Class Notes 1-5-2010

Subjects Covered

1) Linear Systems

2) Functions as Vectors

Functions graphed in vector form.

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

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

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

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

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

$⟨v_x,v_y⟩$

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

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

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

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

${\delta }_{\mathrm{i},\mathrm{m}}=\{\begin{array}{ll}1& \text{if }i=m,\\ 0& \text{else}\end{array}$

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

1) Use vector analogy

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

$\underset{\frac{-T}{2}}{\overset{\frac{T}{2}}{\int }}x(t)\sin \left(\frac{2\pi mt}{T}\right)dt=v_m$

External Links

• [Class Notes.].

Authors

Colby Fullerton

Brian Roath