Class Notes 1-5-2010

Subjects Covered

1) Linear Systems

2) Functions as Vectors

Functions graphed in vector form.

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

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

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

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

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

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

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

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

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

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

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

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

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