User:Egm6321.f10.team4.Yoon/Mtg4

= Mtg 4: tue, 31 Aug 10 =

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Ex: $$\displaystyle F(\cdot) = \frac{d(\cdot)}{dx}$$

$$\displaystyle u(x) \ _\text{and} \ v(x) = $$ any 2 functions of x, $$\displaystyle \ \ \ \forall \ \alpha, \beta \ \ \in \ \ \mathbb{R} $$

Hereafter, $$\displaystyle \ \ \forall$$represents "for any", $$\displaystyle \in $$ represents "belongs to", $$\displaystyle \mathbb{R} $$ represents "set of real numbers"

Ref: Eqs.[[media: 2010_08_26_14_53_13.djvu |(1)p.3-2]]: King et al. (hereafter "K.") 2003 p.3 Eqs.[[media: 2010_08_26_14_53_13.djvu |(1)p.3-4]]: K. 2003 p3 Eq.(1.1) End Ref

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Ex: Matrix algebra

NOTE: Underbar designates bold face 'mathbf'

where $$\displaystyle \mathbf{x} \in \mathbb{R}^m \equiv \mathbb R ^{m \times 1}$$, $$\displaystyle \mathbf{y} \in \mathbb{R}^n \equiv \mathbb R ^{n \times 1}$$, and $$\displaystyle \mathbf A \in \mathbb{R}^{n \times m} $$ = set of n(rows) $$\displaystyle \times $$ m(columns) matrices with real coefficients.

End Example

Nonlinear = not linear

Ex:

Eqs.[[media: 2010_08_26_13_57_13.djvu | (3),p.2-1]] First term $$\displaystyle{{c}_{3}}({{Y}^{1}},t) \ddot Y^1$$ is a nonlinear 2nd-order with respect to $$\displaystyle Y^1 $$, thus making [[media: 2010_08_26_13_57_13.djvu | (3),p.2-1]] an N2-ODE.

End Example

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where the curvature is $$\displaystyle \chi = \frac{1}{R}$$, and $$[\chi]=\frac{1}{L}= [u^2,_{SS}] $$.

Back to general L2-ODE-VC Eq.[[media: 2010_08_26_14_53_13.djvu | (1),p.3-2]] $$\displaystyle \forall \ x$$ such that $$\displaystyle P(x) \neq 0$$, then

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Singular point $$\displaystyle\hat{x} \Rightarrow P(\hat{x})=0$$