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Quick Volume

  1. Nov 27, 2006 #1
    (Wow...it's been over three months since I posted anything...:eek:)
    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 Rt from [itex]t=a[/itex] to [itex]t=b[/itex] if
    [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]

    --------------------------------------------------------------------------------------------------------
    *Edit: it may be simpler to describe Rt in words: Rt is the "union of all closed triangular regions defined by vertices fj,fk,fm for all combinations of j,k,m at any [itex]t \in \left[ {a,b} \right] [/itex]."
     
    Last edited: Nov 27, 2006
  2. jcsd
  3. Dec 25, 2006 #2
    Anybody? :redface:

    Well, let's take the simplest case (n=3), where we have three continuous functions
    [tex]\begin{gathered}
    f_1 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\
    f_2 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\
    f_3 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\
    \end{gathered} [/tex]
    and
    [tex]\forall t \in \left[ {a,b} \right] [/tex], let [tex]R_t[/tex] be the open triangular region with vertices f1(t), f2(t), and f3(t).

    To simplify matters, assume that [itex]\forall p,q \in \left( {a,b} \right),\;\left( {R_p \cap R_q \ne \emptyset } \right) \to \left( {p = q} \right)[/itex].
    Find the net volume traversed by [tex]R_t[/tex] from t=a to t=b :smile:
    ----------------------------------------------------------------------------------
    Initially, one might guess the net volume to simply the sum of the areas of Rt:
    [tex]V_{net} = \frac{1}{2}\int\limits_a^b {\left\| {\left( {f_1 \left( t \right) - f_2 \left( t \right) } \right) \times \left( {f_1 \left( t \right) - f_3 \left( t \right) } \right)} \right\|dt} [/tex]
    But, that is false!
    Consider
    [tex]\begin{gathered}
    f_1 \left( t \right) = \left( {t,0,0} \right) \hfill \\
    f_2 \left( t \right) = \left( {0,t,0} \right) \hfill \\
    f_3 \left( t \right) = \left( {0,0,t} \right) \hfill \\
    \end{gathered} [/tex]
    for
    [tex]0 \leqslant t \leqslant 1 [/tex]

    The region traversed by Rt is the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1).
    Its volume is simply
    [tex]\int\limits_0^1 {\int\limits_0^{1 - x} {\left( {1 - x - y} \right)dy} dx} = \frac{1}{6} [/tex]
    However,
    [tex]\frac{1}{2}\int\limits_0^1 {\left\| {\left( {f_1 \left( t \right) - f_2 \left( t \right)} \right) \times \left( {f_1 \left( t \right) - f_3 \left( t \right)} \right)} \right\|dt} = \frac{{\sqrt 3 }}{6} \ne \frac{1}{6}[/tex]
    -----------------------------------------------------------------------------------
    So, given three continuous functions
    [tex]\begin{gathered}
    f_1 :\left[ {a,b} \right] \subset \mathbb{R} \to \mathbb{R}^3 \hfill \\
    f_2 :\left[ {a,b} \right] \subset \mathbb{R} \to \mathbb{R}^3 \hfill \\
    f_3 :\left[ {a,b} \right] \subset \mathbb{R} \to \mathbb{R}^3 \hfill \\
    \end{gathered} [/tex]

    How can I calculate the net volume traversed by Rt?
    (That is, is there a general formula by which one can calculate this volume?)
     
    Last edited: Dec 25, 2006
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