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In mathematics and physics, the plate trick, also known as Dirac's string trick (after Paul Dirac, who introduced and popularized it), [1] [2] the belt trick, or the Balinese cup trick (it appears in the Balinese candle dance), is any of several demonstrations of the idea that rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while ...
Fundamental theorem of calculus; Integration by parts; Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the integral sign; Trigonometric substitution; Partial fractions in integration. Quadratic integral; Proof that 22/7 exceeds π; Trapezium rule; Integral of the secant function ...
The vertical distance between each vertical set of pivot points must be exactly the same; In order to be balanced front-to-back the balance must either have two sets of two arms located around a central fulcrum or must have two fulcra supporting a single set of arms; The weight of the arms on each side of the fulcrum must be equal (unless see ...
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These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. [1] It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. [2]
The arc length of one branch between x = x 1 and x = x 2 is a ln y 1 / y 2 . The area between the tractrix and its asymptote is π a 2 / 2 , which can be found using integration or Mamikon's theorem .
Gottfried Wilhelm von Leibniz (1646–1716), German philosopher, mathematician, and namesake of this widely used mathematical notation in calculus.. In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively ...
Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to ...
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