Area of a circle trigonometry, uses the coarea formula. Define a function formula_10 by formula_11. Note ρ is a Lipschitz function whose gradient is a unit vector formula_12 (almost everywhere). Let "D" be the disc formula_13 in formula_14. We will show that formula_15, where formula_16 is the two-dimensional Lebesgue measure in formula_14. We shall assume that the one-dimensional Hausdorff measure of the circle formula_18 is formula_19, the circumference of the circle of radius "r". (This can be taken as the definition of circumference.) Then, by the coarea formula, Similar to the onion proof outlined above, we could exploit calculus in a different way in
The Doll that Came Straight from Fairyland a cannon and a magical balloon, but only replies that his godmother should give him what she thinks best. In response, Fairy Zigzag instead asks Princess Pansy what she would like. Princess Pansy asks for a fine clay doll. The Fairy Zigzag creates a magical doll straight from fairyland and then leaves the party. The Prince is left without a gift from his fairy godmother. Princess Pansy names the doll Lady Emmelina and in the days following the party becomes bewitched by the doll. Prince Perfection gets jealous of Lady Emmelina when his sister no longer plays with him. Missing