... 3- space: d(x, y) = Minkowski 4- space: d(x, y) = (y1 - x1)2 + (y2 - x2)2 + (y3 - x3)2 (y1 - x1)2 + (y2 - x2)2 + (y3 - x3)2 - c2(y4 - x4)2 Geometrically, the set of all points in Euclidean 3-space ... sin x4 … cos xn-1 yn = r sin x1 sin x2 sin x3 sin x4 … sin xn-1 cos xn yn+1 = r sin x1 sin x2 sin x3 sin x4 … sin xn-1 sin xn (Notice that x1 is playing the role of ˙ and the x2, x3, , xn-1 ... sin xn-1 sin xn) ∂ 3 = (0, 0, -r sin x sin x sin x , r sin x sin x cos x cos x , ∂x r sin x1 sin x2 cos x3 sin x4… sin xn-1 cos xn, r sin x1 sin x2 cos x3 sin x4 … sin xn-1 sin xn), and so...