Emmy, the Algebra System: Differential Geometry Chapter Five

Functional Differential Geometry: Chapter 5

Author

Affiliation

Published

February 8, 2026

5 Integration

We know how to integrate real-valued functions of a real variable. We want to extend this idea to manifolds, in such a way that the integral is independent of the coordinate system used to compute it.

3-Dimensional Euclidean Space

(define-coordinates (up x y z) R3-rect)

ERR

WARNING: R3-rect already refers to: #'emmy.env/R3-rect in namespace: mentat-collective.emmy.fdg-ch05, being replaced by: #'mentat-collective.emmy.fdg-ch05/R3-rect

(define R3-rect-point ((point R3-rect) (up 'x0 'y0 'z0)))

(define u (+ (* 'u↑0 d:dx) (* 'u↑1 d:dy)))

(define v (+ (* 'v↑0 d:dx) (* 'v↑1 d:dy)))

(print-expression
  (((wedge dx dy) u v) R3-rect-point))

(+ (* u↑0 v↑1) (* -1 u↑1 v↑0))

(define-coordinates (up r theta z) R3-cyl)

ERR

WARNING: R3-cyl already refers to: #'emmy.env/R3-cyl in namespace: mentat-collective.emmy.fdg-ch05, being replaced by: #'mentat-collective.emmy.fdg-ch05/R3-cyl

(define a (+ (* 'a↑0 d:dr) (* 'a↑1 d:dtheta)))

(define b (+ (* 'b↑0 d:dr) (* 'b↑1 d:dtheta)))

(print-expression
  (((wedge dr dtheta) a b) ((point R3-cyl) (up 'r0 'theta0 'z0))))

(+ (* a↑0 b↑1) (* -1 a↑1 b↑0))

(define u (+ (* 'u↑0 d:dx) (* 'u↑1 d:dy) (* 'u↑2 d:dz)))

(define v (+ (* 'v↑0 d:dx) (* 'v↑1 d:dy) (* 'v↑2 d:dz)))

(define w (+ (* 'w↑0 d:dx) (* 'w↑1 d:dy) (* 'w↑2 d:dz)))

(print-expression
  (((wedge dx dy dz) u v w) R3-rect-point))

(+ (* u↑0 v↑1 w↑2) (* -1N u↑0 v↑2 w↑1) (* -1N u↑1 v↑0 w↑2) (* u↑1 v↑2 w↑0) (* u↑2 v↑0 w↑1) (* -1N u↑2 v↑1 w↑0))

(print-expression
  (- (((wedge dx dy dz) u v w) R3-rect-point)
     (determinant
       (matrix-by-rows (list 'u↑0 'u↑1 'u↑2)
                       (list 'v↑0 'v↑1 'v↑2)
                       (list 'w↑0 'w↑1 'w↑2)))))

Computing Exterior Derivatives

(define a (literal-manifold-function 'alpha R3-rect))

(define b (literal-manifold-function 'beta R3-rect))

(define c (literal-manifold-function 'gamma R3-rect))

(define theta (+ (* a dx) (* b dy) (* c dz)))

(define X (literal-vector-field 'X-rect R3-rect))

(define Y (literal-vector-field 'Y-rect R3-rect))

(print-expression
  (((- (d theta)
       (+ (wedge (d a) dx)
          (wedge (d b) dy)
          (wedge (d c) dz))) X Y)
       R3-rect-point))

(define omega
  (+ (* a (wedge dy dz))
     (* b (wedge dz dx)) (* c (wedge dx dy))))

(define Z (literal-vector-field 'Z-rect R3-rect))

(print-expression
  (((- (d omega)
       (+ (wedge
            (d a) dy dz) (wedge (d b) dz dx)
          (wedge (d c) dx dy))) X Y Z)
       R3-rect-point))

Properties of Exterior Derivatives

(print-expression
  (((d (d theta)) X Y Z) R3-rect-point))

5.4 Vector Integral Theorems

(define v (literal-vector-field 'v-rect R2-rect))

(define w (literal-vector-field 'w-rect R2-rect))

(define alpha (literal-function 'alpha R2->R))

(define beta (literal-function 'beta R2->R))

(define R2-rect-basis (coordinate-system->basis R2-rect))

(print-expression
  (let-scheme
    ((dx (ref (basis->oneform-basis R2-rect-basis) 0))
     (dy (ref (basis->oneform-basis R2-rect-basis) 1)))
    (((- (d (+ (* (compose alpha (chart R2-rect)) dx)
               (* (compose beta (chart R2-rect)) dy)))
         (* (compose (- ((partial 0) beta)
                        ((partial 1) alpha))
                     (chart R2-rect))
            (wedge dx dy)))
         v w)
         R2-rect-point)))

(define a (literal-manifold-function 'a-rect R3-rect))

(define b (literal-manifold-function 'b-rect R3-rect))

(define c (literal-manifold-function 'c-rect R3-rect))

(define flux-through-boundary-element
  (+ (* a (wedge dy dz))
     (* b (wedge dz dx))
     (* c (wedge dx dy))))

(define production-in-volume-element
        (* (+ (d:dx a) (d:dy b) (d:dz c))
           (wedge dx dy dz)))

(define X (literal-vector-field 'X-rect R3-rect))

(define Y (literal-vector-field 'Y-rect R3-rect))

(define Z (literal-vector-field 'Z-rect R3-rect))

(print-expression
  (((- production-in-volume-element
       (d flux-through-boundary-element))
       X Y Z)
       R3-rect-point))

(repl/scittle-sidebar)

source: src/mentat_collective/emmy/fdg_ch05.clj