z=f(x,y)$, is continuous at (a,b) if\newline
$f(a,b) = \lim_{(x,y) \to (a,b)} f(x,y) $\newline
Which means:\newline
1. The limit exists\newline
2. The fn value is defined\newline
3. They are the same value

\subsection{Directional Derivatives}
Let z=f(x,y) be a fuction, (a,b) ap point in the domain (a valid input point) and $ \hat{u} $ a unit vector (2D).\newline
The Directional Derivative is then the derivative at the point (a,b) in the direction of $ \hat{u} $ or:\newline
$ D_{\vec{u}}f(a,b) = \hat{u} \cdot \nabla f(a,b)$\newline
This will return a \textit{scalar}.
4-D version: $ D_{\vec{u}}f(a,b,c) = \hat{u} \cdot \nabla f(a,b,c) $

\subsection{Tangent Planes}
let F(x,y,z) = k be a surface and P = $(x_0, y_0, z_0)$ be a point on that surface.\newline
Equation of a Tangent Plane:\newline
$ \nabla F(x_0,y_0,z_0) \cdot <x-x_0, y-y_0, z-z_0> $

\subsection {Approximations}
let $ z=f(x,y) $ be a differentiable function
total differential of f = dz\newline
$dz= \nabla f \cdot < dx, dy >$\newline
This is the \textit{approximate} change in z\newline
The actual change in z is the difference in z values:\newline
$ \Delta z = z-z_1 $

\subsection {Maxima and Minima}
\textbf{Internal Points}\newline
1. Take the Partial Derivatives with respect to X and Y ($ f_x $ and $ f_y $) (Can use gradient)\newline
2. Set derivatives equal to 0 and use to solve system of equations for x and y\newline
3. Plug back into original equation for z.\newline
Use Second Derivative Test for whether points are local max, min, or saddle\newline

\textbf{Second Partial Derivative Test}\newline
1. Find all (x,y) points such that $ \nabla f(x,y) = \vec{0} $\newline
2. Let $ D=f_{xx}(x,y)f_{yy}(x,y)-f_{xy}^2(x,y) $\newline
IF (a) D $>$ 0 AND $ f_{xx} < 0, $ f(x,y) is local max value\newline
(b) D $>$ 0 AND $f_{xx}(x,y) > 0$ f(x,y) is local min value\newline
(c) D $<$ 0, (x,y,f(x,y)) is a saddle point\newline
(d) D $=$ 0, test is inconclusive\newline
3. Determine if any boundary point gives min or max. Typically, we have to parametrize boundary and then reduce to a Calc 1 type of min/max problem to solve.

\textbf{The following only apply only if a boundary is given}\newline
1. check the corner points\newline
2. Check each line (0 $\le$ x $\le$ 5 would give x=0 and x=5 )\newline
On Bounded Equations, this is the global min and max...second derivative test is not needed.

\subsection{Lagrange Multipliers}
Given a function f(x,y) with a constraint g(x,y), solve the following system of equations to find the max and min points on the constraint (NOTE: may need to also find internal points.):\newline
$ \nabla f = \lambda \nabla g $\newline
$ g(x,y) = 0 (or k if given) $

\subsection{Double Integrals}
With Respect to the xy-axis, if taking an integral,\newline
$ \int\int dy dx $ is cutting in vertical rectangles,\newline
$ \int\int dx dy $ is cutting in horizontal rectangles\newline

\textbf{Polar Coordinates}\newline
When using polar coordinates, $ dA = r dr  d\theta $

\subsection{Surface Area of a Curve}
let z = f(x,y) be continuous over S (a closed Region in 2D domain)\newline
Then the surface area of z = f(x,y) over S is:\newline
$ SA = \int\int_S\sqrt{f_x^2+f_y^2+1} dA $

\subsection {Triple Integrals}
$ \int\int\int_s f(x,y,z)dv = \int_{a_1}^{a_2}\int_{\phi_1(x)}^{\phi_2(x)}\int_{\psi_1(x,y)}^{\psi_2(x,y)} f(x,y,z)dzdydx $\newline
Note: $ dv $ can be exchanged for $ dxdydz $ in any order, but you must then choose your limits of integration according to that order

\subsection {Jacobian Method}
$ \int\int_Gf(g(u,v),h(u,v)) | J(u,v) | du dv = \int\int_R f(x,y) dx dy $\newline
\[
J(u,v) =
\begin{vmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{vmatrix}
\]\newline
Common Jacobians:\newline
Rect. to Cylindrical: $r$\newline
Rect. to Spherical: $\rho^2 \sin(\phi)$

\subsection {Vector Fields}
%note, this one should be moved around so that it is on the last column...it's kinda big.
let $ f(x,y,z) $ be a scalar field and $ \vec{F}(x,y,z) = M(x,y,z)\hat{i} + N(x,y,z)\hat{j} + P(x,y,z)\hat{k} $ be a vector field,\newline
Grandient of f = $ \nabla f = < \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} >$\newline
Divergence of $ \vec{F}$:\newline $ \nabla\cdot\vec{F} = \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} + \frac{\partial P}{\partial z} $\newline
Curl of $ \vec{F}$:\newline $ \nabla\times\vec{F} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} & \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
M & N & P
\end{vmatrix} $

\subsection{Line Integrals}
C given by $x = x(t), y = y(t), t\in [a,b]$\newline
$ \int_cf(x,y)ds = \int_a^bf(x(t), y(t)) ds $\newline
where $ ds = \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} dt $\newline
or $ \sqrt{1+(\frac{dy}{dx})^2} dx $\newline
or $ \sqrt{1+(\frac{dx}{dy})^2} dy $\newline
To evaluate a Line Integral,\newline
$\cdot$ get a paramaterized version of the line (usually in terms of t, though in exclusive terms of x or y is ok)\newline
$\cdot$ evaluate for the derivatives needed (usually dy, dx, and/or dt)\newline
$\cdot$ plug in to original equation to get in terms of the independant variable\newline
$\cdot$ solve integral\newline

\textbf{Work}\newline
Let $ \vec{F} = M\hat{i}+N\hat{j}+P\hat{k} $ (force) $ M=M(x,y,z), N=N(x,y,z), P=P(x,y,z) $\newline
(Literally)$ d\vec{r} = dx\hat{i}+dy\hat{j}+dz\hat{k} $\newline
Work $ w = \int_c\vec{F}\cdot d\vec{r} $\newline
(Work done by moving a particle over curve C with force $ \vec{F} $)

\subsection{Independence of Path}
\textbf{Fund Thm of Line Integrals}\newline
C is curve given by $ \vec{r}(t), t\in[a,b];\newline \vec{r}\prime(t) $ exists. If $ f(\vec{r}) $ is continuously differentiable on an open set containing C, then $ \int_c\nabla f(\vec{r}) \cdot d\vec{r} = f(\vec{b}) - f(\vec{a}) $\newline
\textbf{Equivalent Conditions}\newline
$ \vec{F}(\vec{r}) $ continuous on open connected set D. Then,\newline
$ (a) \vec{F} = \nabla f $ for some fn f. (if $ \vec{F} $ is conservative)
$ \Leftrightarrow (b) \int_c\vec{F}(\vec{r})\cdot d\vec{r} is indep. of path in D $\newline
$ \Leftrightarrow (c) \int_c\vec{F}(\vec{r})\cdot d\vec{r} = 0 $ for all closed paths in D.\newline
\textbf{Conservation Theorem}\newline
$ \vec{F} = M\hat{i}+N\hat{j}+P\hat{k} $ continuously differentiable on open, simply connected set D.\newline
$ \vec{F} $ conservative $ \Leftrightarrow \nabla\times\vec{F} = \vec{0} $ \newline
(in 2D $ \nabla\times\vec{F}=\vec{0} $ iff $ M_y = N_x$)

\subsection{Green's Theorem}
(method of changing line integral for double integral - Use for Flux and Circulation across 2D curve and line integrals over a closed boundary)\newline
$ \oint Mdy - Ndx = \int\int_R(M_x+N_y)dxdy $\newline
$ \oint Mdx + Ndy = \int\int_R (N_x-M_y) dxdy $\newline
Let:\newline
$\cdot$R be a region in xy-plane\newline
$\cdot$C is simple, closed curve enclosing R (w/ paramerization $ \vec{r}(t) $)\newline
$ \cdot\vec{F}(x,y) = M(x,y)\hat{i} + N(x,y)\hat{j} $ be continuously differentiable over R$\cup$C. \newline
\textbf{Form 1: Flux Across Boundary}\newline
$ \vec{n} = $ unit normal vector to C\newline
$ \oint_c \vec{F}\cdot\vec{n} = \int\int_R \nabla\cdot\vec{F} dA $\newline
$ \Leftrightarrow\oint Mdy - Ndx = \int\int_R(M_x+N_y)dxdy $\newline
\textbf{Form 2: Circulation Along Boundary}\newline
$ \oint_c\vec{F}\cdot d\vec{r} = \int\int_R \nabla\times\vec{F}\cdot\hat{u} dA $\newline
$ \Leftrightarrow \oint Mdx + Ndy = \int\int_R (N_x-M_y) dxdy $\newline
\textbf{Area of R}\newline
$ A = \oint(\frac{-1}{2}y dx + \frac{1}{2}x dy) $

\subsection{Gauss' Divergence Thm}
(3D Analog of Green's Theorem - Use for Flux over a 3D surface)
Let:\newline
$ \cdot\vec{F}(x,y,z) $ be vector field continuously differentiable in solid S\newline
$ \cdot $S is a 3D solid
$ \cdot\partial S $ boundary of S (A Surface)\newline
$ \cdot\hat{n} $unit outer normal to $ \partial S $\newline
Then,\newline
$ \int\int_{\partial S}\vec{F}(x,y,z)\cdot\hat{n}dS = \int\int\int_S\nabla\cdot\vec{F} dV $\newline
(dV = dxdydz)

\vfill\columnbreak

\subsection{Surface Integrals}
Let\newline
$\cdot$R be closed, bounded region in xy-plane\newline
$\cdot$f be a fn with first order partial derivatives on R\newline
$\cdot$G be a surface over R given by $ z=f(x,y) $\newline
$\cdot g(x,y,z) = g(x,y,f(x,y))$ is cont. on R\newline
Then,\newline$ \int\int_G g(x,y,z) dS = \int\int_R g(x,y,f(x,y)) dS $\newline
where $ dS = \sqrt{f_x^2+f_y^2+1}dydx $\newline
\textbf{Flux of $ \vec{F} $ across G}\newline
$ \int\int_G\vec{F}\cdot{n} dS = \int\int_R[-Mf_x-Nf_y+P]dxdy $\newline
where:\newline
$\cdot\vec{F}(x,y,z) = M(x,y,z)\hat{i} + N(x,y,z)\hat{j} + P(x,y,z)\hat{k} $\newline
$\cdot$G is surface f(x,y)=z\newline
$\cdot\vec{n}$ is upward unit normal on G.\newline
$\cdot$f(x,y) has continuous $1^{st}$ order partial derivatives
\vfill
\subsection{Unit Circle}
(cos, sin)\newline
\includegraphics[scale=0.64]{unit_circle}
\newline\newline\newline\newline
\vfill
\columnbreak
\subsection {Other Information}
$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $\newline
Where a Cone is defined as $ z = \sqrt{a(x^2+y^2)}, $\newline
In Spherical Coordinates, $ \phi = \cos^{-1}(\sqrt{\frac{a}{1+a}}) $\newline
Right Circular Cylinder:\newline
$ V=\pi r^2h, SA=\pi r^2+2\pi rh $\newline
$ \lim_{n\to \inf} (1+\frac{m}{n})^{pn} = e^{mp} $\newline
Law of Cosines:\newline
$ a^2 = b^2 + c^2 - 2bc(\cos(\theta)) $

\subsection{Stokes Theorem}
Let:\newline
$\cdot$S be a 3D surface\newline
$\cdot\vec{F}(x,y,z) = M(x,y,z)\hat{i}+N(x,y,z)\hat{j}+P(x,y,z)\hat{l} $\newline
$\cdot$M,N,P have continuous $1^{st}$ order partial derivatives\newline
$\cdot$C is piece-wise smooth, simple, closed, curve, positively oriented\newline
$\cdot\hat{T}$ is unit tangent vector to C.\newline
Then,\newline
$ \oint\vec{F}_c\cdot\hat{T}dS = \int\int_s (\nabla\times\vec{F})\cdot\hat{n} dS = \int\int_R(\nabla\times\vec{F})\cdot\vec{n}dxdy $\newline
Remember:\newline
$ \oint\vec{F}\cdot\vec{T}ds = \int_c (Mdx + Ndy + Pdz) $

%The following can go away as needed, it is here so I can put my Accreditation message/source code message at the bottom right.
\vfill
{\fontsize{.5}{1}\selectfont Originally Written By Daniel Kenner for MATH 2210 at the University of Utah.\newline Source code available at https://github.com/keytotime/Calc3\_CheatSheet\newline Thanks to Kelly Macarthur for Teaching and Providing Notes.}
\end{multicols*}
\end{document}


\chapter{Objectives}
This study primarily aims to develop an approach that integrates a relatively new metaheuristic, Grey Wolf Optimization, to solve the unequal area static facility layout problem. Other objectives of this study are:

\begin{enumerate}
	\item To evaluate the performance of the proposed approach in generating solutions to the unequal area static facility layout problem.

	\item To evaluate the performance of the proposed approach with varying parameters for various aspects of the approach.

	\item To compare the performance of the proposed approach to the performance of a genetic algorithm-based approach.
\end{enumerate}


\documentclass[8pt]{beamer}
\usepackage{packages}
\usepackage{commands}
\usepackage{beamer_options}
\graphicspath{{img/}}
% \setbeamertemplate{footline}[page number]
% Magic comment for latex workshop
% !TeX root = ./on_two_minimax_theorems_in_graph_presentation.tex
% !TEX program = xelatex

% We need page numbers
%*************************************************************
\title{On Two Minimax Theorems in Graph}
\author{Daniel Blévin - George Apaaboah}
\date\today

%*************************************************************
\begin{document}
\maketitle
\begin{frame}
\tableofcontents
\end{frame}

%************************************************************* THEOREM
\newtheorem{edmonds_theorem}{THEOREM}
\newtheorem{Lucchesi_and_Younger}{THEOREM}
\newtheorem{ly_lemma}{LEMMA}
\newtheorem{hypergraph}{Theorem}

%=============================================================
\section{First theorem}
%-------------------------------------------------------------
\subsection{Definitions}
%*************************************************************
\begin{frame}
\frametitle{Branching}
\begin{block}{branching (r-arborescence) :}
In a directed graph $D = (V, A)$ a branching is a set of arcs not containing
circuits s.t. each node of $D$ is entered at most by one arc in $A'$.
(So the arc in $A'$ make up a forest). % A or A' ???
\end{block}

%\pause
~

\splitpage{
\insertTikzFigure{1}{branchingExemple_base}
{A digraph $D=(A, V)$}
{tikz:branchingExemple_base}
}{
\insertTikzFigure{1}{branchingExemple_simple}
{A branching of $D$ routed at $a$.}
{tikz:branchingExemple_simple}
}
\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Edge disjooint branching}

\splitpage{
    \insertTikzFigure{1}{branchingExemple_disjoint}
    {Two edge-disjoint branching routed respectively at $a$ and $d$.}
    {tikz:branchingExemple_disjoint}
}{
    \insertTikzFigure{1}{branchingExemple_arouted}
    {Two edge-disjoint $a$-routed branching.}
    {tikz:branchingExemple_arouted}
}
\end{frame}

%*************************************************************
\begin{frame}
\frametitle{a-cut}

\begin{block}{a-cut :}
A $a-$cut of $G$ determined by a set $S \subset V(G)$ is the set of
edges going from $S$ to $V(G) - S$.
It will be denoted by $\Delta_G(S)$.
We also set that $\delta_G(S) = |\Delta_G(S)|$.
\end{block}
~

\threesplitpage
{
    \insertTikzFigure{0.6}{cutExemple_cut1}
    {A a-cut of $D$,\\ $S = \{a\}$,\\ $\Delta_D(S)=\{ab, ad\}$,\\ $\delta_D(S) = 2$.}
    {tikz:cutExemple_cut1}
}{
    \insertTikzFigure{0.6}{cutExemple_cut2}
    {A minimal a-cut of $D$,\\ $S = \{a, b, c, d, f\}$,\\ $\Delta_D(S)=\{be\}$,\\ $\delta_D(S) = 1$.}
    {tikz:cutExemple_cut2}
}{
    \insertTikzFigure{0.6}{cutExemple_cut3}
    {A minimal a-cut of $D$,\\ $S = \{a, b, d, e, f\}$,\\ $\Delta_D(S)=\{bc\}$,\\ $\delta_D(S) = 1$.}
    {tikz:cutExemple_cut3}
}
\end{frame}

%-------------------------------------------------------------
\subsection{Theorem}
%*************************************************************
\begin{frame}
\frametitle{Theorem}

\begin{edmonds_theorem}[Edmonds]
The maximum number of edge-disjoint branchings (rooted at $a$)
equals the minimum number of edges in $a-$cuts.
\end{edmonds_theorem}
\end{frame}

%-------------------------------------------------------------
\subsection{Proof}
%*************************************************************
\begin{frame}
\frametitle{Proof}

Let $k$ be the minimum cardinality of an $a$-cut.\\
\textbf{We prove by induction on $k$.}
The case $k = 0$ is trivial.\\~\\

% Maybe useless statement
% If $\Delta_G(S) \ge k,\ \forall S \subset V(G),\ a\in S$ then there are $k$ edge-disjoint branchings.

% + arborescence

~

Let $F$ be a set of edges such that $F$ is an arborescence rooted at $a$.
\begin{equation}
    \Delta_{G-F}(S) \ge k-1,\ \forall\ S \subset V(G),\ a\in S  \label{eq:hyp}
\end{equation}

If \eqref{eq:hyp}, we are done by induction.
Suppose that \eqref{eq:hyp} holds.
\end{frame}
%*************************************************************
\begin{frame}
\frametitle{Proof}

If $F = E(G)$ then it is a branching and we are done ($G-F$ contains $k-1$ edge-disjoint branchings and $F$ is in the $k^{th}$ edge-disjoint branching).

~

If $F \neq E(G)$ and only cover a set $T \subset V(G)$
we show we can add an edge $e \in \Delta_G(T)$ to $F$ so that $F + e$ still satisfies \eqref{eq:hyp}.

% ~

% Following from this, we can see that $F$ only covers a set of vertices $T \subset V(G)$. Therefore, there exist a vertex $v$ that is not connected to $F$ and we can add an edge $e\in \Delta_G(T)$. This will yield an arborescence $F+e$ such that $F$ covers all points of $G$, satisfying $\Delta_{G-F}(S) \ge k-1$ $\forall$ $S\subset V(G),$ $a\in S$.
\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Proof}
Consider a maximal set $A\subset V(G)$ s.t :
\begin{subequations}
\begin{align}
    & a \in A \\
    & A \cup T \ne V(G) \\
    & \delta_{G-F}(S) = k - 1
\end{align}
\end{subequations}

If no such $A$ exists, any edge of $\Delta_G(T)$ can be added to $F$.

\splittextfig{
    \vskip 0.7cm
    Otherwise let $e = (x, y) \in \Delta_{G-F}(A\cup T) - \Delta_{G-F}(A)$ s.t. $x\in T-A$ and $y\in V(G) -T-A$
    as shown in \textbf{figure \ref{tikz:p1edgeexist}}.\\
    Such an edge must exist because :
    \begin{equation*}
    \delta_{G-F}(A \cup T) = \delta_G(A \cup T) \geq k > k - 1 = \delta_{G-F}(A)
    \end{equation*}
}{
    \insertTikzFigure{0.9}{p1edgeexist}
    {}
    {tikz:p1edgeexist}
}
\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Proof}
We claim $e$ can be added to $F$ s.t. $F + e$ is an arborescence (trivial) and \eqref{eq:hyp} holds.

% Now we want to show that that we can add $e$ to $F$ and $F+e$ will be an arborescence and satisfy $\Delta_{G-F}(S) \ge k-1$. For sure, we can see that $F$ is an arborescence.

If $e\not\in \Delta_G(S)$ then $$\delta_{G-F-e}(S) = \delta_{G-F}(S) \ge k-1$$
Now, if we have that $e\in \Delta_G(S)$ then $x\in S$ and $y\in V(G)-S$. That is, $x$ and $y$ are in different set of vertices of $G$. Next, we will use the inequality;
\begin{equation}\delta_{G-F}(S\cup A) + \delta_{G-F}(S\cap A) \le \delta_{G-F}(S) + \delta_{G-F}(A)\end{equation}
From the maximality of $A$ : $\delta_{G-F}(A)=k-1$, $\delta_{G-F}(S\cap A) \ge k-1$ and $\delta_{G-F}(S\cup A) \ge k$.\\
Then from $(1)$ we have $\delta_{G-F}(S) \ge k$.\\
Therefore $\delta_{G-F-e}(S) \ge k-1$.
\eofproof
\end{frame}

%=============================================================
\section{Second theorem}
%-------------------------------------------------------------
\subsection{Definitions}
%*************************************************************
\begin{frame}
\frametitle{Definitions}
\begin{block}
\textbf{Directed cut (dicuts) :}
A directed cut of a weakly connect graph $G$ is the set $D = \Delta_G(S),\ (S \subset V(G),\ S \neq \varnothing)$ provided $\Delta_G(S),\ (V(G) - S) = \varnothing$.

\end{block}

%~
%\pause
\begin{block}
\textbf{Dijoin :} A dijoin is a subset $A' \subset A$ which covers all dicuts. Let $\Omega(D)$ denote the maximum number of arc-disjoint dicuts in $D$
and let $\epsilon(D)$ be the minimum cardinality of a dijoin.

\end{block}

%\pause
\threesplitpage
{
    \insertTikzFigure{0.7}{dicuts_0}
    {A weakly connected digraph containing 3 dicuts}
    {tikz:dicuts_0}
}{
    \insertTikzFigure{0.7}{dicuts_1}
    {Arc-disjoint dicuts\\$\Omega(G) = 2$}
    {tikz:dicuts_1}
}{
    \insertTikzFigure{0.7}{dicuts_2}
    {Dijoin of minimal cardinality\\$\epsilon(G) = 2$}
    {tikz:dicuts_2}
}

\end{frame}

%-------------------------------------------------------------
\subsection{Theroem}
%*************************************************************
\begin{frame}
\frametitle{Theorem}

\begin{Lucchesi_and_Younger}[Lucchesi and Younger]
The maximum number of disjoint directed cuts equals the minimum number of edges which cover all directed cuts ($\Omega(G) = \epsilon(G)$).
\end{Lucchesi_and_Younger}

\end{frame}

%*************************************************************

\subsection{Proof}
%*************************************************************
\begin{frame}
\frametitle{Proof}
%We will use induction on the number of edges. \\
\begin{block}{Case 1 :}
If there are no edges, then the theorem is trivially true.
\end{block}

%\pause
\begin{block}{Case 2 :}
Let $k$ be the maximum directed cuts in $G$, $G_e^{''}$ a digraph resulting from contraction of $e$, $e\in E$.
\end{block}

%\pause
\begin{block}\\
If $G_e^{''}$ contains at most $k-1$ edge-disjoint directed cuts then, by the induction hypothesis, $\exists$ $k-1$ edges covering all directed cuts of $G_e^{''}$. Hence if we add $e$ to $G_e^{''}$, then $G_e^{''}$ covers all directed cuts of $G$
\end{block}

%If $G_e"$ contains at most $k-1$ edge-disjoint directed cuts then, by the induction hypothesis, $\exists$ $k-1$ edges $e_1,\dots,e_{k-1}$ covering all directed cuts of $G_e"$. \\
%From above, we observe that adding $e$ yields $e, e_1,\dots,e_{k-1}$ which is $k$ edges and cover all directed cuts of $G$. Since we need at least $k$ edges, the assertion is proved and we may assume that $G_e"$ contains $k$ disjoint directed cuts for each edge $e$.\\

\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Proof}
\begin{block}\\
If we subdivide all the edges of $G$ by a point, then the graph that we will obtain contains $k+1$ disjoint directed cuts. Hence we can find a subdivision $H$ of $G$ that contains at most $k$ disjoint directed cuts and $f$ is an edge of $H$.
\end{block}

%If we subdivide all the edges of $G$ by a point, then the graph that we will obtain contains $k+1$ disjoint directed cuts. Hence we can find a subdivision $H$ of $G$ that contains $k$ at most $k$ disjoint directed cuts.
\begin{block}\\
If we again subdivide an edge $f$ of $H$ by a point, then it will contain $k+1$ disjoint directed cuts. Hence $H$ contains $k+1$ directed cuts $D_1,\dots,D_{k+1}$ such that only two of them have $f$ as a common edge.
\end{block}

%\pause
\begin{block}\\
From above, $H_f^{''}$ arises from either $G$ or $G_f^{''}$ by subdivision. Hence from the assumption made above, $H$ contains $k$ disjoint directed cuts $c_1,\dots,c_k$ and $f$ not in $c_i$. Thus $D_1,\dots,D_{k+1},c_1,\dots,c_k$ is a collection of directed cuts of $G_0$ and any edge belongs to at most two of them.
\end{block}

\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Proof}

Next, we have the lemma:\\
\begin{ly_lemma}[]
If a digraph $G$ contains at most $k$ disjoint directed cuts, and $F$ is any collection of directed cuts in $G$ such that any edge belongs to at most two of them, then $|F| \le 2k$.
\end{ly_lemma}

%\pause
\begin{block}\\
%Let $D=\Delta_G(S)$, $(S\subset V(G), S\ne \emptyset$) provided $\Delta_G(V(G)-S)=\emptyset$.
\textbf{Proof of Lemma}\\
Let $S_D$ be a set uniquely determined by a directed cut $D$\\
%%First we replace $F$ by a collection of a simple structure.
Let $D_1, D_2 \in F$ be called a laminar if $S_{D_1}\cap S_{D_2}=\emptyset \text{ or } S_{D_1}\subseteq S_{D_2}, \text{ or } S_{D_2}\subseteq S_{D_1} \text{ or } S_{D_1}\cup S_{D_2} = V(G)$. Else $D_1, D_2$ is a crossing\\
\end{block}

\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Proof}
\begin{block}\\
Let $D_1, D_2$ be a crossing pair and set the following:
$$D_1'=\Delta_G(S_{D_1} \cup S_{D_2})$$
$$D_2'=\Delta_G(S_{D_1} \cap S_{D_2})$$
$$F'=F\cup \{D_1',D_2' \}-\{D_1,D_2 \}$$

From these, we can see that $D_1', D_2'$ are directed cuts and cover any edge the same number of times as $D_1, D_2$. Hence $|F'| = |F|$.
\end{block}

%\pause
\begin{block}\\
Also,
$$\underset{D\in F}{\sum}|S_D|^2 < \underset{D\in F'}{\sum}|S_D|^2$$
since
$$|S_{D_1}\cup S_{D_2}|^2 + |S_{D_1}\cap S_{D_2}|>|S_{D_1}|^2+|S_{D_2}|^2$$
\end{block}

\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Proof}
\begin{block}\\
Hence, we cannot repeat the same process of $F$ for $F'$ and achieve a cycle.
We get a collection $F_0$ of directed cuts such that any $e$ belongs to 2 of them, any 2 are laminar and $|F_0| = |F|$.
\end{block}

%\pause
\begin{block}\\
Next, we prove the \textit{Lemma} for the case when $F$ consist of pairwise laminar cuts.
\end{block}

%\pause
\begin{block}

Let $F=\{D_1,\dots,D_N\}$, then we construct a graph $G'$ as follows:\\
$V(G')=\{v_1,\dots,v_N\}$ and join $v_i$ to $v_j$ \textit{iff} $D_i \cap D_j \ne \emptyset$.\\
Then $G'$ contains at most $k$ independent points.
\end{block}
\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Proof}
\begin{block}\\
Now we will show that $G'$ is bipartite. \\
Consider a circuit $(v_1,\dots,v_m)$ in $G'$ and the corresponding sets $S_{D_1},\dots,S_{D_m}$.\\
Then $D_1,\dots,D_m$ must be different. \\If $D_v = D_\mu$ then each edge of $D_v$ also belongs to $D_\mu$ and no other member of $F$. Hence $v_v$ has degree 1 and cannot occur in any circuit of $G'$.
\end{block}

%\pause
\begin{block}\\
Since $D_i \cap D_{i+1} \ne \emptyset$, we have either $S_{D_i}\subset S_{D_{i+1}}$ or $S_{D_i}\supset S_{D_{i+1}}$

We claim that both possibilities occur alternatively and this will prove that $m$ is even.
\end{block}

%\pause
\begin{block}
\textbf{Proof of claim}\\
Suppose not, example, $S_{D_0} \subset S_{D_1}\subset S_{D_2}$.

We say $D_i$ is to the left from $D_j$ if either $S_{D_i}\subset S_{D_j}$ or $V(G)-S_{D_i}\subset S_{D_j}$.

$D_i$ is to the right from $D_j$ if $S_{D_i}\subset V(G)-S_{D_j}$ or $V(G)-S_{D_i}\subset V(G)-S_{D_j}$.
\end{block}

%\eofproof
\end{frame}

%*************************************************************
\begin{frame}
\frametitle{Proof}
\begin{block}\\
Now, since $F$ consist of laminar cuts, each $D_i\ne D_j$ is either to the left or to the right from $D_j$.
\end{block}

%\pause
\begin{block}\\
Since $D_2$ is to the right from $D_1$ but $D_0 = D_m$ is to the left from $D_1$, there is a $j$, $1\le j\le m-1$ such that $D_j$ is to the right from $D_1$ but $D_{j+1}$ is to the left from $D_1$.
\end{block}

%\pause
\begin{block}\\
Since $D_j$ and $D_{j+1}$ have a common edge $e$, which must belong to $D_1$. Hence $e$ belongs to three cuts, which is a contradiction.
\end{block}

\eofproof
\end{frame}

\iffalse
%*************************************************************
\section{Hypergraphs}
%*************************************************************
\begin{frame}
\frametitle{Definitions}
A \textit{hypergraph} H is a finite collection of finite sets. These sets are called edges and the elements of the edges are called vertices. Let $V(H)$ denote the set of vertices. If $E_1,\dots,E_m$ are the edges and $v_1,\dots,v_n$ are the vertices of H, then we define
$$
    a_{ij}=\left\{
    \begin{array}{@{}ll@{}}
    1, & \text{if}\ v_i \in E_j \\
    0, & \text{otherwise}
    \end{array}\right.
$$

\begin{hypergraph}[Hypergraph]
If any hypergraph H' arising from H by multiplication of the vertices satisfies $v_2(H') = 2v(H')$ then $\tau(H) = v(H)$.
\end{hypergraph}

\begin{proof}
First we show that\\
\textbf{1.} If $F$ is a collection of pairwise laminar directed cut then its incidence matrix $A$ is totally unimodular.

\end{proof}
\end{frame}\fi
%%%%%%%%%%%% \fi ??????

\begin{frame}
\frametitle{Figure for you george}

\insertTikzFigure{1}{rightleftcut}
{Illustration of the deﬁnition of being to the right and left for cuts.\\
In (a) the dicuts ??? is on right, in (b) ... ???}
{tikz:rightleftcut}

To be cite as \textbf{figure \ref{tikz:rightleftcut}}.

\end{frame}

\begin{frame}
\frametitle{Bibliography}
Lovasz, L. (1976). On two minimax theorems in graph. Journal of Combinatorial Theory, Series B, 21(2), 96-103.
\end{frame}

{
\metroset{background=dark}
\begin{frame}
\begin{center}
  Thank you, questions ?
\end{center}
\end{frame}
}

\begin{frame}
\frametitle{Appendix}
\textbf{Definitions:}\\
A \textit{hypergraph} H is a finite collection of finite sets. These sets are called edges and the elements of the edges are called vertices. Let $V(H)$ denote the set of vertices. If $E_1,\dots,E_m$ are the edges and $v_1,\dots,v_n$ are the vertices of H, then we define
$$
    a_{ij}=\left\{
    \begin{array}{@{}ll@{}}
    1, & \text{if}\ v_i \in E_j \\
    0, & \text{otherwise}
    \end{array}\right.
$$

\begin{hypergraph}[Hypergraph]
If any hypergraph H' arising from H by multiplication of the vertices satisfies $v_2(H') = 2v(H')$ then $\tau(H) = v(H)$.
\end{hypergraph}

\begin{proof}
First we show that\\
\textbf{1.} If $F$ is a collection of pairwise laminar directed cut then its incidence matrix $A$ is totally unimodular.

\end{proof}

\end{frame}

\end{document}


%
% API Documentation for Peach - Computational Intelligence for Python
% Module peach.optm.quasinewton
%
% Generated by epydoc 3.0beta1
% [Mon Dec 21 08:51:38 2009]
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                          Module Description                           %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \index{peach \textit{(package)}!peach.optm \textit{(package)}!peach.optm.quasinewton \textit{(module)}|(}
\section{Module peach.optm.quasinewton}

    \label{peach:optm:quasinewton}

This package implements basic quasi-Newton optimizers. Newton optimizer is very
efficient, except that inverse matrices need to be calculated at each
convergence step. These methods try to estimate the hessian inverse iteratively,
thus increasing performance.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                               Variables                               %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \subsection{Variables}

\begin{longtable}{|p{.30\textwidth}|p{.62\textwidth}|l}
\cline{1-2}
\cline{1-2} \centering \textbf{Name} & \centering \textbf{Description}& \\
\cline{1-2}
\endhead\cline{1-2}\multicolumn{3}{r}{\small\textit{continued on next page}}\\\endfoot\cline{1-2}
\endlastfoot\raggedright \_\-\_\-d\-o\-c\-\_\-\_\- & \raggedright \textbf{Value:}
{\tt \texttt{...}}&\\
\cline{1-2}
\end{longtable}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                           Class Description                           %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \index{peach \textit{(package)}!peach.optm \textit{(package)}!peach.optm.quasinewton \textit{(module)}!peach.optm.quasinewton.DFP \textit{(class)}|(}
\subsection{Class DFP}

    \label{peach:optm:quasinewton:DFP}
\begin{tabular}{cccccccc}
% Line for object, linespec=[False, False]
\multicolumn{2}{r}{\settowidth{\BCL}{object}\multirow{2}{\BCL}{object}}
&&
&&
  \\\cline{3-3}
  &&\multicolumn{1}{c|}{}
&&
&&
  \\
% Line for peach.optm.optm.Optimizer, linespec=[False]
\multicolumn{4}{r}{\settowidth{\BCL}{peach.optm.optm.Optimizer}\multirow{2}{\BCL}{peach.optm.optm.Optimizer}}
&&
  \\\cline{5-5}
  &&&&\multicolumn{1}{c|}{}
&&
  \\
&&&&\multicolumn{2}{l}{\textbf{peach.optm.quasinewton.DFP}}
\end{tabular}

DFP (\emph{Davidon-Fletcher-Powell}) search

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                Methods                                %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \subsubsection{Methods}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_init\_\_}(\textit{self}, \textit{f}, \textit{df}=\texttt{None}, \textit{B}=\texttt{None}, \textit{h}=\texttt{0.1}, \textit{emax}=\texttt{1e-05}, \textit{imax}=\texttt{1000})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

Initializes the optimizer.

To create an optimizer of this type, instantiate the class with the
parameters given below:
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{xxxx}

          \item[f]

A multivariable function to be optimized. The function should have
only one parameter, a multidimensional line-vector, and return the
function value, a scalar.
          \item[df]

A function to calculate the gradient vector of the cost function
\texttt{f}. Defaults to \texttt{None}, if no gradient is supplied, then it is
estimated from the cost function using Euler equations.
          \item[B]

A first estimate of the inverse hessian. Note that, differently from
the Newton method, the elements in this matrix are numbers, not
functions. So, it is an estimate at a given point, and its values
\emph{should} be coherent with the first estimate (that is, \texttt{B} should
be the inverse of the hessian evaluated at the first estimate), or
else the algorithm might diverge. Defaults to \texttt{None}, if none is
given, it is estimated. Note that, given the same reasons as before,
the estimate of \texttt{B} is deferred to the first calling of the
\texttt{step} method, where it is handled automatically.
          \item[h]

Convergence step. This method does not takes into consideration the
possibility of varying the convergence step, to avoid Stiefel cages.
          \item[emax]

Maximum allowed error. The algorithm stops as soon as the error is
below this level. The error is absolute.
          \item[imax]

Maximum number of iterations, the algorithm stops as soon this
number of iterations are executed, no matter what the error is at
the moment.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.\_\_init\_\_

    \end{boxedminipage}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{step}(\textit{self}, \textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

One step of the search.

In this method, the result of the step is dependent of parameters
calculated before (namely, the estimate of the inverse hessian), so it
is not recomended that different investigations are used with the same
optimizer in the same cost function.
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{x}

          \item[x]

The value from where the new estimate should be calculated. This can
of course be the result of a previous iteration of the algorithm.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      \textbf{Return Value}
      \begin{quote}

This method returns a tuple \texttt{(x, e)}, where \texttt{x} is the updated
estimate of the minimum, and \texttt{e} is the estimated error.
      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.step

    \end{boxedminipage}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_call\_\_}(\textit{self}, \textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

Transparently executes the search until the minimum is found. The stop
criteria are the maximum error or the maximum number of iterations,
whichever is reached first. Note that this is a \texttt{{\_}{\_}call{\_}{\_}} method, so
the object is called as a function. This method returns a tuple
\texttt{(x, e)}, with the best estimate of the minimum and the error.
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{x}

          \item[x]

The initial triplet of values from where the search must start.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      \textbf{Return Value}
      \begin{quote}

This method returns a tuple \texttt{(x, e)}, where \texttt{x} is the best
estimate of the minimum, and \texttt{e} is the estimated error.
      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.\_\_call\_\_

    \end{boxedminipage}

    \label{object:__delattr__}
    \index{object.\_\_delattr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_delattr\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}delattr{\_}{\_}('name') {\textless}=={\textgreater} del x.name
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__getattribute__}
    \index{object.\_\_getattribute\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_getattribute\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}getattribute{\_}{\_}('name') {\textless}=={\textgreater} x.name
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__hash__}
    \index{object.\_\_hash\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_hash\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

hash(x)
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__new__}
    \index{object.\_\_new\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_new\_\_}(\textit{T}, \textit{S}, \textit{...})

      \textbf{Return Value}
      \begin{quote}
\begin{alltt}
a new object with type S, a subtype of T
\end{alltt}

      \end{quote}

    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__reduce__}
    \index{object.\_\_reduce\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_reduce\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

helper for pickle
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__reduce_ex__}
    \index{object.\_\_reduce\_ex\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_reduce\_ex\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

helper for pickle
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__repr__}
    \index{object.\_\_repr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_repr\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

repr(x)
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__setattr__}
    \index{object.\_\_setattr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_setattr\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}setattr{\_}{\_}('name', value) {\textless}=={\textgreater} x.name = value
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__str__}
    \index{object.\_\_str\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_str\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

str(x)
    \vspace{1ex}

    \end{boxedminipage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                              Properties                               %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \subsubsection{Properties}

\begin{longtable}{|p{.30\textwidth}|p{.62\textwidth}|l}
\cline{1-2}
\cline{1-2} \centering \textbf{Name} & \centering \textbf{Description}& \\
\cline{1-2}
\endhead\cline{1-2}\multicolumn{3}{r}{\small\textit{continued on next page}}\\\endfoot\cline{1-2}
\endlastfoot\raggedright \_\-\_\-c\-l\-a\-s\-s\-\_\-\_\- & \raggedright \textbf{Value:}
{\tt {\textless}attribute '\_\_class\_\_' of 'object' objects{\textgreater}}&\\
\cline{1-2}
\end{longtable}

    \index{peach \textit{(package)}!peach.optm \textit{(package)}!peach.optm.quasinewton \textit{(module)}!peach.optm.quasinewton.DFP \textit{(class)}|)}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                           Class Description                           %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \index{peach \textit{(package)}!peach.optm \textit{(package)}!peach.optm.quasinewton \textit{(module)}!peach.optm.quasinewton.BFGS \textit{(class)}|(}
\subsection{Class BFGS}

    \label{peach:optm:quasinewton:BFGS}
\begin{tabular}{cccccccc}
% Line for object, linespec=[False, False]
\multicolumn{2}{r}{\settowidth{\BCL}{object}\multirow{2}{\BCL}{object}}
&&
&&
  \\\cline{3-3}
  &&\multicolumn{1}{c|}{}
&&
&&
  \\
% Line for peach.optm.optm.Optimizer, linespec=[False]
\multicolumn{4}{r}{\settowidth{\BCL}{peach.optm.optm.Optimizer}\multirow{2}{\BCL}{peach.optm.optm.Optimizer}}
&&
  \\\cline{5-5}
  &&&&\multicolumn{1}{c|}{}
&&
  \\
&&&&\multicolumn{2}{l}{\textbf{peach.optm.quasinewton.BFGS}}
\end{tabular}

BFGS (\emph{Broyden-Fletcher-Goldfarb-Shanno}) search

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                Methods                                %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \subsubsection{Methods}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_init\_\_}(\textit{self}, \textit{f}, \textit{df}=\texttt{None}, \textit{B}=\texttt{None}, \textit{h}=\texttt{0.1}, \textit{emax}=\texttt{1e-05}, \textit{imax}=\texttt{1000})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

Initializes the optimizer.

To create an optimizer of this type, instantiate the class with the
parameters given below:
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{xxxx}

          \item[f]

A multivariable function to be optimized. The function should have
only one parameter, a multidimensional line-vector, and return the
function value, a scalar.
          \item[df]

A function to calculate the gradient vector of the cost function
\texttt{f}. Defaults to \texttt{None}, if no gradient is supplied, then it is
estimated from the cost function using Euler equations.
          \item[B]

A first estimate of the inverse hessian. Note that, differently from
the Newton method, the elements in this matrix are numbers, not
functions. So, it is an estimate at a given point, and its values
\emph{should} be coherent with the first estimate (that is, \texttt{B} should
be the inverse of the hessian evaluated at the first estimate), or
else the algorithm might diverge. Defaults to \texttt{None}, if none is
given, it is estimated. Note that, given the same reasons as before,
the estimate of \texttt{B} is deferred to the first calling of the
\texttt{step} method, where it is handled automatically.
          \item[h]

Convergence step. This method does not takes into consideration the
possibility of varying the convergence step, to avoid Stiefel cages.
          \item[emax]

Maximum allowed error. The algorithm stops as soon as the error is
below this level. The error is absolute.
          \item[imax]

Maximum number of iterations, the algorithm stops as soon this
number of iterations are executed, no matter what the error is at
the moment.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.\_\_init\_\_

    \end{boxedminipage}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{step}(\textit{self}, \textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

One step of the search.

In this method, the result of the step is dependent of parameters
calculated before (namely, the estimate of the inverse hessian), so it
is not recomended that different investigations are used with the same
optimizer in the same cost function.
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{x}

          \item[x]

The value from where the new estimate should be calculated. This can
of course be the result of a previous iteration of the algorithm.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      \textbf{Return Value}
      \begin{quote}

This method returns a tuple \texttt{(x, e)}, where \texttt{x} is the updated
estimate of the minimum, and \texttt{e} is the estimated error.
      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.step

    \end{boxedminipage}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_call\_\_}(\textit{self}, \textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

Transparently executes the search until the minimum is found. The stop
criteria are the maximum error or the maximum number of iterations,
whichever is reached first. Note that this is a \texttt{{\_}{\_}call{\_}{\_}} method, so
the object is called as a function. This method returns a tuple
\texttt{(x, e)}, with the best estimate of the minimum and the error.
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{x}

          \item[x]

The initial triplet of values from where the search must start.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      \textbf{Return Value}
      \begin{quote}

This method returns a tuple \texttt{(x, e)}, where \texttt{x} is the best
estimate of the minimum, and \texttt{e} is the estimated error.
      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.\_\_call\_\_

    \end{boxedminipage}

    \label{object:__delattr__}
    \index{object.\_\_delattr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_delattr\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}delattr{\_}{\_}('name') {\textless}=={\textgreater} del x.name
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__getattribute__}
    \index{object.\_\_getattribute\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_getattribute\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}getattribute{\_}{\_}('name') {\textless}=={\textgreater} x.name
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__hash__}
    \index{object.\_\_hash\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_hash\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

hash(x)
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__new__}
    \index{object.\_\_new\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_new\_\_}(\textit{T}, \textit{S}, \textit{...})

      \textbf{Return Value}
      \begin{quote}
\begin{alltt}
a new object with type S, a subtype of T
\end{alltt}

      \end{quote}

    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__reduce__}
    \index{object.\_\_reduce\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_reduce\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

helper for pickle
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__reduce_ex__}
    \index{object.\_\_reduce\_ex\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_reduce\_ex\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

helper for pickle
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__repr__}
    \index{object.\_\_repr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_repr\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

repr(x)
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__setattr__}
    \index{object.\_\_setattr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_setattr\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}setattr{\_}{\_}('name', value) {\textless}=={\textgreater} x.name = value
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__str__}
    \index{object.\_\_str\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_str\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

str(x)
    \vspace{1ex}

    \end{boxedminipage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                              Properties                               %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \subsubsection{Properties}

\begin{longtable}{|p{.30\textwidth}|p{.62\textwidth}|l}
\cline{1-2}
\cline{1-2} \centering \textbf{Name} & \centering \textbf{Description}& \\
\cline{1-2}
\endhead\cline{1-2}\multicolumn{3}{r}{\small\textit{continued on next page}}\\\endfoot\cline{1-2}
\endlastfoot\raggedright \_\-\_\-c\-l\-a\-s\-s\-\_\-\_\- & \raggedright \textbf{Value:}
{\tt {\textless}attribute '\_\_class\_\_' of 'object' objects{\textgreater}}&\\
\cline{1-2}
\end{longtable}

    \index{peach \textit{(package)}!peach.optm \textit{(package)}!peach.optm.quasinewton \textit{(module)}!peach.optm.quasinewton.BFGS \textit{(class)}|)}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                           Class Description                           %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \index{peach \textit{(package)}!peach.optm \textit{(package)}!peach.optm.quasinewton \textit{(module)}!peach.optm.quasinewton.SR1 \textit{(class)}|(}
\subsection{Class SR1}

    \label{peach:optm:quasinewton:SR1}
\begin{tabular}{cccccccc}
% Line for object, linespec=[False, False]
\multicolumn{2}{r}{\settowidth{\BCL}{object}\multirow{2}{\BCL}{object}}
&&
&&
  \\\cline{3-3}
  &&\multicolumn{1}{c|}{}
&&
&&
  \\
% Line for peach.optm.optm.Optimizer, linespec=[False]
\multicolumn{4}{r}{\settowidth{\BCL}{peach.optm.optm.Optimizer}\multirow{2}{\BCL}{peach.optm.optm.Optimizer}}
&&
  \\\cline{5-5}
  &&&&\multicolumn{1}{c|}{}
&&
  \\
&&&&\multicolumn{2}{l}{\textbf{peach.optm.quasinewton.SR1}}
\end{tabular}

SR1 (\emph{Symmetric Rank 1} )search method

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                Methods                                %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \subsubsection{Methods}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_init\_\_}(\textit{self}, \textit{f}, \textit{df}=\texttt{None}, \textit{B}=\texttt{None}, \textit{h}=\texttt{0.1}, \textit{emax}=\texttt{1e-05}, \textit{imax}=\texttt{1000})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

Initializes the optimizer.

To create an optimizer of this type, instantiate the class with the
parameters given below:
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{xxxx}

          \item[f]

A multivariable function to be optimized. The function should have
only one parameter, a multidimensional line-vector, and return the
function value, a scalar.
          \item[df]

A function to calculate the gradient vector of the cost function
\texttt{f}. Defaults to \texttt{None}, if no gradient is supplied, then it is
estimated from the cost function using Euler equations.
          \item[B]

A first estimate of the inverse hessian. Note that, differently from
the Newton method, the elements in this matrix are numbers, not
functions. So, it is an estimate at a given point, and its values
\emph{should} be coherent with the first estimate (that is, \texttt{B} should
be the inverse of the hessian evaluated at the first estimate), or
else the algorithm might diverge. Defaults to \texttt{None}, if none is
given, it is estimated. Note that, given the same reasons as before,
the estimate of \texttt{B} is deferred to the first calling of the
\texttt{step} method, where it is handled automatically.
          \item[h]

Convergence step. This method does not takes into consideration the
possibility of varying the convergence step, to avoid Stiefel cages.
          \item[emax]

Maximum allowed error. The algorithm stops as soon as the error is
below this level. The error is absolute.
          \item[imax]

Maximum number of iterations, the algorithm stops as soon this
number of iterations are executed, no matter what the error is at
the moment.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.\_\_init\_\_

    \end{boxedminipage}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{step}(\textit{self}, \textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

One step of the search.

In this method, the result of the step is dependent of parameters
calculated before (namely, the estimate of the inverse hessian), so it
is not recomended that different investigations are used with the same
optimizer in the same cost function.
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{x}

          \item[x]

The value from where the new estimate should be calculated. This can
of course be the result of a previous iteration of the algorithm.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      \textbf{Return Value}
      \begin{quote}

This method returns a tuple \texttt{(x, e)}, where \texttt{x} is the updated
estimate of the minimum, and \texttt{e} is the estimated error.
      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.step

    \end{boxedminipage}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_call\_\_}(\textit{self}, \textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

Transparently executes the search until the minimum is found. The stop
criteria are the maximum error or the maximum number of iterations,
whichever is reached first. Note that this is a \texttt{{\_}{\_}call{\_}{\_}} method, so
the object is called as a function. This method returns a tuple
\texttt{(x, e)}, with the best estimate of the minimum and the error.
    \vspace{1ex}

      \textbf{Parameters}
      \begin{quote}
        \begin{Ventry}{x}

          \item[x]

The initial triplet of values from where the search must start.
        \end{Ventry}

      \end{quote}

    \vspace{1ex}

      \textbf{Return Value}
      \begin{quote}

This method returns a tuple \texttt{(x, e)}, where \texttt{x} is the best
estimate of the minimum, and \texttt{e} is the estimated error.
      \end{quote}

    \vspace{1ex}

      Overrides: peach.optm.optm.Optimizer.\_\_call\_\_

    \end{boxedminipage}

    \label{object:__delattr__}
    \index{object.\_\_delattr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_delattr\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}delattr{\_}{\_}('name') {\textless}=={\textgreater} del x.name
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__getattribute__}
    \index{object.\_\_getattribute\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_getattribute\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}getattribute{\_}{\_}('name') {\textless}=={\textgreater} x.name
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__hash__}
    \index{object.\_\_hash\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_hash\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

hash(x)
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__new__}
    \index{object.\_\_new\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_new\_\_}(\textit{T}, \textit{S}, \textit{...})

      \textbf{Return Value}
      \begin{quote}
\begin{alltt}
a new object with type S, a subtype of T
\end{alltt}

      \end{quote}

    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__reduce__}
    \index{object.\_\_reduce\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_reduce\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

helper for pickle
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__reduce_ex__}
    \index{object.\_\_reduce\_ex\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_reduce\_ex\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

helper for pickle
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__repr__}
    \index{object.\_\_repr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_repr\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

repr(x)
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__setattr__}
    \index{object.\_\_setattr\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_setattr\_\_}(\textit{...})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

x.{\_}{\_}setattr{\_}{\_}('name', value) {\textless}=={\textgreater} x.name = value
    \vspace{1ex}

    \end{boxedminipage}

    \label{object:__str__}
    \index{object.\_\_str\_\_ \textit{(function)}}

    \vspace{0.5ex}

    \begin{boxedminipage}{\textwidth}

    \raggedright \textbf{\_\_str\_\_}(\textit{x})

    \vspace{-1.5ex}

    \rule{\textwidth}{0.5\fboxrule}

str(x)
    \vspace{1ex}

    \end{boxedminipage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                              Properties                               %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  \subsubsection{Properties}

\begin{longtable}{|p{.30\textwidth}|p{.62\textwidth}|l}
\cline{1-2}
\cline{1-2} \centering \textbf{Name} & \centering \textbf{Description}& \\
\cline{1-2}
\endhead\cline{1-2}\multicolumn{3}{r}{\small\textit{continued on next page}}\\\endfoot\cline{1-2}
\endlastfoot\raggedright \_\-\_\-c\-l\-a\-s\-s\-\_\-\_\- & \raggedright \textbf{Value:}
{\tt {\textless}attribute '\_\_class\_\_' of 'object' objects{\textgreater}}&\\
\cline{1-2}
\end{longtable}

    \index{peach \textit{(package)}!peach.optm \textit{(package)}!peach.optm.quasinewton \textit{(module)}!peach.optm.quasinewton.SR1 \textit{(class)}|)}
    \index{peach \textit{(package)}!peach.optm \textit{(package)}!peach.optm.quasinewton \textit{(module)}|)}


\documentclass[12pt]{article}
\usepackage{fancyhdr}
\usepackage{color}
\usepackage{multicol}
\usepackage{enumitem}
\usepackage{graphicx}
\usepackage{sectsty}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{array}
\newcommand{\sectionbreak}{\clearpage}
\usepackage{lscape}

\usepackage{tikz}
	\usetikzlibrary{arrows,shapes,trees}

\usepackage{tkz-euclide}
\usetkzobj{all}

\allsectionsfont{\centering}

\usepackage{draftwatermark}
	\SetWatermarkText{wolf-math.com}
	\SetWatermarkScale{5}
	\SetWatermarkAngle{90}
	\SetWatermarkLightness{1}

\usepackage[margin=.75in, headsep=0pt]{geometry}
\setlength{\parindent}{0cm}
\pagestyle{empty}

\begin{document}

\section{Graphing Trigonometric Functions}

\subsection{Goals}

\textbf{I will be able to} graph a trigonometric functions using a unit circle as a guide.\\

\textbf{I will be able to} translate and transform trigonometric funtions.\\

\subsection{Standards}

\subsection{Connections}

\clearpage

\section*{Graphing a Sine Wave}

Recall that the radius of the unit circle is 1, and the points on the unit circle $(x,y)$ also correspond to the trig functions $(\cos, \sin)$. \\

We can now draw a graph where the $x$ axis is the \emph{angle} and the $y$ axis is the \emph{trig function}: $f(x)=\sin(x)$.\\

\section*{Graphing a Cosine Wave}

Now draw a graph where the $x$ axis is the \emph{angle} and the $y$ axis is the \emph{trig function}: $f(x)=\cos(x)$.\\

\begin{landscape}

\section*{Graphing a Sine Wave}

Recall that the radius of the unit circle is 1, and the points on the unit circle $(x,y)$ also correspond to the trig functions $(\cos, \sin)$. \\

We can now draw a graph where the $x$ axis is the \emph{angle} and the $y$ axis is the \emph{trig function}: $f(x)=\sin(x)$.\\

\vspace{1in}

\begin{tiny}
    \begin{tikzpicture}[scale=3,cap=round,>=latex]
        % draw the coordinates
        \draw[->] (-1.5cm,0cm) -- (1.5cm,0cm) node[right,fill=white] {$x$};
        \draw[->] (0cm,-1.5cm) -- (0cm,1.5cm) node[above,fill=white] {$y$};

        % draw the unit circle
        \draw[thick] (0cm,0cm) circle(1cm);

        \foreach \x in {0,30,...,360} {
                % lines from center to point
                \draw[gray] (0cm,0cm) -- (\x:1cm);
                % dots at each point
                \filldraw[black] (\x:1cm) circle(0.4pt);
                % draw each angle in degrees
                \draw (\x:0.6cm) node[fill=white] {$\x^\circ$};
        }

		\foreach \x in {45,135,225,315}{
				\draw[gray](0cm,0cm) -- (\x:1cm);
				\filldraw[black] (\x:1cm) circle(0.4pt);
				\draw (\x:0.6cm) node[fill=white] {$\x^\circ$};
		}

        % draw each angle in radians
        \foreach \x/\xtext in {
            30/\frac{\pi}{6},
            45/\frac{\pi}{4},
            60/\frac{\pi}{3},
            90/\frac{\pi}{2},
            120/\frac{2\pi}{3},
            135/\frac{3\pi}{4},
            150/\frac{5\pi}{6},
            180/\pi,
            210/\frac{7\pi}{6},
            225/\frac{5\pi}{4},
            240/\frac{4\pi}{3},
            270/\frac{3\pi}{2},
            300/\frac{5\pi}{3},
            315/\frac{7\pi}{4},
            330/\frac{11\pi}{6},
            360/2\pi}
                \draw (\x:0.85cm) node[fill=white] {$\xtext$};

        \foreach \x/\xtext/\y in {
            % the coordinates for the first quadrant
            30/\frac{\sqrt{3}}{2}/\frac{1}{2},
            45/\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2},
            60/\frac{1}{2}/\frac{\sqrt{3}}{2},
            % the coordinates for the second quadrant
            150/-\frac{\sqrt{3}}{2}/\frac{1}{2},
            135/-\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2},
            120/-\frac{1}{2}/\frac{\sqrt{3}}{2},
            % the coordinates for the third quadrant
            210/-\frac{\sqrt{3}}{2}/-\frac{1}{2},
            225/-\frac{\sqrt{2}}{2}/-\frac{\sqrt{2}}{2},
            240/-\frac{1}{2}/-\frac{\sqrt{3}}{2},
            % the coordinates for the fourth quadrant
            330/\frac{\sqrt{3}}{2}/-\frac{1}{2},
            315/\frac{\sqrt{2}}{2}/-\frac{\sqrt{2}}{2},
            300/\frac{1}{2}/-\frac{\sqrt{3}}{2}}
                \draw (\x:1.25cm) node[fill=white] {$\left(\xtext,\y\right)$};

        % draw the horizontal and vertical coordinates
        % the placement is better this way
        \draw (-1.25cm,0cm) node[above=1pt] {$(-1,0)$}
              (1.25cm,0cm)  node[above=1pt] {$(1,0)$}
              (0cm,-1.25cm) node[fill=white] {$(0,-1)$}
              (0cm,1.25cm)  node[fill=white] {$(0,1)$};

        % draw the coordinates
        \draw[<->] (1.8cm,0cm) -- (6.5cm,0cm) node[right,fill=white] {$x$};
        \draw[<->] (2cm,-1.5cm) -- (2cm,1.5cm) node[above,fill=white] {$y$};

        \draw [fill=black] (2cm,1cm) circle (.5pt);
        \draw (2cm,1cm) node[above=5pt, fill=white] {$(1,0)$};

        \draw [fill=black] (2cm,-1cm) circle (.5pt);
        \draw (2cm,-1cm) node[below=5pt, fill=white] {$(1,0)$};

        \draw [fill=black] (3cm,0cm) circle (.5pt);
        \draw (3cm,0cm) node[below=5pt, fill=white] {$90^\circ$};

        \draw [fill=black] (4cm,0cm) circle (.5pt);
        \draw (4cm,0cm) node[below=5pt, fill=white] {$180^\circ$};

        \draw [fill=black] (5cm,0cm) circle (.5pt);
        \draw (5cm,0cm) node[below=5pt, fill=white] {$270^\circ$};

        \draw [fill=black] (6cm,0cm) circle (.5pt);
        \draw (6cm,0cm) node[below=5pt, fill=white] {$360^\circ$};
\end{tikzpicture}
\end{tiny}

\clearpage

\section*{Graphing a Cosine Wave}

Now draw a graph where the $x$ axis is the \emph{angle} and the $y$ axis is the \emph{trig function}: $f(x)=\cos(x)$.\\

\vspace{1in}

\begin{tiny}
    \begin{tikzpicture}[scale=3,cap=round,>=latex]
        % draw the coordinates
        \draw[->] (-1.5cm,0cm) -- (1.5cm,0cm) node[right,fill=white] {$x$};
        \draw[->] (0cm,-1.5cm) -- (0cm,1.5cm) node[above,fill=white] {$y$};

        % draw the unit circle
        \draw[thick] (0cm,0cm) circle(1cm);

        \foreach \x in {0,30,...,360} {
                % lines from center to point
                \draw[gray] (0cm,0cm) -- (\x:1cm);
                % dots at each point
                \filldraw[black] (\x:1cm) circle(0.4pt);
                % draw each angle in degrees
                \draw (\x:0.6cm) node[fill=white] {$\x^\circ$};
        }

		\foreach \x in {45,135,225,315}{
				\draw[gray](0cm,0cm) -- (\x:1cm);
				\filldraw[black] (\x:1cm) circle(0.4pt);
				\draw (\x:0.6cm) node[fill=white] {$\x^\circ$};
		}

        % draw each angle in radians
        \foreach \x/\xtext in {
            30/\frac{\pi}{6},
            45/\frac{\pi}{4},
            60/\frac{\pi}{3},
            90/\frac{\pi}{2},
            120/\frac{2\pi}{3},
            135/\frac{3\pi}{4},
            150/\frac{5\pi}{6},
            180/\pi,
            210/\frac{7\pi}{6},
            225/\frac{5\pi}{4},
            240/\frac{4\pi}{3},
            270/\frac{3\pi}{2},
            300/\frac{5\pi}{3},
            315/\frac{7\pi}{4},
            330/\frac{11\pi}{6},
            360/2\pi}
                \draw (\x:0.85cm) node[fill=white] {$\xtext$};

        \foreach \x/\xtext/\y in {
            % the coordinates for the first quadrant
            30/\frac{\sqrt{3}}{2}/\frac{1}{2},
            45/\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2},
            60/\frac{1}{2}/\frac{\sqrt{3}}{2},
            % the coordinates for the second quadrant
            150/-\frac{\sqrt{3}}{2}/\frac{1}{2},
            135/-\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2},
            120/-\frac{1}{2}/\frac{\sqrt{3}}{2},
            % the coordinates for the third quadrant
            210/-\frac{\sqrt{3}}{2}/-\frac{1}{2},
            225/-\frac{\sqrt{2}}{2}/-\frac{\sqrt{2}}{2},
            240/-\frac{1}{2}/-\frac{\sqrt{3}}{2},
            % the coordinates for the fourth quadrant
            330/\frac{\sqrt{3}}{2}/-\frac{1}{2},
            315/\frac{\sqrt{2}}{2}/-\frac{\sqrt{2}}{2},
            300/\frac{1}{2}/-\frac{\sqrt{3}}{2}}
                \draw (\x:1.25cm) node[fill=white] {$\left(\xtext,\y\right)$};

        % draw the horizontal and vertical coordinates
        % the placement is better this way
        \draw (-1.25cm,0cm) node[above=1pt] {$(-1,0)$}
              (1.25cm,0cm)  node[above=1pt] {$(1,0)$}
              (0cm,-1.25cm) node[fill=white] {$(0,-1)$}
              (0cm,1.25cm)  node[fill=white] {$(0,1)$};

        % draw the coordinates
        \draw[<->] (1.8cm,0cm) -- (6.5cm,0cm) node[right,fill=white] {$x$};
        \draw[<->] (2cm,-1.5cm) -- (2cm,1.5cm) node[above,fill=white] {$y$};

        \draw [fill=black] (2cm,1cm) circle (.5pt);
        \draw (2cm,1cm) node[above=5pt, fill=white] {$(1,0)$};

        \draw [fill=black] (2cm,-1cm) circle (.5pt);
        \draw (2cm,-1cm) node[below=5pt, fill=white] {$(1,0)$};

        \draw [fill=black] (3cm,0cm) circle (.5pt);
        \draw (3cm,0cm) node[below=5pt, fill=white] {$90^\circ$};

        \draw [fill=black] (4cm,0cm) circle (.5pt);
        \draw (4cm,0cm) node[below=5pt, fill=white] {$180^\circ$};

        \draw [fill=black] (5cm,0cm) circle (.5pt);
        \draw (5cm,0cm) node[below=5pt, fill=white] {$270^\circ$};

        \draw [fill=black] (6cm,0cm) circle (.5pt);
        \draw (6cm,0cm) node[below=5pt, fill=white] {$360^\circ$};
\end{tikzpicture}
\end{tiny}

\clearpage

\section*{Multipliers}

What happens if we multiply something in front of the wave?\\

$$f(x)=2\cdot \sin(x)$$

\begin{tiny}
    \begin{tikzpicture}[scale=3,cap=round,>=latex]

        \draw[<->] (1.8cm,0cm) -- (6.5cm,0cm) node[right,fill=white] {$x$};
        \draw[<->] (2cm,-1.5cm) -- (2cm,1.5cm) node[above,fill=white] {$y$};

        \draw [fill=black] (2cm,1cm) circle (.5pt);
        \draw (2cm,1cm) node[above=5pt, fill=white] {$(1,0)$};

        \draw [fill=black] (2cm,-1cm) circle (.5pt);
        \draw (2cm,-1cm) node[below=5pt, fill=white] {$(1,0)$};

        \draw [fill=black] (3cm,0cm) circle (.5pt);
        \draw (3cm,0cm) node[below=5pt, fill=white] {$90^\circ$};

        \draw [fill=black] (4cm,0cm) circle (.5pt);
        \draw (4cm,0cm) node[below=5pt, fill=white] {$180^\circ$};

        \draw [fill=black] (5cm,0cm) circle (.5pt);
        \draw (5cm,0cm) node[below=5pt, fill=white] {$270^\circ$};

        \draw [fill=black] (6cm,0cm) circle (.5pt);
        \draw (6cm,0cm) node[below=5pt, fill=white] {$360^\circ$};
\end{tikzpicture}
\end{tiny}

\end{landscape}

\end{document}

07_Trigonometry/07.6_Graphing_Trig_Functions/graphs of trig functions.tex


\section{Conducting the Experiments}
\label{sec:Conducting_the_Experiments}
This section describes the experimental arrangement with the device list and all the measurement objects. Furthermore, it contains information about the measuring process.

\subsection{Experimental Arrangement}
\label{subsec:experimental_arrangement}
The experiment was arranged as shown below in figure \ref{fig:experimental_arrangement}:
\begin{figure}[H]
	\centering
	\includegraphics[scale=1]{experimental_arrangement}
	\caption{Experimental Arrangement \cite{interference} - partially modified}
	\label{fig:experimental_arrangement}
\end{figure}
The red laser and the apertures were placed on a rail. This allowed the apertures to be moved alog the x-axis and thus positioned in an ideal spot. The laser with a wavelength of 633 nm then shines through the aperture and the diffraction gets visible on the projection surface. It \textbf{is possible to use a variety of lenses} to optimize the sharpness of the diffraction on the projection surface. In this experiment \textbf{no lenses were used} tough. The diffraction pattern was already sharp enough.

The following equipment was used:

\begin{table}[H]
	\centering
	\renewcommand{\arraystretch}{1.2}
	\begin{tabular}{l l l}
		\hline
		\textbf{Device} & \textbf{Information} & \textbf{Uncertainty} \\
		\hline
		Laser & $\lambda = 633$ nm & N/A \\
		Double Meter & x-Measurements & $\pm1$ mm \\
		Ruler & y-Measurements & $\pm1$ mm \\ \hline
	\end{tabular}
	\caption{Equipment}
	\label{tab:equipment}
\end{table}

The red laser pointer is used to shine through the aperture. The double meter is used to measure the x-distance (alone the x-axis) and the ruler is used to measure the y-position of the maxima and minima. Even tough it is possible to read the position on the ruler to half a millimeter, the uncertainty is still one millimeter due to the difficulty to find the exact center of a minima or maxima.

\subsection{Measuring Procedure}
\label{subsec:measuring_procedure}
First, the aperture is moved along the x-axis to find an ideal spot (sharp diffraction pattern on the projection surface). Then, the x-distance is written down. After that, the absolute value of the center is measured with the ruler. The last step consists of measuring as much absolute y-values of minima or maxima as possible.

\subsection{Measurement Objects}
\label{subsec:measurement_objects}
The measuring objects are a range of different apertures (slits, anti-slits, circular apertures and cross-grids). The different apertures and their specifications are listed in the table below:

\begin{table}[H]
	\centering
	\renewcommand{\arraystretch}{1.2}
	\begin{tabular}{r|c|c|c}
		 & \textbf{Width $w$} & \textbf{Diameter $d$} & \textbf{Grid Constant $g$} \\
		\hline
		\textbf{Slits} & 40 $\mu$m \& 100 $\mu$m & - & - \\
		\textbf{Anti-Slits} & 230 $\mu$m \& 124 $\mu$m & - & - \\
		\textbf{Circular Apertures} & - & 150 $\mu$m \& 100 $\mu$m & - \\
		\textbf{Cross-Grids} & - & - & 28 $\mu$m \& 50 $\mu$m \\ \hline
	\end{tabular}
	\caption{Specifications of the Apertures}
	\label{tab:Specifications_Apertures}
\end{table}

\subsection{Measurements}
\label{subsec:Measurements}
The performed measurements are almost identical, no matter what kind of aperture it is. This is due to the fact, that no lenses were used during this experiment (see section \ref{subsec:experimental_arrangement}).

The maxima are measured for the slits, the anti-slits and the cross-grids. Whilst for the circular apertures the minima are being measured. This is due to the fact, that it is generally easier to measure the maxima. The problem with circular apertures is, that there just is no easy equation to accurately relate the maxima to an angle.