information:highlighting, quality:corruption
Translation to vector notation

The calculation for α = 2, 3 (force components in the y and z directions) yields similar results, so collecting the 3 equations into one: d p d τ = q γ (E + v × B), {\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} \tau }}=q\gamma \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right),} and since differentials in coordinate time dt and proper time dτ are related by the Lorentz factor, d t = γ (v) d τ, {\displaystyle dt=\gamma (v)\,d\tau,} so we arrive at d p d t = q (E + v × B). {\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right).}