Elements of Differential and Integral Calculus - Bookswagon
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Elements of Differential and Integral Calculus

Elements of Differential and Integral Calculus


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1887 Excerpt: ... word implies that we conceive the magnitude to be made up of infinitesimal parts. Element of an Arc. Let us put Case of Polar Co-ordinates. To express the element of a curve referred to polar co-ordinates, differentiate the equa s = the length of any arc of a curve; ds = an element of this arc. If P and P' be two points of a curve, we shall have (chord PP'Y = Ax' + Ay. When PP' becomes infinitesimal, As the ratio of ds to PP' becomes unity ( 77), and we have M Ay 80. Equations of certain Noteworthy Curves. The Cycloid. The cycloid is a curve described by a point on the circumference of a circle rolling on a straight line. A point on the circumference of a carriage-wheel, as the carriage moves, describes a series of cycloids, one for each revolution of the wheel. To find the equation of the cycloid, let P be the generating point. Let us take the line on which the circle rolls as the axis of X, and let us place the origin at the point 0 where P is in contact with the line OX. Also put a = the radius of the circle; u = the angle through which the circle has rolled, expressed in terms of unit-radius. Then, when the circle has rolled through any distance OR, this distance will be equal to the length of the arc PR of the circle between P and the point of contact R, that is, to au. We thus have, for the co-ordinates of the centre, C, of the circle, and for the co-ordinates of the point P on the cycloid, which are the equations of the cycloid with u as an independent variable. This gives sin u = VI--cos' u =----. a Then, by substituting in the first equation x = a cos---V2ay-y', (2) which is the equation of the cycloid in the usual form. 81. The Lemniscate is the locus of a point, the product of whose distances from two fixed points (called foci) is equ...


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Product Details
  • ISBN-13: 9781152246362
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 50
  • Spine Width: 3 mm
  • Width: 189 mm
  • ISBN-10: 1152246364
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 109 gr


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