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Derivatives of a multivariative function and corresponding Jacobian with vector-space package

f:(0,oo) x [0,2pi] -> R²

(r,phi) -> (r*cos(phi),r*sin(phi))

       

polar :: Double -> Double -> ((Double,Double) :~> (Double,Double))

polar r phi = (r,phi) ->  (((idD) r)*cos( idD phi),((idD) r)*sin( idD phi))

       

Couldn't match expected type `(Double, Double) :> (Double, Double)'

            with actual type `(t0, t1)'

In the expression:

  (((idD) r) * cos (idD phi), ((idD) r) * sin (idD phi))

In the expression:

   (r, phi)

    -> (((idD) r) * cos (idD phi), ((idD) r) * sin (idD phi))

In an equation for `polar':

    polar r phi

      =  (r, phi)

          -> (((idD) r) * cos (idD phi), ((idD) r) * sin (idD phi))

       

polarx :: Double -> Double -> ((Double,Double) :~> Double)

polarx r phi = (r,phi) ->  ((idD) r)*cos( idD phi)

       

Couldn't match expected type `Double'

            with actual type `(Double, Double)'

Expected type: (Double, Double) :> Double

  Actual type: (Double, Double) :> (Double, Double)

In the return type of a call of `idD'

In the first argument of `(*)', namely `((idD) r)'

       

polarCoordD :: ((Double,Double) :~> (Double,Double))

polarCoordD = (r,phi) ->  pairD (polarx (r,phi), polary (r,phi))

where polarx :: (Double,Double) :~> Double

      polarx = (r,phi) -> (fst . unpairD $ (idD) (r,phi))*cos( snd . unpairD $ idD (r, phi))

      polary :: (Double,Double) :~> Double

      polary = (r,phi) -> (fst . unpairD $ (idD) (r,phi))*sin( snd . unpairD $ idD (r, phi))

       

cylCoordD :: (Vec3 Double :~> Vec3 Double)

cylCoordD = (r,phi,z) ->  tripleD (cylx (r,phi,z), cyly (r,phi,z),cylz (0,0,z))

where cylx :: (Double,Double,Double) :~> Double

      cylx = (r,phi,z) -> (fst' . untripleD $ (idD) (r,phi,z))*cos( snd' . untripleD $ idD (r, phi,z))

      cyly :: (Double,Double,Double) :~> Double

      cyly = (r,phi,z) -> (fst' . untripleD $ (idD) (r,phi,z))*sin( snd' . untripleD $ idD (r, phi,z))

      cylz :: (Double,Double,Double) :~> Double

      cylz = (_,_,z) -> third . untripleD $ idD (0,0,z)

      fst' :: (a,b,c) -> a

      fst' (x,_,_) = x

      snd' :: (a,b,c) -> b

      snd' (_,y,_) = y

      third :: (a,b,c) -> c

      third (_,_,z) = z

       

let transmat = (r,phi,z) -> powVal $ liftD3 (,,) (normalized $ derivAtBasis (cylCoordD (r,phi,z)) (Left ())) (normalized $ derivAtBasis (cylCoordD (r,phi,z)) (Right (Left ()))) (normalized $ derivAtBasis (cylCoordD (r,phi,z)) (Right (Right ())))

 

*Main> transmat (2, rad 0, 0)

((1.0,0.0,0.0),(0.0,1.0,0.0),(0.0,0.0,1.0))

 

*Main> transmat (2, rad 90, 0)

((6.123233995736766e-17,1.0,0.0),(-1.0,6.123233995736766e-17,0.0),(0.0,0.0,1.0))

       

rad :: Double -> Double

rad = (pi*) . (*recip 180)