a finite number of vector fields.

Then, here we are going to show how to approximate a given family of diffeomorphisms $\Phi$, up to order $\lambda^k\Omega^{k'}$ with $k+k'=4$, in terms of some differential operators that we will show later can be identified with Lie derivative operators with respect to some vector field. In other words, we are looking for an expansion of the following type: \^\_[,]{}T & & = \_

[c]{}k,k’=0\
k+k’=4

\^4 \^k\^[k’]{}()\^\_[k,k’]{}T. \[mainidea\] We have performed this expansion term by term. For the sake of brevity we present here only the most relevant developments and results. We start by introducing the set of differential operators that we have used to build such an expansion by giving their actions on a general tensorial quantity $T$:[^3] \_[(1,0)]{}T & := & \_ , \[operator10\]\
\_[(0,1)]{}T & := & \_ ,\
\_[(2,0)]{}T & := & \_[ ]{}-\^2\_[(1,0)]{}T,\
\_[(1,1)]{}T & := & \_ -(\_0\_[(1,0)]{}\_[(0,1)]{}+ \_1\_[(0,1)]{}\_[(1,0)]{} )T\[L11\],\
\_[(0,2)]{}T & := & \_[ ]{}-\^2\_[(0,1)]{}T,\
\_[(3,0)]{}T & := & \_[ ]{}-3\_[(1,0)]{}\_[(2,0)]{}T-\^3\_[(1,0)]{}T,\
\_[(2,1)]{}T & := & \_\
& & -2\_[(1,0)]{}\_[[(1,1)]{}]{}T-\_[[(0,1)]{}]{}\_[[(2,0)]{}]{}T- 2\_2\_[[(1,0)]{}]{}\_[[(0,1)]{}]{}\_[[(1,0)]{}]{}T\
& & -(\_1-\_2)\_[[(0,1)]{}]{}\^2\_[[(1,0)]{}]{}T -(\_0-\_2)\^2\_[[(1,0)]{}]{}\_[[(0,1)]{}]{}T \[L21\],\
\_[(1,2)]{}T & := & \_\
& & -2\_[[(0,1)]{}]{}\_[[(1,1)]{}]{}T-\_[[(1,0)]{}]{}\_[[(0,2)]{}]{}T- 2\_3\_[[(0,1)]{}]{}\_[[(1,0)]{}]{}\_[[(0,1)]{}]{}T\
& & -(\_0-\_3)\_[[(1,0)]{}]{}\^2\_[[(0,1)]{}]{}T -(\_1-\_3)\^2\_[[(0,1)]{}]{}\_[[(1,0)]{}]{}T \[L12\],\
\_[(0,3)]{}T & := & \_[ ]{}-3\_[(0,1)]{}\_[(0,2)]{}T-\^3\_[(0,1)]{}T,\
\_[(4,0)]{}T 