In many cases, yes! As a start, we'll restrict ourselves for now to symbolic/exact regions and won't concern ourselves with discretizing regions for numerical integrations (though this is possible).
### Parametric Regions
We'll begin by noticing that `RegionConvert` can give us a parametric representation of many regions. For example,
```
RegionConvert[Circle[], "Parametric"]
(* Out: *)
(* ParametricRegion[{{Cos[x], Sin[x]}, 0 <= x <= 2 Pi}, {x}] *)
```
(Those `x`'s are actually `\[FormalX]`s, by the way, but that would just clutter this explanation up.)
Our strategy for parametric regions will simply be to get the tangent information of our region from that function. (It would be nice if Mathematica had built-in ways to extract tangent info—there's a lot of potential for differential geometry here.)
### Implicit Regions
We can also tackle implicit regions! This is the (only) other capability of `RegionConvert`: to convert a region to `ImplicitRegion` form.
There's a fundamental piece of missing information needed for integrating over Mathematica's regions: orientation. Currently, the sign can't be guaranteed one way or the other for parametric regions. (I'm considering changing the functionality so that you always get a positive answer. More on this issue later.) For implicit regions we face the same obstacle. The best we can do is $\int_{\vec{\gamma} |\vec{v}\cdot d\vec{\gamma}|$.
That disclaimed, here's our strategy:
* Put the defining constraint in disjunctive normal form with `BooleanConvert`.
* Separate the disjuncts from each other to be added together later. (Potential issue: overlapping conditions.)
* Look at the regions defined by each conjunctions, and keep only the ones that are 1-dimensional (not zero-dimensional; sorry, physicists with `DiracDelta`-laden vector fields.) (Possible issue: currently we check the apparent codimension given by equality count. Perhaps we should instead be turning these back into regions and using the built-in `RegionDimension`.)
* Extract the equalit(y/ies) from the conjunction; subtract one side from the other to obtain a list of $n-1$ functions which specify the curve as their mutual zero set ($n$ the embedding dimension)
* Look at the gradients of these functions and use those to "project off" components of our vector field at each point until only the part pointing along the curve remains.
* Integrate the norm of those vectors over the region.
* Add together the resulting integrals.
### Multi-segment lines
Also, I figured it would be nice to check for `Line` regions first, and handle them separately. After all, these are fairly unambiguously directed! For some reason, `RegionConvert` doesn't like converting them to parametric form—but that means we *definitely* lose all orientation info. Instead, I check for regions with head `Line` and homebrew the parametrizations. No change needs to be made to the input; the code will just react differently to `Line` regions.
### Explicit parametrization
I've also included support for the explicit, non-region parametric form used in the other nice answer to this post! Credit to @E. Chan-López for the argument pattern! Mine's a bit different, though: I demand, in that mode, an actual function to be given as a first argument, to avoid both a var and an iterator argument.
----------
The new sections are not very well commented. I'll come back and edit them when I can.
```
(*Helper function for extracting parts of ParametricRegion:*)
ParametricRegionDestructure[
ParametricRegion[{x_, cons_}, params_]] := {x, cons, params}
(* Helper functions for ImplicitRegions: *)
(* This could probably be more compact. *)
ImplicitRegionDestructure[
ImplicitRegion[cond_, params_]] := {ExtractJuncts[cond], params}
ExtractDisjuncts[HoldPattern[Or[x__]]] := {x}
ExtractDisjuncts[x : Except[_Or]] := {x}
ExtractConjuncts[HoldPattern[And[x__]]] := {x}
ExtractConjuncts[x : Except[_And]] := {x}
ExtractJuncts[cond_] :=
ExtractConjuncts /@ ExtractDisjuncts[BooleanConvert[cond]]
ApparentCodim[conjunctList_] := Count[conjunctList, _Equal]
FilterJunctsByCodim[disjunctList_, codim_] :=
Select[disjunctList, ApparentCodim[#] == codim &]
ConjunctsToGrads[conjunctList_, params_] :=
Cases[conjunctList, x_ == y_ :> Grad[x - y, params]]
ProjectOff[vf_, nv_] := Simplify[vf - nv (vf . nv/(nv . nv))]
ProjectOffAll[vf_, nvs_] := Fold[ProjectOff, vf, nvs]
(* Helper functions for lines: *)
LineDestructure[x : HoldPattern[Line[{{__} ..}, {{__} ..} ..]]] :=
Flatten[(LineDestructure@*Line /@ (List @@ x)), 1]
LineDestructure[HoldPattern[Line[x : {{__} ..}]]] :=
Partition[x, 2, 1]
LineSegmentToFormal[{x0 : {__},
x1 : {__}}] := ((1 - \[FormalT]) x0 + \[FormalT] x1)
LineSegmentToTangent[{x0 : {__}, x1 : {__}}] := x1 - x0
(*Make it hold its arguments,and make it look like Integrate on \
input:*)
SetAttributes[LineIntegrate, HoldAll]
SyntaxInformation[LineIntegrate] = SyntaxInformation[Integrate];
LineIntegrate[v0 : {__}, Element[(vars : {__}), region_?RegionQ]] :=
Module[{(*Vector field expression turned into a function:*)
v = Construct[Function, Unevaluated[vars], Unevaluated[v0]],
(*variables to hold the region:*)
regiontype, cregion, nEmbed = RegionEmbeddingDimension[region],
(*variables to hold the components of a parameterized region:*)
x, cons, params,
(*more variables to hold the components of an implicit region*)
juncts,
(* for lines: *)
lines,
(*variable to hold the tangent vector to our curve:*)
tangentVector},
(*The regiontype and cregion are set in the \
Condition (/;) guarding the module expression,
so we only ever have to compute them once.*)
Switch[regiontype,
"Line",
lines = LineDestructure[cregion];
Plus @@
Table[
With[{vf = v @@ LineSegmentToFormal[line],
dl = LineSegmentToTangent[line]},
Integrate[vf . dl, {\[FormalT], 0, 1}]], {line, lines}],
"Parametric",
{x, cons, params} = ParametricRegionDestructure[cregion];
tangentVector = D[x, params];
(*Apply our vector field to points in the region (x);
insert params into the integral syntactically*)
With[{f = v @@ x, params0 = params},
Integrate[f . tangentVector,
params0 \[Element] ImplicitRegion[cons, params0]]],
"Implicit",
{juncts, params} = ImplicitRegionDestructure[cregion];
juncts = FilterJunctsByCodim[juncts, nEmbed - 1];
With[{vf = v @@ params, params0 = params},
Plus @@ Table[
With[{projected =
ProjectOffAll[vf, ConjunctsToGrads[conjuncts, params]],
conjunction = And @@ conjuncts},
With[{integrand =
Simplify[Sqrt[projected . projected],
params0 \[Element] Reals]},
Integrate[integrand,
params0 \[Element]
ImplicitRegion[conjunction, params0]]]],
{conjuncts, juncts}]
]] /;(*Check that RegionConvert succeeded;
set variables.*)(MatchQ[
cregion = region, _Line] && (regiontype = "Line"; True)) ||
MatchQ[
cregion =
RegionConvert[region,
regiontype = "Parametric"], _ParametricRegion] ||
MatchQ[
cregion =
RegionConvert[region, regiontype = "Implicit"], _ImplicitRegion]
] /;(*Also check that our dimensions match up.*)
Length[Unevaluated[v0]] == Length[Unevaluated[vars]] ==
RegionEmbeddingDimension[region] && RegionDimension[region] == 1
(* For parametrically explicit arguments: *)
LineIntegrate[v_, r : {rs___}, iterator : {t_Symbol, _, _}] :=
Construct[Module, Unevaluated[{t}],
Unevaluated@Module[{dr = D[r, t]}, Integrate[v[rs] . dr, iterator]]]
```
This is just a start. There are a lot of issues:
* No support for the cases where we can't convert to parametric regions—which includes some basic ones (like lines joining multiple points)!
* No way to choose orientation—except for `Line` regions or explicit parametrizations, integrals are only unique up to a sign, and you may get the opposite sign than the one you bargained for without any indication of this.
* We put a *lot* of trust in formal variables being definitionless, and also we allow expressions like `v` and `f` to evaluate fully with those formal variables inside. (Not to mention the newly-exposed conditions in `ImplicitRegion`!) This could be risky, in part because they *might* have a definition, but mostly because parts of our given expression might evaluate to something undesirable when acting on them outside of an integral—we don't know, and don't want to risk it.
* No error messages yet; it just returns unevaluated. Ideally we should say what's wrong if we can't evaluate.
* We should also have messages for alerting the user as to what kind of integral will be taken (absolute or not)
* We could generalize to higher dimensions! Why only integrate on curves? (Note that the current implementation supports any embedding dimension, by the way.)
* Options! To, e.g., specify the preferred representation of our region, or include extra assumptions in our internal `Simplify`s.
* Mathematica has access to a great curated collection of curves via `SpaceCurve`. These often come with tangent vectors included as properties! We could extend this to use those too—not just regions—but that's pretty straightforward.
Here are some tests!
```
(* (naturally) Parametric regions: *)
LineIntegrate[{-y, x}, {x, y} \[Element] Circle[{0, 0}, r]]
(* Out: *)
(* 2 Pi r^2 *)
(* Naturally implicit region: *)
reg = ImplicitRegion[(a == 5 || a == 0) && 0 <= b <= 1, {a, b}];
LineIntegrate[{x, y}, {x, y} \[Element] reg]
(* Out: 1 *)
(* Explicitly parametrized region: *)
LineIntegrate[{x,y} |-> {-y,x}, {Sin[t], Cos[t]}, {t, 0, 2 Pi}]
(* Out: -2 Pi *)
(* Maple multisegment line tests: *)
LineIntegrate[{x, y}, {x, y} \[Element] Line[{{1, 2}, {3, -4}}]]
LineIntegrate[{x, y}, {x, y} \[Element]
Line[{{0, 0}, {1, 1}, {1, -1}}]]
LineIntegrate[{4 y^3, -2 x^2}, {x, y} \[Element]
Line[{{-1, -1}, {1, -1}, {1, 1}, {-1, 1}, {-1, -1}}]]
(* Out: 10, 1, -16 *)