;;; ;;; Parameters ;;; (define \$x [|θ|]) (define \$X [|(* r (sin θ)) ; = x (* r (cos θ)) ; = y |]) ;; ;; Local basis ;; (define \$e ((flip ∂/∂) x~# X_#)) e;[| [| (* r (cos θ)) (* -1 r (sin θ)) |] |]_#~# ;; ;; Metric tensor ;; (define \$g__ (generate-tensor 2#(V.* e_%1 e_%2) {1 1})) (define \$g~~ (M.inverse g_#_#)) g_#_#;[| [| r^2 |] |]_#_# g~#~#;[| [| (/ 1 r^2) |] |]~#~# ;; ;; Christoffel symbols of the first kind ;; (define \$Γ___ (with-symbols {j k l} (* (/ 1 2) (+ (∂/∂ g_j_k x~l) (∂/∂ g_j_l x~k) (* -1 (∂/∂ g_k_l x~j)))))) Γ_#_#_#;(tensor {1 1 1} {0} )_#_#_# ;; ;; Christoffel symbols of the second kind ;; (define \$Γ~__ (with-symbols {i j k l} (. g~i~j Γ_j_k_l))) Γ~#_#_#;(tensor {1 1 1} {0} )~#_#_# ;; ;; Riemann curvature tensor ;; (define \$R~i_j_k_l (with-symbols {m} (+ (- (∂/∂ Γ~i_j_l x~k) (∂/∂ Γ~i_j_k x~l)) (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l))))) R~#_#_#_#;(tensor {1 1 1 1} {0} )~#_#_#_# (define \$R____ (with-symbols {i} (. g_i_# R~i_#_#_#))) R_#_#_#_#;(tensor {1 1 1 1} {0} )_#_#_#_# ;; ;; Ricci curvature ;; (define \$Ric__ (with-symbols {i j k} (contract + R~i_j_k_i))) Ric_#_#;[| [| 0 |] |]_#_# ;; ;; Scalar curvature ;; (define \$scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k))) scalar-curvature;0