Let P be a vector and let the two Cartesian reference frames, E and B, be defined by their respective unit vectors (e1, e2, e3) and (b1, b2, b3). Then P can be expressed using the unit vectors of E or B:
P = BP1 b1+ BP2 b2 + BP3 b3 The time derivatives of P in E and in B are defined as
EdP/di = (dEP1/di) e1 + (dEP2/di) e2 + (dEP3/di) e3 BdP/di = (dBP1/di) b1 + (dBP2/di) b2 + (dBP3/di) b3
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The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.