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\title{8.012 Cheat Sheet(s)}
\author{Marc Horowitz}
\begin{document}
\maketitle

\section{Introduction}

This is my attempt at a comprehensive list of all of the equations
used in 8.012.  It is also an attempt to learn \LaTeX\ math mode.
Therefore, it probably isn't complete, and it also probably has
mistakes, but use it in good health.

\section{Vectors}
\subsection{Geometric Definition}

\sec{Magnitude}

$\mag{\A} = $ magnitude(length) of $\A$ 


\sec{Unit vector}

\[ \hat{\A} = \frac{\A}{\mag{\A}} \]


\sec{Scalar Multiplication}

$c\A = $ make \A\ $c$ times longer in the same direction.


\sec{Addition}

Head to tail (you want me to draw it!?!?)


\sec{Subtraction}

\[ \A - \B = \A + (-1\B) \]


\sec{Scalar Multiplication (Dot Product)}

\[ \A \cdot \B = \mag{\A} \mag{\B} \cos \theta \]


\sec{Vector Multiplication (Cross Product)}

\[ \C = \A \times \B \]

where:

\[ \mag{\C} = \mag{\A} \mag{\B} \sin \theta \]

The direction of \bvec{C} is defined according to the right hand rule.
That is, If you curl the fingers on your right hand from \bvec{A} to
\bvec{B}, your thumb will point in the direction of \bvec{C}.  Stupid
definition, isn't it?

Note:

The area of the parallelogram enclosed by \bvec{A} and \bvec{B} is
\mag{\A \times \B}.


\subsection{Components, or how to compute the damn things.}

In three dimensions, because two is for 8.01 wimps.

\sec{Definition}

\[ \A = (A_{x},A_{y},A_{z}) \]


\sec{Magnitude}

\[ \mag{\A} = \sqrt{A_{x}^{2}+A_{y}^{2}+A_{z}^{2}} \]


\sec{Scalar Multiplication}

\[ c\A = (cA_{x},cA_{y},cA_{z}) \]


\sec{Addition}

\[ \A + \B = (A_{x}+B_{x},A_{y}+B_{y},A_{z}+B_{z}) \]


\sec{Scalar Multiplication}

\[ \A \cdot \B = A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z} \]


\sec{Vector Multiplication}

\[ \mag{
\begin{array}{ccc}
	\ih	& \jh	& \kh \\
	A_{x}	& A_{y}	& A_{z}	\\
	B_{x}	& B_{y}	& B_{z}
\end{array}}
\]

Where \mag{\ldots} is the determinant of a matrix.

\subsection{Properties of Vectors}

\[\A+\B=\B+\A\]

\[\A+(\B+\C)=(\A+\B)+\C\]

\[c(d\A)=(cd)\A\]

\[(c+d)\A=c\A+d\A\]

\[c(\A+\B)=c\A+c\B\]

\[\A\cdot\B=\B\cdot\A\]

\[c\A\cdot\B=c(\A\cdot\B)\]

\[\A\times\B=-\B\times\A\]

\[c\A\times\B=c(\A\cdot\B)\]

\section{Kinematics}

\subsection{General Motion}

\sec{position}

\[ \bvec{r} = (x,y,z) \]


\sec{velocity and speed}

\[ \bvec{v} = \frac{d\bvec{r}}{dt} \]
\[ s = \mag{\bvec{v}} \]

\sec{acceleration}

\[ \bvec{a} = \frac{d\bvec{v}}{dt} = \frac{d^{2}\bvec{r}}{dt^{2}} \]

\subsection{Uniform Circular Motion}

\sec{angular position}

\[ \bvec{r} = r(\cos\omega t \,\ih + \sin\omega t \,\jh) \]


\sec{angular velocity}

\[ \bvec{v} = r\omega(-\sin\omega t \,\ih + \cos\omega t \,\jh) \]

\[ \rh = r\omega \]


\sec{angular acceleration}

\[ \bvec{a} = r\omega^{2}(-\cos\omega t \,\ih - \sin\omega t \,\jh) \]

\[ \mag{\bvec{a}} = -\omega^{2}\bvec{r} \]


\subsection{Polar Coordinates}

\sec{definitions}

\[ \rh = \ih\cos\theta + \jh\sin\theta \]

\[ \th = -\ih\sin\theta + \jh\cos\theta \]


\sec{derivatives}

\[ \dot{\rh} = \dot{\theta}\th \]

\[ \dot{\th} = -\dot{\theta}\rh \]

\sec{motion equations}

\[ \bvec{r} = r\rh \]

\[ \bvec{v} = \dot{r}\rh + r\dot{\theta}\th \]

\[ \bvec{a} = (\ddot{r}-r\dot{\theta}^{2})\rh + (r\ddot{\theta}+2\dot{r}\dot{\theta})\th \]

\section{Newton's Laws}

\end{document}
