diff options
| -rw-r--r-- | chapters/6-Over-a-Map.scm | 104 | ||||
| -rw-r--r-- | log/2022-09-29.md | 6 | ||||
| -rw-r--r-- | log/2022-10-04.md | 17 |
3 files changed, 127 insertions, 0 deletions
diff --git a/chapters/6-Over-a-Map.scm b/chapters/6-Over-a-Map.scm new file mode 100644 index 0000000..748316b --- /dev/null +++ b/chapters/6-Over-a-Map.scm @@ -0,0 +1,104 @@ + +(define ((vector-field->vector-field-over-map mu:N->M) v-on-M) + (procedure->vector-field + (lambda (f-on-M) + (compose (v-on-M f-on-M) mu:N->M)))) + +(define (((differential mu) v) f) + (v (compose f mu))) + +(define ((form-field->form-field-over-map mu:N->M) w-on-M) + (define (make-fake-vector-field V-over-mu n) + (define ((u f) m) + ((V-over-mu f) n)) + (procedure->vector-field u)) + (procedure->nform-field + (lambda vectors-over-map + (lambda (n) + ((apply w-on-M + (map (lambda (V-over-mu) + (make-fake-vector-field V-over-mu n)) + vectors-over-map)) + (mu:N->M n)))) + (get-rank w-on-M))) + +(define S2 (make-manifold S^2 2 3)) +(define S2-spherical + (coordinate-system-at 'spherical 'north-pole S2)) +(define-coordinates (up theta phi) S2-spherical) +(define S2-basis (coordinate-system->basis S2-spherical)) + +(define-coordinates t R1-rect) + +(define mu + (compose (point S2-spherical) + (up (literal-function 'theta) + (literal-function 'phi)) + (chart R1-rect))) + +(define S2-basis-over-mu + (basis->basis-over-map mu S2-basis)) + +(define h + (literal-manifold-function 'h-spherical S2-spherical)) + +(((basis->vector-basis S2-basis-over-mu) h) + ((point R1-rect) 't0)) +; (down (((partial 0) h-spherical) (up (theta t0) (phi t0))) +; (((partial 1) h-spherical) (up (theta t0) (phi t0)))) + + +(((basis->1form-basis S2-basis-over-mu) + (basis->vector-basis S2-basis-over-mu)) + ((point R1-rect) 't0)) +; (up (down 1 0) (down 0 1)) + +(((basis->1form-basis S2-basis-over-mu) + ((differential mu) d/dt)) + ((point R1-rect) 't0)) +; (up ((D theta) t0) ((D phi) t0)) + +(define ((pullback-function mu:N->M) f-on-M) + (compose f-on-M mu:N->M)) + +(define ((pushforward-vector mu:N->M mu^-1:M->N) v-on-N) + (procedure->vector-field + (lambda (f) + (compose (v-on-N (compose f mu:N->M)) mu^-1:M->N)))) + +(define (pullback-vector-field mu:N->M mu^-1:M->N) + (pushforward-vector mu^-1:M->N mu:N->M)) + +(define ((pullback-form mu:N->M) omega-on-M) + (let ((k (get-rank omega-on-M))) + (if (= k 0) + ((pullback-function mu:N->M) omega-on-M) + (procedure->nform-field + (lambda vectors-on-N + (apply ((form-field->form-field-over-map mu:N->M) + omega-on-M) + (map (differential mu:N_>M) vectors-on-N))) + k)))) + +(define mu (literal-manifold-map 'MU R2-rect R3-rect)) + +(define f (literal-manifold-function 'f-rect R3-rect)) +(define X (literal-vector-field 'X-rect R2-rect)) + +(((- ((pullback mu) (d f)) (d ((pullback mu) f))) X) + ((point R2-rect) (up 'x0 'y0))) +; 0 + +(define theta (literal-1form-field 'THETA R3-rect)) +(define Y (literal-vector-field 'Y-rect R2-rect)) + +(((- ((pullback mu) (d theta)) (d ((pullback mu) theta))) X Y) + ((point R2-rect) (up 'x0 'y0))) +; 0 + + +;;; Exersize 6.1: Velocities on a Globe + +; a) + +; b) diff --git a/log/2022-09-29.md b/log/2022-09-29.md new file mode 100644 index 0000000..977e300 --- /dev/null +++ b/log/2022-09-29.md @@ -0,0 +1,6 @@ + +Didn't get much done at all today! Had a short conversation with Josh after +lunch about 1-forms and tangent bundles. Started reading "over a map" chapter, +but didn't work through it. + +Tomorrow planning to fly in the afternoon. diff --git a/log/2022-10-04.md b/log/2022-10-04.md new file mode 100644 index 0000000..8be5440 --- /dev/null +++ b/log/2022-10-04.md @@ -0,0 +1,17 @@ + +Read through Chapter 6 ("Over a Map"), but felt like I wasn't absorbing too +much. Maybe not that bad of a problem at this point? + +Took a look at "Visual Differential Geometry and Forms" (VDGF), and the +definition of a 1-form was short and much more readable. I think that book is a +good complement! Carroll's "Spacetime and Geometry" is also decent. VDGF led to +a tangent to flip through Penrose's "The Road to Reality". Sort of surprising +myself at level of interest in abstract geometry and physics... but need to +focus back down on the plan for now. + +The next chapter (7. Directional Derivatives) is particularly long, so probably +won't finish it this week. Would also like to finish reading the intro/overview +content in VDGF, and compare some of the definitions there against FDG. + +Started working through (short) exersize on paper, but should do it +computationally. |