# Absolute Differential Calculus

• Antonio Romano
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

## Abstract

In this chapter, we address the fundamental problem of extending the differential calculus to manifolds. To understand the problem we are faced with, consider a C 1 vector field Y(t) assigned along the curve x(t) on the manifold V n . We recall that on an arbitrary manifold the components Y i (t) of Y(t) are evaluated with respect to the local natural bases of local charts (U, x i ), UV n . Consequently, when we try to define the derivative of Y along x(t), we must compare the vector $${\bf{Y}}(t + \Delta t) \in {T}_{x(t+\Delta t)}{V }_{n}$$, referred to the local basis e i (tt), with the vector Y(t)∈T x(t) V n , referred to the local basis e i (t). Since we do not know how to relate the basis e i (tt) to the basis e i (t), we are not in a position to compare the two preceding vectors; consequently, we cannot assign a meaning to the derivative of the vector field Y(t) along the curve x(t) of V n .

## Keywords

Vector Field Parallel Transport Differential Calculus Coordinate Form Local Chart

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