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.

The individual component representation of vector $\overrightarrow{v}$ in the x-direction.

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

The two-dimensional example of a vector in its components with the vector designations.

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

This is the summation used to represent the vector $\overrightarrow{v}$ as having as many dimensions as needed to express its full value:

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

$⟨v_x,v_y⟩$

This is the equation for finding the distance between the two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ who are separated by angle $\theta$.

$\overrightarrow{u}\cdot \overrightarrow{v}=|\overrightarrow{u}||\overrightarrow{v}|\cos \theta$

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

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

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

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

Example

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

Given function: $x(t)=x(t+T)$

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

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

Reviewed By

• Ryan Gratias