Question

Derivative of a union.

Answer 1

The context is, I am looking for the smallest area that holds all curves of length 1.

So I label each of these length 1 curves as \(C_i\) and make each of them functions of two continuous variables \(\theta_i\) and \(t_i\) since we can rotate them and translate them around. So I call each curve (set of points). Each curve has a length and infinitesimal width, which gives them an infinitesimal area.

I'd like to now take the union of all of these infinitely many curves,

\[A = \bigcup_i C_I (t_i, \theta_i)\]

Now since we have infinitely many infinitesimally small areas, seems fair that we could get a finite area out of it.

And for this, I think we have for all the variables some minimum point,

\[\frac{\partial A}{\partial t_j} = 0\]
and
\[\frac{\partial A}{\partial \theta_j} = 0\]

I dunno though, like maybe this isn't gonna work out but it's something I came up with a few minutes ago and thought I'd share.

Answer 2

thanks for sharing

Answer 3

The Union included the states of Maine, New York, New Hampshire, Vermont, Massachusetts, Connecticut, Rhode Island, Pennsylvania, New Jersey, Ohio, Indiana, Illinois, Kansas, Michigan, Wisconsin, Minnesota, Iowa, California, Nevada, and Oregon. Abraham Lincoln was their President.

Answer 4

now find the area :P

Answer 5

What you have given us is insufficient to calculate area. We need to know how the lines are placed in space. Instead of lines, let me use points as an example. If all points were discrete/disjoint, then the length of the union would be 0. On the other hand, if all points were "next" to each other extending from minus infinity to positive infinity, then the length would be infinite. If we placed the points such that they make up the unit closed interval, then obviously the length would be one.
tl;dr area of the union of lines depends on the position of lines in space

Answer 6

Also, do you count curves which are congruent to each other being the same curve?

Answer 7

@SnuggieLad I'm glad I'm on the Confederate side.