Rectifiable curve
A rectifiable curve is a curve which has a well-defined finite length. Rectifiable curves are mainly important in complex analysis because they are needed to define the path integral.Suppose γ : [a, b] -> C is a continuous function from an interval into the complex plane. This curve γ is called rectifiable if the following supremum is finite:
- sup {∑i=1n |γ(ti)-γ(ti-1)| : n in N and a≤t0<t1<...<tn≤b}
In an analogous manner (by replacing the absolute value with the Euclidean distance or a norm), one can define rectifiable curves γ : [a, b] -> Rn and, more generally, γ : [a, b] -> V where V is a normed vector space.
Every continuous and piecewise continuously differentiable[?] curve γ is rectifiable, and its length can be computed as the ordinary Riemann integral
- L = ∫ab |γ'(t)| dt
I have kept our early copy-books,
landmarks of nursery life.html">life.html">life. I am/am.html">am/am.html">am/am.html">am painfully punctual, tidy and
that they involve. I am a little stern and severe except with
of my faults are physical. If I could have chosen my own life--
better and might have been less overwrought and disturbable. But
once said to me, which is better than being on a pirate-ship and
morals, talents, defects, temptations, and appearance as
If I had to confess and expose one opinon of myself which might
my power.html">power of love.html">love.html">love coupled with my power of criticism, but what I
moderation, self.html">self-control and the authority that comes from a
or too stationary to acquire these.
MARGOT ASQUITH.
"P.S. This is my second attempt to write about myself and I am not
of the two--it is more external--but, after all, what can one say
what one's friends think one is? Just now I am within a few weeks
inclined to sum up my life in this way:
"'An unfettered childhood and triumphant youth; a lot of love-
man and great happiness; the love of children and seventh heaven;
this diary and a better description of an improved self."
THE END OF BOOK TWO
End of the Project Gutenberg Etext of Margot Asquith, An Autobiography: Volumes I.
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