deletions | additions       diff --git a/vec_x_t_Delta_t__1.tex b/vec_x_t_Delta_t__1.tex  index 8a009e2..eb667d0 100644  --- a/vec_x_t_Delta_t__1.tex  +++ b/vec_x_t_Delta_t__1.tex 
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$$\vec{x}(t) = \vec{x}(t-\Delta t) + \vec{v}(t -\Delta t) \Delta t + \frac{1}{2} \vec{a}(t-\Delta t) \Delta t^2 + \frac{1}{6} \vec{b}(t-\Delta t) \Delta t^3 + O (\Delta t^4) $$  $$\vec{v}(t) = \vec{v}(t-\Delta t) + \frac{1}{2}(\vec{a}(t-\Delta t) + \vec{a}(t)) \Delta t + \frac{1}{4}(\vec{b}(t-\Delta t) - \vec{b}(t)) \Delta t^2 + O(\Delta t^3)$$  Substituiting  $$\vec{x}(t + \Delta t) = \vec{x}(t) + (\vec{v}(t-\Delta t) + \frac{1}{2}(\vec{a}(t-\Delta t) + \vec{a}(t)) \Delta t + \frac{1}{4}(\vec{b}(t-\Delta t) - \vec{b}(t)) \Delta t^2 + O(\Delta t^3)) \Delta t + \frac{1}{2} \vec{a}(t) \Delta t^2 + \frac{1}{6} \vec{b}(t) \Delta t^3 + O (\Delta t^4) $$