Recombining vectors after taking derivatives

Question

I want to map two expansions order by order and to solve to get unknowns coefficient. We define $X_{\mu}= \frac{(x-y)_{\mu}}{(x-y)^2}$. The quantity I need to expand in the end is $t_{\mu\nu}= \frac{X_{\mu}X_{\nu}}{X^2}$. as $x \rightarrow y$. The way I do this is by defining $\vec{x} = \vec{y}+\epsilon \vec{s}$ where $\epsilon$ is the parameter that is going to go to zero. In Mathematica, I coded this as

CapXp[x_, y_, z_] := ( (x - z)/((x - z).(x - z)));

And then, I can take the expansion (for the first component of the vector (x-y) as

Series[CapXp[{y1 + ϵ s1, y2 + ϵ s2, 
y3 + ϵ s3, y4 + ϵ s4}, {y1, y2, y3, y4}, {z1, 
z2, z3, z4}][[1]], {ϵ, 0, 1}]

This works sort of fine but produces a outcome where all the norms as explicit sums of squares and so on. For example, the order 0 term is

(y1 - z1)/(y1^2 + y2^2 + y3^2 + y4^2 - 2 y1 z1 + z1^2 - 2 y2 z2 + z2^2 - 2 y3 z3 + z3^2 - 2 y4 z4 + z4^2)

After going to higher order, when trying to map the two expansions, Mathematica is not able to recognize the term that are the same anymore such as $(x-y)^2$. Would you know a way to deal with that sort of explicit differentiation of vector but then recombining them in term of norm?

Answer 1

Why not work with symbolic vectors? For instance, let:

x = y - z + ϵ s;

e = x / x.x;
e //TeXForm

$(y - z + s ϵ)/(y - z + s ϵ).(y - z + s ϵ)$

Then, compute the series by hand:

e0 = e /. ϵ -> 0;
e1 = D[e, ϵ] /. ϵ -> 0;
e2 = D[e, ϵ, ϵ]/2 /. ϵ -> 0;
e0 //TeXForm
e1 //TeXForm
e2 //TeXForm

$\frac{y-z}{(y-z).(y-z)}$

$\frac{s}{(y-z).(y-z)}-\frac{(y-z) (s.(y-z)+(y-z).s)}{((y-z).(y-z))^2}$

$\frac{1}{2} \left(\frac{2 (y-z) (s.(y-z)+(y-z).s)^2}{((y-z).(y-z))^3}-\frac{2 s (s.(y-z)+(y-z).s)}{((y-z).(y-z))^2}-\frac{2 s.s (y-z)}{((y-z).(y-z))^2}\right)$

Now, comparing e1 with your version:

r1 = (e1 /. {y->{y1,y2,y3,y4}, z->{z1,z2,z3,z4}, s->{s1,s2,s3,s4}})[[1]] //FullSimplify;
r1 //TeXForm

$\frac{\operatorname{s1} \left(-(\operatorname{y1}-\operatorname{z1})^2+(\operatorname{y2}-\operatorname{z2})^2+(\operatorname{y3}-\operatorname{z3})^2+(\operatorname{y4}-\operatorname{z4})^2\right)+2 (\operatorname{y1}-\operatorname{z1}) (\operatorname{s2} (\operatorname{z2}-\operatorname{y2})-\operatorname{s3} \operatorname{y3}+\operatorname{s3} \operatorname{z3}-\operatorname{s4} \operatorname{y4}+\operatorname{s4} \operatorname{z4})}{\left((\operatorname{y1}-\operatorname{z1})^2+(\operatorname{y2}-\operatorname{z2})^2+(\operatorname{y3}-\operatorname{z3})^2+(\operatorname{y4}-\operatorname{z4})^2\right)^2}$

r2 = SeriesCoefficient[
    CapXp[{y1+ϵ s1,y2+ϵ s2,y3+ϵ s3,y4+ϵ s4},{y1,y2,y3,y4},{z1,z2,z3,z4}][[1]],
    {ϵ,0,1}
] //FullSimplify;
r2 //TeXForm

$\frac{\operatorname{s1} \left(-(\operatorname{y1}-\operatorname{z1})^2+(\operatorname{y2}-\operatorname{z2})^2+(\operatorname{y3}-\operatorname{z3})^2+(\operatorname{y4}-\operatorname{z4})^2\right)+2 (\operatorname{y1}-\operatorname{z1}) (\operatorname{s2} (\operatorname{z2}-\operatorname{y2})-\operatorname{s3} \operatorname{y3}+\operatorname{s3} \operatorname{z3}-\operatorname{s4} \operatorname{y4}+\operatorname{s4} \operatorname{z4})}{\left((\operatorname{y1}-\operatorname{z1})^2+(\operatorname{y2}-\operatorname{z2})^2+(\operatorname{y3}-\operatorname{z3})^2+(\operatorname{y4}-\operatorname{z4})^2\right)^2}$

r1 === r2

True

Note that Series doesn't work correctly with Dot, but you can see this answer for a way to get Series to work properly with Dot, in which case you can just use Series instead. Here is the workaround from the answer:

protect = Unprotect[System`Private`InternalSeries];
System`Private`InternalSeries[a_Dot, {x_, x0_, n_Integer?NonNegative}] := Module[
    {d = NestList[D[#, x]&, a, n], res},

    res = Quiet @ Check[d /. x->x0, $Failed];
		SeriesData[x, x0, TensorExpand @ res, 0, n+1, 1] /; res =!= $Failed
]
Protect @@ protect;

Then:

Series[e, {ϵ, 0, 1}] //Simplify //TeXForm

$\frac{y-z}{-y.z-z.y+y.y+z.z}+\frac{\epsilon ((y-z) (-s.y-y.s+s.z+z.s)+s (-y.z-z.y+y.y+z.z))}{(-y.z-z.y+y.y+z.z)^2}+O\left(\epsilon ^2\right)$

Update

For your revised expression you can use TensorProduct to represent $X_\mu X_\nu$. So:

e = With[{v = y - z + ϵ s}, TensorProduct[v,v]/v.v];

e/.ϵ->0
D[e,ϵ]/.ϵ->0//Simplify
D[e,ϵ,ϵ]/2/.ϵ->0//Simplify

I didn't use MathJax because it doesn't work well with \[TensorProduct].