deletions | additions       diff --git a/vec_a_t_vec_a__.tex b/vec_a_t_vec_a__.tex  index f012228..9fdfcc5 100644  --- a/vec_a_t_vec_a__.tex  +++ b/vec_a_t_vec_a__.tex 
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$$(\vec a(t) + \vec a(t+\Delta t))\Delta t^2= \vec x(t-\Delta t) - \vec x(t) + \vec{x}(t+2 \Delta t) - \vec x(t+\Delta t) + O(4) $$  $$ \vec{v}(t \widetilde{\vec{v}}(t  + \Delta t) = \frac{\vec{x}(t+2\Delta t) - \vec{x}(t)}{2 \Delta t}$$ $$ \vec{v}(t) \widetilde{\vec{v}}(t)  = \frac{\vec{x}(t+\Delta t) - \vec{x}(t-\Delta t)}{2 \Delta t}$$ $$\frac{1}{2}(\vec $$ \frac{1}{2}(\vec  a(t) + \vec a(t+\Delta t))\Delta t = \vec{v}(t \widetilde{\vec{v}}(t  + \Delta t) - \vec{v}(t) \widetilde{\vec{v}}(t) + O(3) $$  $$ \widetilde{\vec{v}}(t + \Delta t) = \widetilde{\vec{v}}(t) + \frac{1}{2}(\vec a(t) + \vec a(t+\Delta t))\Delta t  + O(3) $$