Vector Analysis

Nov 2 2017
--
-- Tensor Arithmetics
--
1 + [| 1, 2, 3 |]
-- [|2, 3, 4|]

[| 1, 2, 3 |] + 1
-- [|2, 3, 4|]

[| 1, 2, 3 |]_i + [| 1, 2, 3 |]_i
-- [|2, 4, 6|]_i

[| 10, 20, 30 |] + [| 1, 2, 3 |]
-- [| [| 11, 12, 13 |], [| 21, 22, 23 |], [| 31, 32, 33 |] |]

[| 100, 200, 300 |]_i + [|[| 1, 2, 3 |],
                          [| 10, 20, 30 |]|]_j_i
-- [| [| 101, 110 |], [| 202, 220 |], [| 303, 330 |] |]_i_j

[|[| 11, 12 |],
  [| 21, 22 |],
  [| 31, 32 |]|]_i_j + [| 100, 200, 300 |]_i
-- [| [| 111, 112 |], [| 221, 222 |], [| 331, 332 |] |]_i_j

[| 100, 200, 300 |]_i + [|[| 11, 12 |],
                          [| 21, 22 |],
                          [| 31, 32 |]|]_i_j
-- [| [| 111, 112 |], [| 221, 222 |], [| 331, 332 |] |]_i_j

--
-- Derivative
--
∂/∂ (f x y z) x
-- (f_1 x y z)

∂/∂ [| (f x), (g x) |] x
-- [| (f_1 x), (g_1 x) |]

∂/∂ (f x y z) [| x y z |]
-- [| (f_1 x y z), (f_2 x y z), (f_3 x y z) |]

[| (∂/∂ $ x), (∂/∂ $ y) |] (f x y)
-- [| (f_1 x y) (f_2 x y) |]

[| (∂/∂ $ x), (∂/∂ $ y) |] [| (f x y), (g x y) |]
-- [| [| (f_1 x y) (g_1 x y) |] [| (f_2 x y) (g_2 x y) |] |]

--
-- Nabla
--
def ∇ := ∂/∂

∇ (f x y) [| x, y |]
-- [| (f_1 x y), (f_2 x y) |]

∇ [| (f x y), (g x y) |] [| x, y |]
-- [| [| (f_1 x y), (f_2 x y) |], [| (g_1 x y), (g_2 x y) |] |]

--
-- Contraction
--
contract ([|1, 2, 3|]~i * [|10, 20, 30|]_i)
-- [10, 40, 90]

def trace %t := withSymbols [i] sum (contract t~i_i)

trace [|[|10, 20, 30|], [|20, 40, 60|], [|30, 60, 90|]|]
-- 140

--
-- Divergence
--
def div %t %x := trace (!(∇ t x))

div [| (f x y z), (g x y z), (h x y z) |] [| x, y, z |]
-- f|1 x y z + g|2 x y z + h|3 x y z

--
-- Taylor Expansion
--
def multivariateTaylorExpansion $f %xs %ys :=
  withSymbols [h]
    let hs := generateTensor (\[x] -> h_x) (tensorShape xs)
     in map2
          (*)
          (map (1)#(1 / fact %1) nats0)
          (map
             (compose
                (1)#(V.substitute xs ys %1)
                (1)#(V.substitute hs (withSymbols [i] xs_i - ys_i) %1))
             (iterate (compose (1)#(∇ %1 xs) (1)#(V.* hs %1)) f))

def taylorExpansion $f $x $a := multivariateTaylorExpansion f [|x|] [|a|]

take 3 (taylorExpansion (f x) x 0)
-- [f 0, x * (f|1 0), x^2 * (f|1|1 0) / 2]

take 3 (multivariateTaylorExpansion (f x y) [| x, y |] [| 0, 0 |])
-- [f 0 0, x * f|1 0 0 + y * f|2 0 0, (x^2 * f|1|1 0 0 + x * y * f|2|1 0 0 + y * x * f|1|2 0 0 + y^2 * f|2|2 0 0) / 2]
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