# 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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