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.
The integral of a real-valued function of a real variable is the limit of a sum of products of the values of the function on subintervals and the lengths of the increments of the independent variable in those subintervals:
| ;; here were the deps I needed for this... I don't know enough vega-lite to go without the Hanami example I had | |
| ;; already built, so excuse me there! | |
| {:paths ["dev"] | |
| :deps {io.github.nextjournal/clerk {:local/root "../clerk"} | |
| notespace-sicmutils {:mvn/version "0.16.2"} | |
| aerial.hanami/aerial.hanami {:mvn/version "0.12.7"} | |
| sicmutils/sicmutils {:git/url "https://github.com/sicmutils/sicmutils" | |
| :sha "8658c0c8883b8225a742b9422061f40b852f375d"}}} |
| (defn install-fn-printers! | |
| "NOTE: If you are using Emacs + cider-nrepl, these two `print-method` | |
| implementations will not be installed. I'm currently working on a fix to get | |
| this enabled in the Emacs environment... But if you want to test this feature | |
| out, simply execute these forms. | |
| https://github.com/clojure-emacs/cider-nrepl/blob/master/src/cider/nrepl/print_method.clj#L42-L71 | |
| TODO write a with-methods instead to do this..." | |
| [] | |
| #?(:clj |
| (defn ^:no-doc careful-def [ns] | |
| #?(:cljs | |
| (fn [sym form] | |
| `(def ~sym ~form)) | |
| :clj | |
| (let [ns-sym (ns-name ns) | |
| nsm (ns-map ns) | |
| remote? (fn [sym] | |
| (when-let [v (nsm sym)] |
| shadow-cljs - running: lein run -m shadow.cljs.devtools.cli --npm clj-repl | |
| Syntax error (UnsatisfiedLinkError) compiling at (/private/var/folders/xw/0lq56zhn4hb4lknppw_k086c0000gn/T/form-init10880055894866318920.clj:1:126). | |
| /private/var/folders/xw/0lq56zhn4hb4lknppw_k086c0000gn/T/jna5901602065421044698.tmp: dlopen(/private/var/folders/xw/0lq56zhn4hb4lknppw_k086c0000gn/T/jna5901602065421044698.tmp, 1): no suitable image found. Did find: | |
| /private/var/folders/xw/0lq56zhn4hb4lknppw_k086c0000gn/T/jna5901602065421044698.tmp: no matching architecture in universal wrapper | |
| /private/var/folders/xw/0lq56zhn4hb4lknppw_k086c0000gn/T/jna5901602065421044698.tmp: no matching architecture in universal wrapper | |
| Full report at: | |
| /var/folders/xw/0lq56zhn4hb4lknppw_k086c0000gn/T/clojure-14930451110325842276.edn |
| (/ | |
| (+ (* -717897987691852588770249/1125899906842624 (expt x 74) (sqrt x) (cos (+ (* x (sqrt x)) (* 3 (sqrt x))))) | |
| (* -17947449692296314719256225/562949953421312 (expt x 73) (sqrt x) (cos (+ (* x (sqrt x)) (* 3 (sqrt x))))) | |
| (* -412791342922815238542893175/1125899906842624 (expt x 73) (sin (+ (* x (sqrt x)) (* 3 (sqrt x))))) | |
| (* -879425034922519421243555025/1125899906842624 (expt x 72) (sqrt x) (cos (+ (* x (sqrt x)) (* 3 (sqrt x))))) | |
| (* -19640492446602933741106062225/1125899906842624 (expt x 72) (sin (+ (* x (sqrt x)) (* 3 (sqrt x))))) | |
| (* 23549048157369686724410751225/281474976710656 (expt x 71) (sqrt x) (cos (+ (* x (sqrt x)) (* 3 (sqrt x))))) | |
| (* -57162627269963762380831076625/140737488355328 (expt x 71) (sin (+ (* x (sqrt x)) (* 3 (sqrt x))))) | |
| (* 1180872394115405245080929164125/281474976710656 (expt x 70) (sqrt x) (cos (+ (* x (sqrt x)) (* 3 (sqrt x))))) | |
| (* 80027678177949267333163507275/8796093022208 (expt x 70) (sin (+ (* x (sqrt x)) (* 3 (sqrt x))))) |
| ;; | |
| ;; [[jacobian]] handles this main logic. [[jacobian]] can only take a structural | |
| ;; input. [[euclidean]] and [[multivariate]] below widen handle, respectively, | |
| ;; optionally-structural and multivariable arguments. | |
| (defn- jacobian | |
| "Takes: | |
| - some function `f` of a single [[s/structure?]] argument | |
| - the unperturbed structural `input` |
| (define ((replace-differential-tag oldtag newtag) object) | |
| (cond ((differential? object) | |
| (terms->differential | |
| (map (lambda (term) | |
| (let ((terms (differential-tags term))) | |
| (cond ((and (memv oldtag terms) | |
| (memv newtag terms)) | |
| (make-differential-term | |
| (remove-differential-tag newtag (remove-differential-tag oldtag terms)) | |
| :zero)) |
| ;; Here is the smoking gun... | |
| (define (f x) | |
| (lambda (cont) | |
| (cont (lambda (y) (* x y)) | |
| (lambda (g) (g x))))) | |
| ;; (D f) is: | |
| (lambda (cont) |
| (ns sicmutils.sci | |
| (:require [sicmutils.env :as env] | |
| [sci.core :as sci])) | |
| (defn ns-bindings | |
| "Returns a pair of | |
| - `(ns-publics sym)` fn entries | |
| - `(ns-publics sym)` macro entries" | |
| [sym] |
