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+
+(define R2 (make-manifold R^n 2))
+
+(define U (patch 'origin R2))
+
+; these are coordinate systems
+(define R2-rect (coordinate-system 'rectangular U))
+(define R2-polar (coordinate-system 'polar/cylindrical U))
+
+; these are charts, and their inverses
+(define R2-rect-chi (chart R2-rect))
+(define R2-rect-chi-inverse (point R2-rect))
+(define R2-polar-chi (chart R2-polar))
+(define R2-polar-chi-inverse (point R2-polar))
+
+((compose R2-polar-chi R2-rect-chi-inverse) (up 'x0 'y0))
+; (up 
+;   (sqrt (+ (expt x0 2) (expt y0 2)))  ; radius
+;   (atan y0 x0))                       ; angle (theta)
+
+((compose R2-rect-chi R2-polar-chi-inverse) (up 'r0 'theta0))
+; (up
+;   (* r0 (cos theta0))     ; x
+;   (* r0 (sin theta0)))    ; y
+
+
+((D (compose R2-rect-chi R2-polar-chi-inverse)) (up 'r0 'theta0))
+; (down
+;   (up
+;       (cos theta0)
+;       (sin theta0))
+;   (up
+;       (* -1 r0 (sin theta0))
+;       (* r0 (cos theta0))))
+
+(define R2->R (-> (UP Real Real) Real))
+(define f (compose (literal-function 'f-rect R2->R) R2-rect-chi))
+
+(define R2-rect-point (R2-rect-chi-inverse (up 'x0 'y0)))
+
+(define corresponding-polar-point
+  (R2-polar-chi-inverse
+    (up (sqrt (+ (square 'x0) (square 'y0)))
+        (atan 'y0 'x0))))
+
+(f R2-rect-point)
+; (f-rect (up x0 y0))
+
+(f corresponding-polar-point)
+; (f-rect (up x0 y0))
+
+; confirms that the CAS simplifies to the same point
+
+(define-coordinates (up x y) R2-rect)
+(define-coordinates (up r theta) R2-polar)
+
+(x (R2-rect-chi-inverse (up 'x0 'y0)))
+; x0
+
+; expect r0 * cos(theta0)
+(x (R2-polar-chi-inverse (up 'r0 'theta0)))
+; (* r0 (cos theta0))
+
+; expect r0 * sin(theta0)
+(y (R2-polar-chi-inverse (up 'r0 'theta0)))
+; (* r0 (sin theta0))
+
+
+;(r (R2-polar-chi-inverse (up 'r0 'theta0)))
+; r0
+
+; expect sqrt(x0^2 + y0^2)
+(r (R2-rect-chi-inverse (up 'x0 'y0)))
+; (sqrt (+ (expt x0 2) (expt y0 2)))
+
+; expect atan(y0, x0)
+(theta (R2-rect-chi-inverse (up 'x0 'y0)))
+; (atan y0 x0)
+
+; h: x * r^2 + y^3
+(define h (+ (* x (square r)) (cube y)))
+
+(h R2-rect-point)
+; (+ (expt x0 3) (* x0 (expt y0 2)) (expt y0 3))
+; aka: x0^3 + x0 y0^2 + y0^3 
+;      x0 (x0^2 + y0^2) + y0^3
+
+(h (R2-polar-chi-inverse (up 'r0 'theta0)))
+; (+ (* (expt r0 3) (expt (sin theta0) 3)) (* (expt r0 3) (cos theta0)))
+
+
+;;; Exersize 2.1a
+
+((- r (* 2 'a (+ 1 (cos theta))))
+ ((point R2-rect) (up 'x 'y)))
+; (/ 
+;   (+ (* -2 a x)
+;      (* -2 a (sqrt (+ (expt x 2) (expt y 2))))
+;      (expt x 2) (expt y 2))
+;   (sqrt (+ (expt x 2) (expt y 2))))
+
+
+; "Lemniscate of Bernoulli"
+; (x^2 + y^2)^2 = 2 a^2 (x^2 - y^2)
+
+((-
+   (square (+ (square x) (square y)))
+   (* 2
+      (square 'a)
+      (- (square x) (square y))))
+ ((point R2-polar) (up 'r0 'theta0)))
+; (+ (* -4 (expt a 2) (expt r0 2) (expt (cos theta0) 2))
+;    (* 2 (expt a 2) (expt r0 2))
+;    (expt r0 4))
+;
+; r^2 / a^2 + 2 = 4 cos^2 (theta)
+; => r^2 = 2 a^2 cos(2 theta)
+
+; this matches Wikipedia, with substitution of a^2 = 2 c^2 (in the rectangular version)
+
+
+;;; Exersize 2.1b
+
+(define R3-rect-chi (chart R3-rect))
+(define R3-rect-chi-inverse (point R3-rect))
+(define R3-cyl-chi (chart R3-cyl))
+(define R3-cyl-chi-inverse (point R3-cyl))
+
+(define-coordinates (up x y z) R3-rect)
+(define-coordinates (up r theta z-cyl) R3-cyl)
+
+; R->R3
+(define (helix-rect t)
+  (up (* 'R (cos t))
+      (* 'R (sin t))
+      (* 's t)))
+
+; R->R3
+(define (helix-cyl t)
+  (up 'R
+      t
+      (* 's t)))
+
+((compose R3-rect-chi R3-cyl-chi-inverse helix-cyl) 't)
+; (up
+;   (* R (cos t))
+;   (* R (sin t))
+;   (* s t))
+
+
+((compose R3-cyl-chi R3-rect-chi-inverse helix-rect) 't)
+; (up R
+;     t
+;     (* s t))
+
+((- helix-rect (compose R3-rect-chi R3-cyl-chi-inverse helix-cyl)) 't)
+; (up 0 0 0)
+
+;;; Exersize 2.2
+
+; given polar coordinates of point on plane, get to spherical coordinates of point on sphere with:
+((compose
+   (chart S2-spherical)
+   (point S2-Riemann)
+   (chart R2-rect)
+   (point R2-polar))
+ (up 'rho 'theta))
+; (up (acos (/ (+ -1 (expt rho 2))
+;              (+ 1 (expt rho 2))))
+;     theta)
+
+; given spherical coordinates of point on sphere, what are the polar
+; coordinates of corresponding point on the plane?
+((compose
+   (chart R2-polar)
+   (point R2-rect)
+   (chart S2-Riemann)
+   (point S2-spherical))
+ (up 'phi 'lambda))
+; (up (/ (sin phi)
+;        (+ -1 (cos phi)))
+;     (atan (* -1 (sin lambda))
+;           (* -1 (cos lambda))))
+
+; this is just the inverse of the above, right?
+
+; huh, I would expect the second term to just be lambda.
+; atan (-sin(lambda), -cos(lambda)) -> tan^-1(tan(lambda)) -> lambda
+; seems like an identity, ok
+