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 straight on, and when he meets other people he must ask what they are thinking. In his quest, he travels across the sea, through a gorse bush and over an iceberg to a castle. Along the way he collects thoughts. These thoughts consist of the Prince's wishes for his gift: a steamboat, a cannon and a magical balloon. The gifts that the Prince had wished for earlier help him reach the castle where Lady Emmelina is waiting. Once he finds her, the Prince quickly returns home. When Princess Pansy sees her brother she hugs him while Lady Emmelina falls to