User:Eml5526.s11.team5.JA/Mtg8

  Mtg 8: Wed, 19 Jan 11

 [[media: Fe1.s11.mtg8.djvu| Page 8-1 ]] 

Note: $$ \mathbf{F} = m \mathbf{a} \ $$ ([[media: Fe1.s11.mtg6.djvu|    page 6-1 ]])

Newton's law is a  claim  - that could have been  wrong  - about the actual relation between the force  F  on a particle with mass m and its acceleration  a . One tests it by calculating the acceleration with a presumed force and comparing it to the measured value. The test will fail if neither newton's law or the presumed force is wrong. Could  F  = m  a  be tested more generally, without recourse to positing forces and looking at actual solutions?  It seems not .

Kane, String theory,  Physics Today , Nov 2010



Epsilon Eridani, closest known planetary system to our solar system, 10 light-years away. (wikimedia Commons) http://commons.wikimedia.org/wiki/File:NASA-JPL-Caltech_-_Double_the_Rubble_%28PIA11375%29_%28pd%29.jpg



EML 4500 Fall 2008: http://commons.wikimedia.org/wiki/File:1bcspring.jpg

 Motivation for WRF  continued

[[media: Fe1.s11.mtg7.djvu| (1)   page 7-4 ]]  more general than [[media: Fe1.s11.mtg7.djvu|  (2)   page 7-2 ]]


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Consider: $$ \mathbf{w}.\mathbf{P(v)}= 0 \ \forall \ \mathbf{w} \in R^n \ $$

$$
 *  $$ \displaystyle {\color{Red}(1)}
 * }

 2 equivalent statements: 

 [[media: Fe1.s11.mtg8.djvu| Page 8-2 ]] 

1) Use $$ \left \{ \mathbf{b}_i \right \} \ $$ (basis), and let $$ \mathbf{w} = \sum_i \alpha_i \mathbf{b}_i \ $$. Then [[media: Fe1.s11.mtg8.djvu| (1)   page 8-1 ]] is equivalent to


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$$ \mathbf{w}.\mathbf{P(v)}= 0 \ \forall \ \mathbf{w} = \sum_i \alpha_i \mathbf{b}_i \in R^n \ $$

$$
 *  $$ \displaystyle {\color{Red}(1)}
 * }


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$$ \mathbf{w}.\mathbf{P(v)}= 0 \ \forall \ \left \{ \alpha_1, ......, \alpha_n \right \} \in R^n \ $$, s.t. $$ \sum_i \alpha_i \mathbf{b}_i \ $$

$$
 *  $$ \displaystyle {\color{Red}(2)}
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$$
 *  $$ \displaystyle {\color{Red}(3)}
 * } How?

Proof:

 A)  

 Choice 1:   $$ {\color{Blue} \alpha_1 = 1, \alpha_1 = .... = \alpha_n = 0 }\ $$   since    $$ {\color{Blue} \left \{ \alpha_1 = .... = \alpha_n  \right \} }\ $$   are arbitrary  , example, $$ \forall \left \{ \alpha_1 = .... = \alpha_n \right \} \ $$


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(2) $$ \Rightarrow \mathbf{b}_1.\mathbf{P(v)}= 0 \ $$

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 *  $$ \displaystyle
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 Choice 2:  $$ \left \{ \alpha_1, ...., \alpha_n \right \} = \left \{ 0, \underbrace_{\color{Blue} \alpha_2} , 0, 0, ...., 0 \right \} \ $$


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(2)  $$ \Rightarrow \mathbf{b}_2.\mathbf{P(v)}= 0 \ $$

$$
 *  $$ \displaystyle
 * }

 [[media: Fe1.s11.mtg8.djvu| Page 8-3 ]] 

Choice n: $$ \left \{ \alpha_1, ...., \alpha_n \right \} = \left \{ 0, 0, ...., \underbrace_{\color{Blue} \alpha_n} \right \} \ $$


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[[media: Fe1.s11.mtg8.djvu| (2)   page 8-2 ]] $$ \Rightarrow \mathbf{b}_n.\mathbf{P(v)}= 0 \ $$

$$
 *  $$ \displaystyle {\color{Red}(1)}
 * }

 Show    (2)  equivalent


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$$ \mathbf{a}_i.\mathbf{P(v)}= 0 \ \forall i = 1, ...., n \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(3)}
 * }
 * }

Scalar(inner) product
Vectors: $$ \mathbf{u}, \mathbf{v} \in R^n \ $$, $$ \left \{ \mathbf{b}_i \right \} \ $$ basis

<span id="(1)">
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$$ < \mathbf{u}, \mathbf{v} > := \mathbf{u} \bullet \mathbf{v} = \sum_i u_iv_i \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(4)}
 * }

Functions: $$ f,g: \left [ a,b  \right ] \in  R \ $$

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$$ <f, g > = \int^b_a f(x)g(x) dx \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(5)}
 * }