Finding Volume Traversed by Rt from a to b

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In summary, the conversation discusses the use of continuous functions to define a set of points, R_t, where the dot product of those points with a given function, g, is equal to the dot product of the same points with a set of other functions. This set of points is defined by a set of inequalities involving the coordinates of the points and the functions.
  • #1
bomba923
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(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:
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  • #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?)
 
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  • #3



Wow, that is quite a complex problem! It seems like you are dealing with multiple continuous functions and finding the volume traversed by their corresponding points. It's great to see that you are still actively working on math problems after three months.

From what I understand, Rt is essentially a collection of closed triangular regions defined by the vertices of the functions f1 to fn at any given t in the interval [a, b]. And you are trying to find the volume of this entire collection.

To find the net volume traversed by Rt from t=a to t=b, it seems like you would need to determine the overlapping regions between the different triangular regions. This is where the conditions of p and q being different and the intersection of their boundaries come into play.

Without actually seeing the specific functions and points involved, it's hard for me to give a concrete answer. But it seems like you have set up a mathematical framework to tackle this problem. Keep up the good work and I hope you are able to find a solution to this problem!
 

Related to Finding Volume Traversed by Rt from a to b

1. What is volume traversed by Rt from a to b?

The volume traversed by Rt from a to b is the amount of space that is enclosed by the graph of the function Rt and the x-axis between the points a and b. It represents the total amount of space that is covered by the function as it moves along the x-axis from a to b.

2. How is the volume traversed by Rt from a to b calculated?

The volume traversed by Rt from a to b can be calculated using the definite integral of the function Rt over the interval [a, b]. This integral represents the area under the curve of the function between the points a and b, which can be interpreted as the volume of the solid formed by the graph of the function and the x-axis.

3. What is the significance of finding volume traversed by Rt from a to b?

Finding the volume traversed by Rt from a to b is important in many scientific and mathematical applications, such as calculating the displacement of an object over a certain time period or determining the total amount of fluid that flows through a pipe. It also helps in understanding the behavior of functions and their relationship to space and time.

4. Can the volume traversed by Rt from a to b be negative?

Yes, the volume traversed by Rt from a to b can be negative. This can occur when the function Rt dips below the x-axis between the points a and b, resulting in a negative area under the curve. It is important to consider the direction and sign of the function when interpreting the volume traversed in these cases.

5. Are there any limitations to finding volume traversed by Rt from a to b?

One limitation to finding the volume traversed by Rt from a to b is that it assumes the function Rt is continuous over the interval [a, b]. If the function has discontinuities or breaks in its graph, the volume calculated may not accurately represent the actual amount of space covered by the function. Additionally, the accuracy of the calculation may be affected by the precision of the measuring tools and techniques used to determine the volume.

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