Integral where upper limit is a dot - meaning?

Question

Here's what I'm looking at (from this book, page 14):

I get that the dot on top of $B$ is the time derivative - but what's that dot near the top of the integral?

Answer 1

The notation

$$\int \limits _0 ^\cdot f(t) \ \Bbb d t$$

is a short-hand for the map

$$T \mapsto \int \limits _0 ^T f(t) \ \Bbb d t .$$

Concretely, you should interpret your formula as defining a map

$$(B,T) \mapsto \int \limits _0 ^T B^i (t) \dot {B^j} (t) \ \Bbb d t .$$