Time Derivative of Vector and Mathematical Objects in Tensor Calculus and its Application to Newtonian Equation of Motion

Abstract

We represent a vector as the contravariant components with respect to covariant basis and we derivative it with respect to time. Now if we put it in the left side of Newtonian equation of motion for acceleration (time derivative of velocity), we get equation of motion in tensorial form. Now finding the acceleration components in spherical polar coordinate system and get the contravariants components of acceleration and changing it to the physical components. Again if we consider two different coordinate system and using the covariant basis formula we get it tensorial rank does not change this type of differentiation.