- #1

bomba923

- 763

- 0

(Wow...it's been over three months since I posted anything...)

Anyhow,

Given continuous functions

[tex]\begin{gathered}

f_1 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\

\vdots \hfill \\

f_n :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\

\end{gathered} [/tex]

for which

[tex]\exists g:\left[ {a,b} \right] \to \mathbb{R}^3 {\text{ such that }}f_1 \cdot g = \cdots = f_n \cdot g[/tex]

define [itex]\forall t \in \left[ {a,b} \right][/itex]

[tex]R_t = \left\{ {\left( {x,y,z} \right)\left| {\left( {x,y,z} \right) \cdot g = f_1 \cdot g} \right.} \right\} \cap [/tex]

[tex]\bigcup\limits_{\begin{subarray}{l}

j < k < m \leqslant n, \\

\left( {j,k,m} \right) \in \mathbb{N}^3

\end{subarray}} {\left\{ {\left( {x,y,z} \right)\left| \begin{gathered}

\left( {y - y_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {y_j - y_k } \\

{x_j - x_m } & {y_j - y_m } \\

\end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {y_j - y_k } \\

{x_j - x_m } & {y_j - y_m } \\

\end{array} } \right| \wedge \hfill \\

\left( {y - y_j } \right)\left( {x_j - x_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {y_j - y_m } \\

{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {y_j - y_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {y_j - y_m } \\

{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \wedge \hfill \\

\left( {y - y_k } \right)\left( {x_k - x_m } \right)\left| {\begin{array}{*{20}c}

{x_k - x_m } & {y_k - y_m } \\

{x_k - x_j } & {y_k - y_j } \\

\end{array} } \right| \leqslant \left( {x - x_k } \right)\left( {y_k - y_m } \right)\left| {\begin{array}{*{20}c}

{x_k - x_m } & {y_k - y_m } \\

{x_k - x_j } & {y_k - y_j } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {z_j - z_k } \\

{x_j - x_m } & {z_j - z_m } \\

\end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {z_j - z_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {z_j - z_k } \\

{x_j - x_m } & {z_j - z_m } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_j } \right)\left( {x_j - x_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {z_j - z_m } \\

{x_j - x_k } & {z_j - z_k } \\

\end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {z_j - z_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {z_j - z_m } \\

{x_j - x_k } & {z_j - z_k } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_k } \right)\left( {x_k - x_m } \right)\left| {\begin{array}{*{20}c}

{x_k - x_m } & {z_k - z_m } \\

{x_k - x_j } & {z_k - z_j } \\

\end{array} } \right| \leqslant \left( {x - x_k } \right)\left( {z_k - z_m } \right)\left| {\begin{array}{*{20}c}

{x_k - x_m } & {z_k - z_m } \\

{x_k - x_j } & {z_k - z_j } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c}

{y_j - y_k } & {z_j - z_k } \\

{y_j - y_m } & {z_j - z_m } \\

\end{array} } \right| \leqslant \left( {y - y_j } \right)\left( {z_j - z_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {z_j - z_k } \\

{x_j - x_m } & {z_j - z_m } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_j } \right)\left( {y_j - y_m } \right)\left| {\begin{array}{*{20}c}

{y_j - y_m } & {z_j - z_m } \\

{y_j - y_k } & {z_j - z_k } \\

\end{array} } \right| \leqslant \left( {y - y_j } \right)\left( {z_j - z_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {z_j - z_m } \\

{x_j - x_k } & {z_j - z_k } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_k } \right)\left( {y_k - y_m } \right)\left| {\begin{array}{*{20}c}

{y_k - y_m } & {z_k - z_m } \\

{y_k - y_j } & {z_k - z_j } \\

\end{array} } \right| \leqslant \left( {y - y_k } \right)\left( {z_k - z_m } \right)\left| {\begin{array}{*{20}c}

{y_k - y_m } & {z_k - z_m } \\

{y_k - y_j } & {z_k - z_j } \\

\end{array} } \right| \hfill \\

\end{gathered} \right.} \right\}} [/tex]

where [tex]\forall i > 0,\;\left( {x_i ,y_i ,z_i } \right) = f_i \left( t \right) [/tex]

Find the

[tex] \exists p,q \in \left( {a,b} \right):\left( {R_p - \partial R_p } \right) \cap \left( {R_q - \partial R_q } \right) \ne \emptyset \, \wedge \, p \ne q [/tex]

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*

Anyhow,

Given continuous functions

[tex]\begin{gathered}

f_1 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\

\vdots \hfill \\

f_n :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\

\end{gathered} [/tex]

for which

[tex]\exists g:\left[ {a,b} \right] \to \mathbb{R}^3 {\text{ such that }}f_1 \cdot g = \cdots = f_n \cdot g[/tex]

define [itex]\forall t \in \left[ {a,b} \right][/itex]

[tex]R_t = \left\{ {\left( {x,y,z} \right)\left| {\left( {x,y,z} \right) \cdot g = f_1 \cdot g} \right.} \right\} \cap [/tex]

[tex]\bigcup\limits_{\begin{subarray}{l}

j < k < m \leqslant n, \\

\left( {j,k,m} \right) \in \mathbb{N}^3

\end{subarray}} {\left\{ {\left( {x,y,z} \right)\left| \begin{gathered}

\left( {y - y_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {y_j - y_k } \\

{x_j - x_m } & {y_j - y_m } \\

\end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {y_j - y_k } \\

{x_j - x_m } & {y_j - y_m } \\

\end{array} } \right| \wedge \hfill \\

\left( {y - y_j } \right)\left( {x_j - x_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {y_j - y_m } \\

{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {y_j - y_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {y_j - y_m } \\

{x_j - x_k } & {y_j - y_k } \\

\end{array} } \right| \wedge \hfill \\

\left( {y - y_k } \right)\left( {x_k - x_m } \right)\left| {\begin{array}{*{20}c}

{x_k - x_m } & {y_k - y_m } \\

{x_k - x_j } & {y_k - y_j } \\

\end{array} } \right| \leqslant \left( {x - x_k } \right)\left( {y_k - y_m } \right)\left| {\begin{array}{*{20}c}

{x_k - x_m } & {y_k - y_m } \\

{x_k - x_j } & {y_k - y_j } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {z_j - z_k } \\

{x_j - x_m } & {z_j - z_m } \\

\end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {z_j - z_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {z_j - z_k } \\

{x_j - x_m } & {z_j - z_m } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_j } \right)\left( {x_j - x_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {z_j - z_m } \\

{x_j - x_k } & {z_j - z_k } \\

\end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {z_j - z_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {z_j - z_m } \\

{x_j - x_k } & {z_j - z_k } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_k } \right)\left( {x_k - x_m } \right)\left| {\begin{array}{*{20}c}

{x_k - x_m } & {z_k - z_m } \\

{x_k - x_j } & {z_k - z_j } \\

\end{array} } \right| \leqslant \left( {x - x_k } \right)\left( {z_k - z_m } \right)\left| {\begin{array}{*{20}c}

{x_k - x_m } & {z_k - z_m } \\

{x_k - x_j } & {z_k - z_j } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c}

{y_j - y_k } & {z_j - z_k } \\

{y_j - y_m } & {z_j - z_m } \\

\end{array} } \right| \leqslant \left( {y - y_j } \right)\left( {z_j - z_k } \right)\left| {\begin{array}{*{20}c}

{x_j - x_k } & {z_j - z_k } \\

{x_j - x_m } & {z_j - z_m } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_j } \right)\left( {y_j - y_m } \right)\left| {\begin{array}{*{20}c}

{y_j - y_m } & {z_j - z_m } \\

{y_j - y_k } & {z_j - z_k } \\

\end{array} } \right| \leqslant \left( {y - y_j } \right)\left( {z_j - z_m } \right)\left| {\begin{array}{*{20}c}

{x_j - x_m } & {z_j - z_m } \\

{x_j - x_k } & {z_j - z_k } \\

\end{array} } \right| \wedge \hfill \\

\left( {z - z_k } \right)\left( {y_k - y_m } \right)\left| {\begin{array}{*{20}c}

{y_k - y_m } & {z_k - z_m } \\

{y_k - y_j } & {z_k - z_j } \\

\end{array} } \right| \leqslant \left( {y - y_k } \right)\left( {z_k - z_m } \right)\left| {\begin{array}{*{20}c}

{y_k - y_m } & {z_k - z_m } \\

{y_k - y_j } & {z_k - z_j } \\

\end{array} } \right| \hfill \\

\end{gathered} \right.} \right\}} [/tex]

where [tex]\forall i > 0,\;\left( {x_i ,y_i ,z_i } \right) = f_i \left( t \right) [/tex]

Find the

*net*volume traversed by**R**from [itex]t=a[/itex] to [itex]t=b[/itex] if_{t}[tex] \exists p,q \in \left( {a,b} \right):\left( {R_p - \partial R_p } \right) \cap \left( {R_q - \partial R_q } \right) \ne \emptyset \, \wedge \, p \ne q [/tex]

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*

**Edit**: it may be simpler to describe**R**in words:_{t}**R**is the "union of all closed triangular regions defined by vertices f_{t}_{j},f_{k},f_{m}for all combinations of j,k,m at any [itex]t \in \left[ {a,b} \right] [/itex]."
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