By David Olsen
<h2>Discover how the world's greatest principles, innovations, and activities replaced the process history!</h2>
What could existence be like if the Age of cause by no means challenged others to imagine another way, if the economic Revolution by no means occurred, or if the recent York inventory alternate by no means got here into existence?
801 stuff you should still Know promises the lowdown on suggestions and occasions that reworked prior civilizations into the cultures that we all know this day. every one access explains a game-changing proposal or second in time, detailing the way it assisted in shaping societies worldwide. You'll discover attention-grabbing info you'd by no means heard earlier than, and be shocked to profit how those significant affects have at once impacted how you live.
From the 6th century B.C. to the current day, you'll observe the beautiful humans, acts, and concepts that experience encouraged change--and revolutionized the world.
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Additional info for 801 Things You Should Know: From Greek Philosophy to Today's Technology, Theories, Events, Discoveries, Trends, and Movements That Matter
10]. A very surprising fact, which can be deduced from Cauchy’s theorem, is that if f is holomorphic then it can be diﬀerentiated twice. ) It follows that f is holomorphic, so it too can be diﬀerentiated twice. Continuing, one ﬁnds that f can I. Introduction be diﬀerentiated any number of times. Thus, for complex functions diﬀerentiability implies inﬁnite diﬀerentiability. ) A closely related fact is that wherever a holomorphic function is deﬁned it can be expanded in a power series. That is, if f is deﬁned and diﬀerentiable everywhere on an open disk of radius R about w, then it will be given by a formula of the form ∞ an (z − w)n f (z) = n=0 valid everywhere in that disk.
Topology can be thought of as the geometry that arises when we use a particularly generous notion of equivalence, saying that two shapes are equivalent, or homeomorphic, to use the technical term, if each can be “continuously deformed” into the other. For example, a sphere and a cube are equivalent in this sense, as ﬁgure 1 illustrates. Because there are very many continuous deformations, it is quite hard to prove that two shapes are not equivalent in this sense. For example, it may seem obvious that a sphere (this means the surface of a ball rather than the solid ball) cannot be continuously deformed into a torus (the shape of the surface of a doughnut of the kind that has a hole in it), since they are fundamentally diﬀerent shapes—one has a “hole” and the other does not.
Suppose that you have a lump of impure rock and wish to calculate its mass from its density. Suppose also that this density is not constant but varies rather irregularly through the rock. Perhaps there are even holes inside, so that the density is zero in places. What should you do? Riemann’s approach would be this. First, you enclose the rock in a cuboid. For each point (x, y, z) in this cuboid there is then an associated density d(x, y, z) (which will be zero if (x, y, z) lies outside the rock or inside a hole).