;; ;; 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 ;; (define \$∇ ∂/∂) (∇ (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)) ;=> 140 (define \$trace (lambda [%t] (with-symbols {i} (contract + t~i_i)))) (trace [|[|10 20 30|] [|20 40 60|] [|30 60 90|]|]) ;=> 140 ;; ;; Divergence ;; (define \$div (compose ∇ (trace \$))) (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 ;; (define \$taylor-expansion (lambda [%f %xs %as] (with-symbols {h} (let {[\$hs (generate-tensor 1#h_%1 (tensor-size xs))]} (map2 * (map 1#(/ 1 (fact %1)) nats0) (map (compose (V.substitute xs as \$) (V.substitute hs (with-symbols {i} (- xs_i as_i)) \$)) (iterate (compose (∇ \$ xs) (V.* hs \$)) f))))))) (take 3 (taylor-expansion (f x) x 0)) ;=> ;{(f 0) ; (* x (f_1 0)) ; (/ (* x^2 (f_1_1 0)) ; 2)} (take 3 (taylor-expansion (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)}