Class Notes 1-5-2010

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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

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Functions graphed in vector form.

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

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

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

$⟨v_x,v_y⟩$

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

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

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

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

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

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

Example

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

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

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