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 Road to Little Dribbling: More Notes from a Small Island The Road to Little Dribbling: More Notes from a Small Island The Road to Little Dribbling: More Notes From a Small Island is a humorous travel book by American author Bill Bryson, first published in 2015. Twenty years after the publication of "Notes From a Small Island", Bryson makes another journey around Great Britain to see what has changed. In the opening chapters he notes that the straight line distance from Bognor Regis on the south coast to Cape Wrath in Scotland is the longest straight line one can travel in the UK without crossing any part of the sea.