Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This can be foound in N. L. White - Matroid Applications page 2.

Unidimensional case

Author considers a set of points $P = \{ p_i | i \in \mathbb{N} \}$. In this case points are to be considered on the real line, so it is really simple, no components, just scalars.

Author considers that two points, arbitrarily chosen in the set, say $p_i,p_j \in P$. Then he wants to evaluate their distance $|| p_i - p_j ||$. No problems till now. All the calculations are done in order to find a good relation and see that, if those points are connected using a rigid bar and they move with a certain speed then their distance as they move must not change. This is possible using derivates.

He also says that the distance and its square must not change. Makes sense of course.

Than he writes the following.

$\frac{d}{dt}[p_i(t)-p_j(t)]^2 = [p_i(t)-p_j(t)][p'_i(t)-p'_j(t)] = 0$

What's this, can you explain?

share|cite|improve this question
If the square of the distance is constant, then the derivative of the square of the distance is zero. Then the author uses the product rule incorrectly; it should be $2 (p_i(t) - p_j(t))(p_i'(t) - p_j'(t)) = 0$. – Qiaochu Yuan Feb 24 '12 at 1:28
ah ok just omitting... of course, sorry I am so specific and strict sometimes... thanks Yuan... post yuor answer I`ll set it as the correct one :) – Andry Feb 24 '12 at 1:35

Note that $(p_{i}(t)-p_{j}(t))^2$ is constant in relation to time then it's derivative is zero, that is,


Or evaluating the derivative with the chain rule


What is the same ( in the sense that one equation holds if and only if the other does) that


Note: Perhaps in the book there is a typing mistake in the first equality but what is important is that $(p_{i}(t)-p_{j}(t))(p_{i}'(t)-p_{j}'(t))=0$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.