# Derivatives

• Gerd Baumann

## Abstract

The symmetry analysis of differential equations is based on several differential operators. Among these operators are the ordinary differentiation, the total differential, the Fréchet derivative, the Euler-Lagrange derivative, and the prolongation, to name the main operators. The basis of the symmetry analysis is the prolongation of a differential equation. Unfortunately, the prolongation as a differential operator is not implemented in Mathematica. This chapter will discuss the different types of derivatives used in the calculus of symmetry analysis and will demonstrate their application by several examples. Another subject of the present chapter is the presentation of the theoretical background for the derivatives. One point we will discuss is the connection of the theory with the practical implementation of these operators in Mathematica. Application of the defined operators to several examples will demonstrate their use. Throughout the text, we use subscripts to denote a differentiation. The subscripted representation in Mathematica is created by the function LieTraditionalForm[]. This MathLie function converts the standard form of differentials in Mathematica to a traditional form frequently used in mathematics.

## Keywords

Vector Field Euler Equation Tangent Vector Total Derivative Symmetry Analysis
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