some questions related to virtual displacement

Question

Could anyone tell me if the virtual displacement between two trajectories is $\delta x$ then

(1) how the squared distance between them is $\delta x^{\top}\delta x$?

(2)And why, $\frac{d}{d t}\left(\delta \mathbf{x}^{T} \delta \mathbf{x}\right)=2 \delta \mathbf{x}^{T} \delta \dot{\mathbf{x}}=2 \delta \mathbf{x}^{T} \frac{\partial \mathbf{f}}{\partial \mathbf{x}} \delta \mathbf{x}$?

(3) Could anyone explain to me how equation (3) comes from?

Reference, page 3: http://web.mit.edu/nsl/www/preprints/contraction.pdf

Thanks!

Answer 1

(1) For any vector $v$: $v^{\top} v = \langle v, v \rangle = \Vert v \Vert^2$.

(2) For any continuous symmetric bilinear form $B$: $\frac{d}{dt} B(v(t),v(t)) = 2 B(v(t),v^\prime(t))$. The result is applied to the inner product here.

(3) This is Grönwall's inequality.