in which db/dt denotes the time derivative of b, and (e1, e2, e3) are the three orthogonal unit vectors associated with reference frame E. Parameters b1, b2, and b3 are the projections of the vector b on the unit vectors of the reference frame E.
In this definition, the time derivative of a vector depends on the coordinate system in which it is taken. To illustrate this point, let us visualize a youngster drawing with red ink a line of 5 cm on his abdomen along the axis of his body. Let us denote by b the vector joining the ends of this red line. Suppose that the youth proceeds to perform somersaults. The time derivative of b will be equal to zero with respect to a coordinate system embedded on the trunk of the youth. In that coordinate system, b is not a function of time. On the other hand, when expressed in terms of unit vectors fixed on earth, b is a function of time and, therefore, its time derivative is not equal to zero. The term acceleration in Newton's second law is the acceleration with respect to a Cartesian coordinate system fixed on earth.
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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.