Equation Sheet

Name	Function
Vectors
Magnitude	$\left\Vert {\vec {a}}\right\\|={\sqrt {a_{x}^{2}+a_{y}^{2}}}$
Dot Product	$\left\langle a,b\right\rangle \,\cdot \,\left\langle c,d\right\rangle =a\,c+b\,d$
Angle between 2 vectors	$\cos {(\theta )}={\frac {{\vec {a}}\cdot {\vec {b}}}{\left\Vert {\vec {a}}\right\\|\,\left\Vert {\vec {b}}\right\\|}}$
Cross Product	$\left\Vert {\vec {a}}\times {\vec {b}}\right\\|=\left\Vert {\vec {a}}\right\\|\,\left\Vert {\vec {b}}\right\\|\,\sin {(\theta )}$
Vector Functions
Velocity	${\vec {v}}={\frac {d}{dt}}\,{\vec {r}}$
Tangent Vector	${\vec {T}}={\frac {\vec {v}}{\left\Vert {\vec {v}}\right\\|}}$
Normal Vector	${\vec {N}}={\frac {\frac {d{\vec {T}}}{dt}}{\left\Vert {\frac {d{\vec {T}}}{dt}}\right\\|}}$
Acceleration	${\vec {a}}={\frac {d}{dt}}\,{\vec {v}}$

Partial Derivatives
A	B
Multiple Integrals
Average Value	${\frac {\iint \limits _{R}\,f(x)\,dA}{\iint \limits _{R}\,dA}}$
Area	$\iint \limits _{R}\,dA$
Volume	$\iiint \limits _{R}\,dV$
Mass	$\iiint \limits _{R}\,\sigma \,dV$
First Moment	$\iiint \limits _{R}\,r\,\sigma \,dV$
Second Moment	$\iiint \limits _{R}\,r^{2}\,\sigma \,dV$

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