User:Eml5526.s11.team01.roark/Mtg07

[[media: Fe1.s11.mtg7.djvu| Mtg 7]]: Mon, 17 Jan 11 [[media: Fe1.s11.mtg7.djvu| Page 7-1]] Motivation for WRF [[media: Fe1.s11.mtg6.djvu| (p. 6-4)]]: Vectors in $\displaystyle {{\mathbb{R}}^{n}}$ |undefined



$$\displaystyle \{\begin{matrix} {{\mathbf b}_{i}}, & i=1, & ..., & n\} \\ \end{matrix}$$ basis for $$\displaystyle {{\mathbb{R}}^{n}}$$, not necessarily orthonormal
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$$\displaystyle {{\mathbf b}_{i}}\cdot {{\mathbf b}_{j}}\ne {{\delta }_{ij}}$$
 *  (1)
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$$\displaystyle \underbrace_{Kronec\ker \,\delta }=\left\{ \begin{matrix} 1 \\   0  \\ \end{matrix} \right.\begin{matrix} for\,i=j \\ for\,i\ne j \\ \end{matrix}$$ Q Find $$\displaystyle \{{{v}_{i}}\}$$ such that
 *  (2)
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$$\displaystyle \mathbf V=\sum\limits_{i=1}^{n}$$ Equation three yields
 *  (3)
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$$\displaystyle \sum\limits_{j=1}^{n}{\underbrace_{known}}\underbrace_{Unk}=\underbrace{\mathbf{v}}_{known}$$
 *  (4)

Successively multiply (4) by $$\displaystyle {{\mathbf b}_{i}},i=1,...,n$$ => n equations for n unknowns $$\displaystyle \{{{v}_{j}}\}$$ [[media: Fe1.s11.mtg7.djvu| Page 7-2]] Equation [[media: Fe1.s11.mtg7.djvu|(4) p. 7-4]] yields
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$$\displaystyle \mathbf P(\mathbf v):=\sum\limits_{j=1}^{n}{{{\mathbf b}_{j}}{{v}_{j}}-\mathbf v=\mathbf 0}$$
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$$\displaystyle {{\mathbf b}_{i}}\cdot \mathbf P(\mathbf v)=0,\,\,i=1,...,n$$
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$$\displaystyle \underbrace{{{\mathbf{b}}_{i}}\cdot \sum\limits_{j=1}^{n}}_{\sum\limits_{j=1}^{n}{\underbrace{\left( {{\mathbf{b}}_{i}}{{\mathbf{b}}_{j}} \right)}_{{v}_{j}}}}=\underbrace{{{\mathbf{b}}_{i}}\cdot v}_$$
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$$\displaystyle \underbrace_{\mathbf K}\underbrace_{\mathbf d}=\underbrace_{\mathbf F}$$
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Where $$\displaystyle \mathbf{K}$$ is the “stiffness” matrix = Gram matrix $$\displaystyle \mathbf \Gamma ({{\mathbf b}_{1}},...,{{\mathbf b}_{n}})$$
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$$\displaystyle \mathbf K \mathbf d=\mathbf F$$ [[media: Fe1.s11.mtg7.djvu| Page 7-3]] Example: Consider $$\displaystyle \left\{ {{\mathbf a}_{i}},i-1,...,n \right\}$$
 *  (5)
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$$\displaystyle {{\mathbf a}_{i}}\cdot {{\mathbf a}_{j}}={{\delta }_{ij}}$$ This is orthonormal basis (Cartesian)
 *  (1)
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$$\displaystyle {{\mathbf b}_{j}}={{b}_{jk}}{{\mathbf a}_{k}}$$ Where k is a repeated index, sum on “k”
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$$\displaystyle \left[ {{b}_{jk}} \right]=\left[ \begin{matrix} 1 & 1 & 1 \\   2 & -1 & 3  \\   3 & 2 & 6  \\ \end{matrix} \right]\begin{matrix} \leftarrow  \\ \leftarrow  \\ \leftarrow  \\ \end{matrix}\begin{matrix} {{\mathbf b}_{1}} \\ {{\mathbf b}_{2}} \\ {{\mathbf b}_{3}} \\ \end{matrix}$$ e.g., $$\displaystyle {{\mathbf b}_{2}}=2{{\mathbf a}_{1}}+(-1){{ \mathbf a}_{2}}+3{{\mathbf a}_{3}}$$ Consider $$\displaystyle \mathbf v=5{{\mathbf a}_{1}}-7{{\mathbf a}_{2}}-4{{\mathbf a}_{3}}$$
 *  (3)
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HW 2.2:
 * 1) Find determinant $$\displaystyle \left[ {{b}_{jk}} \right]$$
 * 2) Find $$\displaystyle \mathbf \Gamma \left( {{\mathbf b}_{1}},{{\mathbf b}_{2}},{{\mathbf b}_{2}} \right)= \mathbf K$$, det $$\displaystyle \mathbf \Gamma $$
 * 3) Find $$\displaystyle \mathbf F=\left\{ {{F}_{i}} \right\}=\left\{ {{\mathbf b}_{i}}\cdot \mathbf v \right\}$$
 * 4) Solve [[media: Fe1.s11.mtg7.djvu|(5) p. 7-3]] for $$\displaystyle \mathbf d=\left\{ {{v}_{j}} \right\}$$

End HW 2.2