Copyright	(c) Nils Alex 2020
License	MIT
Maintainer	nils.alex@fau.de
Safe Haskell	None
Language	Haskell2010

Math.Tensor.Basic.Sym2

Contents

Flat positive-definite metric
Flat Lorentzian metric
Injections from \(S^2V\) into \(V\times V\)
Surjections from \(V\times V\) onto \(S^2V\)
Vertical coefficients for functions on \(S^2V\)
Kronecker delta on \(S^2V\)
Internals

Description

Definitions of symmetric tensors.

Synopsis

gamma :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Cov (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Tensor r v
gamma' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Cov (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Tensor r v
someGamma :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> m (T v)
gammaInv :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Con (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Tensor r v
gammaInv' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Con (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Tensor r v
someGammaInv :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> m (T v)
eta :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Cov (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Tensor r v
eta' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Cov (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Tensor r v
someEta :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> m (T v)
etaInv :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Con (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Tensor r v
etaInv' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Con (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Tensor r v
someEtaInv :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> m (T v)
injSym2Con' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (i :: Symbol) (r :: Rank) v. (InjSym2ConRank id n a b i ~ Just r, SingI r, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Sing i -> Tensor r v
injSym2Cov' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (i :: Symbol) (r :: Rank) v. (InjSym2CovRank id n a b i ~ Just r, SingI r, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Sing i -> Tensor r v
someInjSym2Con :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> m (T v)
someInjSym2Cov :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> m (T v)
surjSym2Con' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (i :: Symbol) (r :: Rank) v. (SurjSym2ConRank id n a b i ~ Just r, SingI r, Fractional v) => Sing id -> Sing n -> Sing a -> Sing b -> Sing i -> Tensor r v
surjSym2Cov' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (i :: Symbol) (r :: Rank) v. (SurjSym2CovRank id n a b i ~ Just r, SingI r, Fractional v) => Sing id -> Sing n -> Sing a -> Sing b -> Sing i -> Tensor r v
someSurjSym2Con :: (Fractional v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> m (T v)
someSurjSym2Cov :: (Fractional v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> m (T v)
someInterSym2Con :: Num v => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> Demote Symbol -> T v
someInterSym2Cov :: Num v => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> Demote Symbol -> T v
someDeltaSym2 :: Num v => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> T v
trianMapSym2 :: forall a. Integral a => a -> Map (Vec (S (S Z)) Int) Int
facMapSym2 :: forall a b. (Integral a, Num b) => a -> Map (Vec (S (S Z)) Int) b
sym2Assocs :: forall (n :: Nat) v. Num v => Sing n -> [(Vec (S (S (S Z))) Int, v)]
sym2AssocsFac :: forall (n :: Nat) v. Fractional v => Sing n -> [(Vec (S (S (S Z))) Int, v)]

Flat positive-definite metric

gamma :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Cov (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Tensor r v Source #

gamma' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Cov (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Tensor r v Source #

someGamma :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> m (T v) Source #

gammaInv :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Con (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Tensor r v Source #

gammaInv' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Con (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Tensor r v Source #

someGammaInv :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> m (T v) Source #

Flat Lorentzian metric

eta :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Cov (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Tensor r v Source #

eta' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Cov (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Tensor r v Source #

someEta :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> m (T v) Source #

etaInv :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Con (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Tensor r v Source #

etaInv' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (r :: Rank) v. ('['(VSpace id n, Con (a :| '[b]))] ~ r, (a < b) ~ True, SingI n, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Tensor r v Source #

someEtaInv :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> m (T v) Source #

Injections from \(S^2V\) into \(V\times V\)

injSym2Con' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (i :: Symbol) (r :: Rank) v. (InjSym2ConRank id n a b i ~ Just r, SingI r, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Sing i -> Tensor r v Source #

injSym2Cov' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (i :: Symbol) (r :: Rank) v. (InjSym2CovRank id n a b i ~ Just r, SingI r, Num v) => Sing id -> Sing n -> Sing a -> Sing b -> Sing i -> Tensor r v Source #

someInjSym2Con :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> m (T v) Source #

someInjSym2Cov :: (Num v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> m (T v) Source #

Surjections from \(V\times V\) onto \(S^2V\)

surjSym2Con' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (i :: Symbol) (r :: Rank) v. (SurjSym2ConRank id n a b i ~ Just r, SingI r, Fractional v) => Sing id -> Sing n -> Sing a -> Sing b -> Sing i -> Tensor r v Source #

surjSym2Cov' :: forall (id :: Symbol) (n :: Nat) (a :: Symbol) (b :: Symbol) (i :: Symbol) (r :: Rank) v. (SurjSym2CovRank id n a b i ~ Just r, SingI r, Fractional v) => Sing id -> Sing n -> Sing a -> Sing b -> Sing i -> Tensor r v Source #

someSurjSym2Con :: (Fractional v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> m (T v) Source #

someSurjSym2Cov :: (Fractional v, MonadError String m) => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> m (T v) Source #

Vertical coefficients for functions on \(S^2V\)

someInterSym2Con :: Num v => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> Demote Symbol -> T v Source #

someInterSym2Cov :: Num v => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> Demote Symbol -> Demote Symbol -> T v Source #

Kronecker delta on \(S^2V\)

someDeltaSym2 :: Num v => Demote Symbol -> Demote Nat -> Demote Symbol -> Demote Symbol -> T v Source #

Internals

trianMapSym2 :: forall a. Integral a => a -> Map (Vec (S (S Z)) Int) Int Source #

facMapSym2 :: forall a b. (Integral a, Num b) => a -> Map (Vec (S (S Z)) Int) b Source #

sym2Assocs :: forall (n :: Nat) v. Num v => Sing n -> [(Vec (S (S (S Z))) Int, v)] Source #

sym2AssocsFac :: forall (n :: Nat) v. Fractional v => Sing n -> [(Vec (S (S (S Z))) Int, v)] Source #