6.5. Parameter derivatives

Given a parameter defined with DefParameter we can take parametric derivatives of expressions which depend on that parameter, using ParamD. For historical reasons xTensor` has a generic parametric derivative called OverDot such that every field is considered to depend on the corresponding (undefined) parameter.

ParamD                Parametric derivative with respect to a defined parameter
OverDot                Generic parametric derivative with respect to an unspecified parameter

Parametric derivatives.

The generic parametric derivative OverDot (Dot is used by Mathematica as inner scalar product) is represented by a (very) small over dot:

In[489]:=

OverDot[r[]^2]

Out[489]=

2 Overscript[r_^, .] r_^

OverDot has been defined to commute with partial derivatives, but not with other covariant derivatives or with Lie derivatives:

In[490]:=

OverDot[PD[-a][r[]]]//InputForm

Out[490]//InputForm=

PD[-a][OverDot[r[]]]

In[491]:=

OverDot[Cd[-a][r[]]]//InputForm

Out[491]//InputForm=

OverDot[Cd[-a][r[]]]

All fields currently defined do not depend on external parameters, so that we define new objects:

In[492]:=

ParamD[time][r[]]

Out[492]=

0

In[493]:=

DefTensor[Q[-a], {M3, time}]

** DefTensor: Defining tensor Q[-a] .

In[494]:=

? Q

Global`Q

Dagger[Q]^=Q
DependenciesOfTensor[Q]^={time,M3}
Info[Q]^={tensor,}
PrintAs[Q]^=Q
SlotsOfTensor[Q]^={-TangentM3}
SymmetryGroupOfTensor[Q]^=StrongGenSet[{},GenSet[]]
TensorID[Q]^={}
xTensorQ[Q]^=True

Currently I have not found a good representation for the parametric derivatives, so that they remain as they are:

In[495]:=

ParamD[time][3Q[-a] v[a]]

Out[495]=

3 v_ ^a ParamD[time][Q_a^ ]

In[496]:=

ParamD[time, time][3Q[-a] v[a]]

Out[496]=

3 v_ ^a ParamD[time, time][Q_a^ ]

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