# Taylor expand around a vector

I have a matrix $$A(\vec{v})$$ whose entries are scalar functions of a vector $$\vec{v}=(v_{x},v_{y})$$. To be concrete I have something like

$$A=\begin{pmatrix}f_{1}(\vec{v})& f_{2}(\vec{v})\\f_{3}(\vec{v}) &f_{4}(\vec{v}) \end{pmatrix}$$

My goal is to Taylor expand the matrix $$A$$ around the point $$\vec{v_{0}}$$ at first oder in $$|\vec{v}-\vec{v_{0}}|$$.

I do that with:

Normal@Series[A[v], {vx ,v0x, 1}, {vy, v0y, 1}]//MatrixForm


The matrix I get is now a function of $$\vec{v}-\vec{v_{0}}$$ but I want to transform it in a matrix in $$\vec{w}=\vec{v}-\vec{v_{0}}$$. I have tried with

A[v]=A[v-v0]


but it doesn't work. I guess my question is how to change the variable of a function.

Also doing the expansion I don't get quadratic terms in $$v_{x}$$ $$v_{y}$$ (because it's first oder), but I get mixed terms like $$v_{x}v_{y}$$. Is there a way to get rid of those?

• I think you want to adapt the accepted answer here. Apr 6, 2022 at 3:19

Normal@Series[A[{vx, vy}], {vx, v0x, 1}, {vy, v0y, 1}]/. Derivative[{1,1}][A] ->0

 Normal@Series[A[{vx, vy}], {vx, v0x, 1}, {vy, v0y, 1}]/. Derivative[{1,1}][A] ->0 /. {vx - v0x -> wx, vy - v0y -> wy}