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 The Doll that Came Straight from Fairyland "The Doll That Came Straight From Fairyland" is a short story for children published in 1898 by the suffragist and children's fairytale writer Evelyn Sharp. The first edition was published by John Lane in London, England, in "All The Way To Fairyland" with seven other short stories. Evelyn Sharp was born in 1869 and was the ninth of eleven children. She had three years of schooling in London where she made her living. Evelyn's first publication was an article about fairy tales which led to "Wymps and other Fairy Tales" (1896) and "All