Calculus
The word calculus means small pebble in Latin, a diminutive of calx which translates to stone. Ancient Romans used these tiny stones for counting distances and tallying votes on an abacus. This physical object gave rise to the modern mathematical term for calculation. English speakers adopted the word at least as early as 1672 before Newton or Leibniz published their texts. Today the term persists in medicine where it refers to actual mineral deposits in the body. Beyond mathematics education, the name applies to specific computational theories like propositional calculus or lambda calculus. Philosophers also use the phrase to describe systems such as Bentham's ethical calculus.
Calculations of volume and area appear in the Egyptian Moscow papyrus without explanation of how they were obtained. Eudoxus of Cnidus developed the method of exhaustion to prove formulas for cone and pyramid volumes during ancient Greece. Archimedes combined this method with indivisibles to solve problems now treated by integral calculus. He calculated the center of gravity of a solid hemisphere in The Method of Mechanical Theorems. Liu Hui discovered the method of exhaustion independently in China during the third century AD. Zu Gengzhi established a method later called Cavalieri's principle to find the volume of a sphere in the fifth century. Hasan Ibn al-Haytham derived a formula for the sum of fourth powers in the Middle East. Bhāskara II suggested that the differential coefficient vanishes at an extremum value of the function. Madhava of Sangamagrama studied series equivalent to Maclaurin expansions more than two hundred years before their introduction in Europe.
Isaac Newton and Gottfried Wilhelm Leibniz each independently formulated infinitesimal calculus in the late 17th century. Newton first published his results in Method of Fluxions while Leibniz published Nova Methodus pro Maximis et Minimis earlier. Newton claimed Leibniz stole ideas from unpublished notes shared with members of the Royal Society. This controversy divided English-speaking mathematicians from continental European mathematicians for many years. Leibniz developed much of the notation used in calculus today including dx and dy symbols. Newton called his calculus the science of fluxions which endured in English schools into the 19th century. The first complete treatise on calculus written in English using Leibniz notation was not published until 1815. John von Neumann later wrote about this work as foundational to modern mathematics. Both men are now given credit for independently inventing and developing calculus despite the priority dispute.
Michel Rolle and Bishop Berkeley fiercely criticized the use of infinitesimal quantities in early calculus. Berkeley famously described infinitesimals as ghosts of departed quantities in his book The Analyst published in 1734. It took approximately 150 years after Newton and Leibniz before Cauchy and Weierstrass found a way to avoid mere notions of infinitely small quantities. Cauchy defined continuity in terms of infinitesimals in his Cours d'Analyse. Weierstrass formalized the concept of limit and eliminated infinitesimals from standard academic practice. Bernhard Riemann used these ideas to give a precise definition of the integral. Henri Lebesgue invented measure theory based on earlier developments by Émile Borel. Laurent Schwartz introduced distributions which can be used to take the derivative of any function whatsoever. Abraham Robinson developed non-standard analysis in the 1960s using technical machinery from mathematical logic.
The derivative represents change concerning time if the input of the function represents time. If f is a function that takes time as input and gives position of a ball at that time, then the derivative is velocity. The slope between two points on a curve is called a secant line while the tangent line is a limit of secant lines. For the squaring function at point three the slope of the tangent line equals six. This means the function goes up six times as fast as it goes to the right. Lagrange's notation uses an apostrophe-like mark called a prime so the derivative of f is denoted f prime. Leibniz intended dx divided by dy to represent the quotient of two infinitesimally small numbers. The derivative is defined by taking the limit as h tends to zero for all small values of h.
The symbol of integration is an elongated S chosen to suggest summation. A Riemann sum approximates distance traveled by breaking time into many short intervals. We multiply the time elapsed in each interval by one of the speeds in that interval. Taking the limit of all such sums finds the exact distance traveled when speed changes. If velocity remains constant at 50 mph for 3 hours the total distance equals 150 miles. The definite integral inputs a function and outputs a number representing algebraic sum of areas between graph and x-axis. Functions differing by only a constant have the same derivative and form a family of functions. The unspecified constant present in the indefinite integral is known as the constant of integration. The fundamental theorem relates antiderivatives to definite integrals providing an algebraic method of computing many integrals without performing limit processes.
Calculus serves as the mathematical backbone for solving problems where variable quantities change with time or another reference value. Newton's second law states that the derivative of momentum concerning time equals net force upon it. Maxwell's theory of electromagnetism and Einstein's theory of general relativity are expressed in language of differential calculus. Chemistry uses calculus to determine reaction rates and study radioactive decay. In biology population dynamics starts with reproduction and death rates to model population changes. Calculus allows determination of maximal profit by calculating marginal cost and marginal revenue in economics. Spacecraft use variation of Euler method to approximate curved courses within zero-gravity environments. Green's theorem applies in instruments called planimeters used to calculate area of flat surfaces on drawings. Medicine uses calculus to find optimal branching angle of blood vessel to maximize flow.
Up Next
Common questions
What is the origin of the word calculus?
The word calculus means small pebble in Latin, a diminutive of calx which translates to stone. Ancient Romans used these tiny stones for counting distances and tallying votes on an abacus.
Who independently formulated infinitesimal calculus in the late 17th century?
Isaac Newton and Gottfried Wilhelm Leibniz each independently formulated infinitesimal calculus in the late 17th century. Both men are now given credit for independently inventing and developing calculus despite the priority dispute.
When was the first complete treatise on calculus written in English using Leibniz notation published?
The first complete treatise on calculus written in English using Leibniz notation was not published until 1815. This work followed decades where Newton called his calculus the science of fluxions which endured in English schools into the 19th century.
How does calculus define the derivative of a function at a specific point?
The derivative is defined by taking the limit as h tends to zero for all small values of h. For the squaring function at point three the slope of the tangent line equals six.
What role does calculus play in modern physics theories like general relativity?
Maxwell's theory of electromagnetism and Einstein's theory of general relativity are expressed in language of differential calculus. Calculus serves as the mathematical backbone for solving problems where variable quantities change with time or another reference value.