Series expansion of a multivariate function

I am new to Mathematica. I would like to understand if this output of $$\text{Series}\left[\left\{u v,\frac{u^2}{2}+w^2,\log \left(\frac{1}{u^2+1}\right)\right\},\{u,0,1\},\{v,0,1\},\{w,0,1\}\right]$$ is correct, because "by hand" I found that the jacobian matrix is $$J=\left( \begin{array}{ccc} v & u & 0 \\ u & 0 & 2 w \\ -\frac{2 u}{u^2+1} & 0 & 0 \\ \end{array} \right)$$ and $$\text{Normal}\left[\text{Series}\left[\left\{u v,\frac{u^2}{2}+w^2,\log \left(\frac{1}{u^2+1}\right)\right\},\{u,0,1\},\{v,0,1\},\{w,0,1\}\right]\right]$$ We get as output $$\{u v,0,0\}$$ but it should be $$\{0,0,0\}$$. Thanks.

\[Delta][u_, v_, w_] := {u*v, 1/2 u^2 + w^2,
Log[1/(u^2 + 1)]}

Normal[
Series[{u*v, 1/2 u^2 + w^2, Log[1/(u^2 + 1)]}, {u, 0, 1}, {v, 0,
1}, {w, 0, 1}]]

Out[124]= {u v, 0, 0}

• Please directly post your Mathematica code instead of the LaTeX formula. Dec 16 '20 at 8:39

I think Series is not suitable for multiple vector value function. So I recommend to create the Taylor expand by hand.

δ[u_, v_, w_] := {u*v, 1/2 u^2 + w^2, Log[1/(u^2 + 1)]}
D[δ[u, v, w], {{u, v, w}, 1}]
D[δ[u, v, w], {{u, v, w}, 2}]


Mathematica must know the "smallness" of u,v,w !

Assuming same order try

Normal[Series[{u*v, 1/2 u^2 + w^2, Log[1/(u^2 + 1)]} /. {u -> eps u, v -> eps v, w -> eps w}
, {eps, 0, 2}]] /. eps -> 1
(*{u v, u^2/2 + w^2, -u^2}*)