{"page_number":231,"title":"Page 231","overview":"This page from \"The Britannica Guide to Analysis and Calculus\" provides definitions and historical context for two fundamental mathematical concepts: \"Fluxion,\" Isaac Newton's original term for the derivative, and the \"Fourier Transform,\" an important integral transform, including their mathematical formulations.","text_summary":"The page is divided into two main sections, each addressing a key concept in analysis and calculus.\n\nThe first section, titled \"FLUXION,\" explains that this was Isaac Newton's original term for what is now known as the derivative, introduced around 1665. Newton referred to a changing quantity as a \"fluent\" and its instantaneous rate of change as a \"fluxion.\" His infinitesimal calculus involved two primary problems: (1) given a fluent (modernly, a function), find its fluxion (modernly, a derivative); and (2) given a fluxion (a function), find a corresponding fluent (an indefinite integral). An example is provided: if y = x³, the fluxion of y is 3x² times the fluxion of x, expressed in modern notation as dy/dt = 3x²(dx/dt). The text notes that Newton's specific terminology and notations were eventually superseded by the concepts of derivatives and differentials developed by G. W. Leibniz.\n\nThe second section, \"FOURIER TRANSFORM,\" defines it as an integral transform applied to an integrable complex-valued function *f* of a single real variable. The mathematical formula for the Fourier transform is given as:\nf̂(ξ) = (2π)^(-1/2) ∫_(-∞)^∞ e^(-ixξ) f(x) dx.\nFollowing this, the page introduces a more general form of an integral equation:\nf(y) = ∫_(-∞)^∞ K(x, y) F(x) dx.\nIt clarifies that in this general form, *f(y)* is an integral transform of *F(x)*, and the function *K(x, y)* is referred to as the kernel. The section concludes by stating that a reciprocal relationship often holds true, providing the inverse transform:\nF(y) = ∫_(-∞)^∞ K'(x, y) f(x) dx.","content_markdown":"# Page 231\n\n### Page Overview\nThis page from \"The Britannica Guide to Analysis and Calculus\" provides definitions and historical context for two fundamental mathematical concepts: \"Fluxion,\" Isaac Newton's original term for the derivative, and the \"Fourier Transform,\" an important integral transform, including their mathematical formulations.\n\n### Text Content Summary\nThe page is divided into two main sections, each addressing a key concept in analysis and calculus.\n\nThe first section, titled \"FLUXION,\" explains that this was Isaac Newton's original term for what is now known as the derivative, introduced around 1665. Newton referred to a changing quantity as a \"fluent\" and its instantaneous rate of change as a \"fluxion.\" His infinitesimal calculus involved two primary problems: (1) given a fluent (modernly, a function), find its fluxion (modernly, a derivative); and (2) given a fluxion (a function), find a corresponding fluent (an indefinite integral). An example is provided: if y = x³, the fluxion of y is 3x² times the fluxion of x, expressed in modern notation as dy/dt = 3x²(dx/dt). The text notes that Newton's specific terminology and notations were eventually superseded by the concepts of derivatives and differentials developed by G. W. Leibniz.\n\nThe second section, \"FOURIER TRANSFORM,\" defines it as an integral transform applied to an integrable complex-valued function *f* of a single real variable. The mathematical formula for the Fourier transform is given as:\nf̂(ξ) = (2π)^(-1/2) ∫_(-∞)^∞ e^(-ixξ) f(x) dx.\nFollowing this, the page introduces a more general form of an integral equation:\nf(y) = ∫_(-∞)^∞ K(x, y) F(x) dx.\nIt clarifies that in this general form, *f(y)* is an integral transform of *F(x)*, and the function *K(x, y)* is referred to as the kernel. The section concludes by stating that a reciprocal relationship often holds true, providing the inverse transform:\nF(y) = ∫_(-∞)^∞ K'(x, y) f(x) dx.\n\n### Visual Elements (Diagrams, Figures, Graphs, Portraits, Illustrations)\n*No visual elements on this page.*","has_visuals":0,"visual_count":0,"visuals":[]}