Class Notes 1-5-2010

Modeling functions as vectors. Using function approximations, the vector path is described.

This article covers the notes given in class on January 5, 2010.

Subjects Covered

1) Linear Systems

2) Functions as Vectors

Functions graphed in vector form.

$v_{\mathrm {x} }={\vec {v}}\cdot \mathbf {\hat {i}}$

$\mathbf {\hat {v}} =v_{\mathrm {x} }\mathbf {\hat {i}} +v_{\mathrm {y} }\mathbf {\hat {j}}$

$\mathbf {\hat {v}} =\sum _{i}v_{\mathrm {i} }\mathbf {\hat {a}} _{\mathrm {i} }$

$\langle v_{x},v_{y}\rangle$

$\mathbf {\hat {u}} \cdot \mathbf {\hat {v}} =|\mathbf {\hat {u}} ||\mathbf {\hat {v}} |\cos \theta$

${\vec {v}}\cdot \mathbf {\hat {i}} =v_{\mathrm {x} }(\mathbf {\hat {i}} \cdot \mathbf {\hat {i}} )+v_{\mathrm {y} }\mathbf {\hat {j}} \cdot \mathbf {\hat {i}}$

${\vec {v}}\cdot \mathbf {\hat {i}} =v_{\mathrm {x} }$

${\vec {v}}\cdot \mathbf {\hat {a}} _{\mathrm {m} }=\sum _{i}v_{\mathrm {i} }\mathbf {\hat {a}} _{\mathrm {i} }\cdot \mathbf {\hat {a}} _{\mathrm {m} }=v_{\mathrm {m} }$

$\delta _{\mathrm {i,m} }\equiv {\begin{cases}1&{\mbox{if }}i=m,\\0&{\mbox{else}}\end{cases}}$

Example

Function waves with varying periods based on the function x(t) = x(t+T)

Given function:

$x(t)=x(t+T)\,$

$x(t)=\sum _{n=1}^{\infty }\left[b_{n}\sin \left(\left({\frac {2\pi n}{T}}\right)t\right)\right]$

1) Use vector analogy

$x(t)\cdot \sin \left({\frac {2\pi mt}{T}}\right)=\sum _{n=1}^{\infty }\left[b_{n}\sin \left(\left({\frac {2\pi n}{T}}\right)t\right)\cdot \sin \left({\frac {2\pi mt}{T}}\right)\right]$

$\int _{\frac {-T}{2}}^{\frac {T}{2}}x(t)\sin \left({\frac {2\pi mt}{T}}\right)\,dt=v_{m}$

External Links

• [Class Notes.].

Authors

Colby Fullerton

Brian Roath