Parametrization Watch

gavak
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#1
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I know it sounds stupid but how can i parametrize a square?:eek:
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davros
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for what purpose, exactly? some sort of double-integral over a region? a contour integral? painting your bedroom????

tell us what's on your mind!
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gavak
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Ehehe, I need it for a line integral..... (So i need the position vector r for the points of the square in some sort of parametrization)..... Thanks!!
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Zhen Lin
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Hmm, I think it would have to be a piecewise parametrisation...
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DFranklin
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Yes. In practical terms, it's simplest to split the integral into 4 pieces, with each piece being 1 side.
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Jake22
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Well here is an example for a parameterization of a square in the plane with corners (0,0), (0,1), (1,1) and (1,0).

f(t):[0,4)\rightarrow \mathbb{R}^2 given by

f(t) = \begin{cases}

                  (0,t), \quad \qquad \; \; \; 0 \leq t \leq 1 \\

                  (t-1,1),  \qquad 1 < t \leq 2 \\

                  (1,3-t),\qquad 2 < t \leq 3 \\

                  (4-t,0),\qquad 3 < t < 4 

                 \end{cases}
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DFranklin
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(Original post by Jake22)
Well here is an example for a parameterization of a square in the plane with corners (0,0), (0,1), (1,1) and (1,0).

f(t):[0,4)\rightarrow \mathbb{R}^2 given by

f(t) = \begin{cases}

                  (0,t), \quad \qquad \; \; \; 0 \leq t \leq 1 \\

                  (t-1,1),  \qquad 1 < t \leq 2 \\

                  (1,3-t),\qquad 2 < t \leq 3 \\

                  (4-t,0),\qquad 3 < t < 4 

                 \end{cases}
But when it comes down to doing the integral, you're pretty certain to end up writing

I = \int_0^1 f(t) ... \,dt + \int_1^2 f(t)... \, dt + \int_2^3 f(t) ... \, dt + \int_3^4 f(t) ... \,dt.

In which case you might as well define

f(t) = (0, t), g(t) = (t,1), h(t)=(1,1-t), k(t) = (1-t, 0) and

I = \int_0^1 f(t) ... \,dt + \int_0^1 g(t) ... \,dt + \int_0^1 h(t) ... \,dt + \int_0^1 k(t) ... \,dt

The definitions are simpler, and you're more likely to be able to spot any useful symmetries in the integrals.
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Jake22
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Yeah, I wasn't thinking of using it for anything, just wanted to give a quick example of a parameterization. If I were to integrate over the curve, I would just use the linear substitutions from my parameterization after splitting it into the sum of four integrals as you have outlined.
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